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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" neuromorphic computing, automtotive radar, spiking neural network, Fourier transform, resonate-and-fire, angle-of-arrival " lang="en" name="keywords"/> <base href="/html/2503.00898v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S1" title="In Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2" title="In Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Neuron model and network architecture</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS1" title="In II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-A</span> </span><span class="ltx_text ltx_font_italic">Angle estimation - Dendritic vector multiplication</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS2" title="In II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-B</span> </span><span class="ltx_text ltx_font_italic">Distance estimation - Neural resonators</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS3" title="In II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-C</span> </span><span class="ltx_text ltx_font_italic">Envelope estimation and gradient estimation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS4" title="In II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-D</span> </span><span class="ltx_text ltx_font_italic">Spiking Functions</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3" title="In Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Evaluation</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS1" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic">Dataset Simulation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS2" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic">Metric</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS3" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-C</span> </span><span class="ltx_text ltx_font_italic">Parameter optimization</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS4" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-D</span> </span><span class="ltx_text ltx_font_italic">Model evaluation for a single chirp</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS5" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-E</span> </span><span class="ltx_text ltx_font_italic">Model evaluation of early detections for a single chirp</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS6" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-F</span> </span><span class="ltx_text ltx_font_italic">Model evaluation for multiple consecutive chirps</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS7" title="In III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-G</span> </span><span class="ltx_text ltx_font_italic">Visual results on real data</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S4" title="In Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S5" title="In Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Future work</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar <br class="ltx_break"/> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Nico Reeb1, Javier Lopez-Randulfe, Robin Dietrich and Alois C. Knoll <br class="ltx_break"/> <span class="ltx_text ltx_font_italic" id="id1.1.id1">Technical University of Munich <br class="ltx_break"/></span>Munich, Germany <br class="ltx_break"/>1<a class="ltx_ref ltx_href" href="mailto:nico.reeb@tum.de" title="">nico.reeb@tum.de</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id2.2.id1">TUM School of Computation, Information and Technology</span> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id3.id1">Automotive radar systems face the challenge of managing high sampling rates and large data bandwidth while complying with stringent real-time and energy efficiency requirements. The growing complexity of autonomous vehicles further intensifies these requirements. Neuromorphic computing offers promising solutions because of its inherent energy efficiency and parallel processing capacity.</p> <p class="ltx_p" id="id4.id2">This research presents a novel spiking neuron model for signal processing of frequency-modulated continuous wave (FMCW) radars that outperforms the state-of-the-art spectrum analysis algorithms in latency and data bandwidth. These spiking neural resonators are based on the resonate-and-fire neuron model and optimized to dynamically process raw radar data while simultaneously emitting an output in the form of spikes. We designed the first neuromorphic neural network consisting of these spiking neural resonators that estimates range and angle from FMCW radar data. We evaluated the range-angle maps on simulated datasets covering multiple scenarios and compared the results with a state-of-the-art pipeline for radar processing.</p> <p class="ltx_p" id="id5.id3">The proposed neuron model significantly reduces the processing latency compared to traditional frequency analysis algorithms, such as the Fourier transformation (FT), which needs to sample and store entire data frames before processing. The evaluations demonstrate that these spiking neural resonators achieve state-of-the-art detection accuracy while emitting spikes simultaneously to processing and transmitting only 0.02% of the data compared to a float-32 FT. The results showcase the potential for neuromorphic signal processing for FMCW radar systems and pave the way for designing neuromorphic radar sensors.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> neuromorphic computing, automtotive radar, spiking neural network, Fourier transform, resonate-and-fire, angle-of-arrival </div> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1"><span class="ltx_text ltx_framed ltx_framed_rectangle" id="p1.1.1" style="border-color: #000000;"> <span class="ltx_inline-block ltx_minipage ltx_align_middle" id="p1.1.1.1" style="width:390.3pt;"> <span class="ltx_p" id="p1.1.1.1.1">This is the version of the article before peer review or editing, as submitted by an author to <span class="ltx_text ltx_font_italic" id="p1.1.1.1.1.1">Neuromorphic Computing and Engineering</span>. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it.</span> </span></span></p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">As society and industry depend on increasingly complex signal processing systems, solutions become more energy-intensive. This trend drives the need for optimized processing pipelines that minimize energy consumption and data bandwidth while improving performance and reliability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib15" title="">15</a>]</cite>. The automotive industry is paradigmatic due to the limited energy availability in cars and the need for precise and reliable low-latency systems to achieve fully autonomous driving <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib28" title="">28</a>]</cite>. One of the most critical operations in advanced driving assistance systems (ADASs) is sensor signal processing. Subsequent crucial tasks, such as navigation, collision avoidance, or adaptive cruise control, depend on accurate and fast sensor data processing. The frequency-modulated continuous-wave radar (FMCW) sensor is a prominent example utilized in ADASs. Its low cost and long-range detection, combined with its robustness to bad lighting and weather conditions, make it a fundamental element for perceiving the car’s surroundings <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib23" title="">23</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">A promising approach to developing low-power processing systems is to mimic biology. The human brain can collect and process sensor data, reason, and decide how to interact with the environment while consuming only 20 W <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib27" title="">27</a>]</cite>. The brain’s superior efficiency arises from parallel, asynchronous, and event-driven processing, and communication via binary spikes. Neuromorphic computing is a research field that aims to replicate these characteristics of the brain to create accurate and energy efficient solutions to real-world problems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib20" title="">20</a>]</cite>. Spiking neural networks (SNNs) are among the most promising neuromorphic algorithms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib19" title="">19</a>]</cite>. Neuromorphic hardware is developed in symbiosis to take advantage of the efficient processing of neuromorphic algorithms. SNNs consist of an asynchronous network of artificial neurons that transmit information via spikes, relying on internal dynamics models driven by a continuous stream of incoming data. In addition to advancing the understanding of the human brain <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib25" title="">25</a>]</cite>, research in SNNs focuses on engineering problems, such as finding an efficient replacement for deep neural networks (DNNs) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib26" title="">26</a>]</cite>, optimizing operations in large data centers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib29" title="">29</a>]</cite>, or designing algorithms for small embedded systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib4" title="">4</a>]</cite>. Most neuron models in SNNs follow integrate-and-fire dynamics, where a neuron charges over time based on input activity and emits a spike once its state crosses a defined threshold. IF models are present in most neuromorphic chips <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib24" title="">24</a>]</cite>, and offer good results in the conversion of DNNs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib26" title="">26</a>]</cite>, applying biologically inspired learning techniques <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib13" title="">13</a>]</cite>, or solving complex optimization problems<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib5" title="">5</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Object detection with FMCW radars is based on the frequency analysis of the continuous analog signal generated by the sensor. Traditional signal processing methods typically convert discrete time sampled data into the frequency domain, with the Fourier transform (FT) being the most widely used technique for frequency analysis. Some neuromorphic algorithms have already emerged for computing the frequency spectrum. Authors in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib12" title="">12</a>]</cite> apply sequential spiking band-pass filters to process audio signals based on their frequency spectrum. The works in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib17" title="">17</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib18" title="">18</a>]</cite> introduced an electric circuit and a neuron model that encode analog signals to temporal spikes and provide a mathematically equivalent representation of the FT. Alternatively, Izhikevich proposed the resonate-and-fire (RF) neuron model, which enables frequency analysis on continuous input data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib10" title="">10</a>]</cite>. The researchers in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib9" title="">9</a>]</cite> studied the application of resonate-and-fire neurons to oscillating signals. Approaches utilizing resonate-and-fire neurons focused on one-dimensional frequency analysis, whereas FMCW radar processing relies on multi-dimensional frequency analysis for angle and velocity information. To the best of our knowledge there is no research on how resonating neurons compare with classic frequency analysis methods, such as the FT, on multi-dimensional FMCW radar data.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">In this work, we advance the resonate-and-fire neuron model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib10" title="">10</a>]</cite> to concurrently process incoming radar data and optimally convey information through spiking. Traditional frequency analysis techniques, such as the FT, calculate the spectrum after storing all data samples in a given time window. By allowing concurrent processing and spiking, we can achieve three benefits over traditional approaches:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1)</span> <div class="ltx_para" id="S1.I1.ix1.p1"> <p class="ltx_p" id="S1.I1.ix1.p1.1">Reduce the latency of target detection, as each data sample incoming updates the estimate.</p> </div> </li> <li class="ltx_item" id="S1.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2)</span> <div class="ltx_para" id="S1.I1.ix2.p1"> <p class="ltx_p" id="S1.I1.ix2.p1.1">Reduce data bandwidth due to sparse spiking.</p> </div> </li> <li class="ltx_item" id="S1.I1.ix3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3)</span> <div class="ltx_para" id="S1.I1.ix3.p1"> <p class="ltx_p" id="S1.I1.ix3.p1.1">Remove the need for memory storage for sensor data due to the immediate processing by the neurons.</p> </div> </li> </ul> <p class="ltx_p" id="S1.p4.2">We arranged these neural resonators into a single layer capable of performing two-dimensional frequency analysis on FMCW radar data allowing to estimate range and angle simultaneously (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS1" title="II-A Angle estimation - Dendritic vector multiplication ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-A</span></span></a>). To maintain the accuracy achieved by state-of-the-art methodologies, we developed neuron dynamics that filter out noise from the incoming signal (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS3" title="II-C Envelope estimation and gradient estimation ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-C</span></span></a>). We implemented three different spike functions including time- and rate-coding for comparison (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS4" title="II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-D</span></span></a>). We simulated radar datasets with exact positional information of point targets, a level of accuracy not available in real radar datasets, validated the approach on these datasets, and compared it with the FT method (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3" title="III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">III</span></a>). Finally, we applied the spiking neural resonators to public radar datasets, allowing us to visually assess the model’s generalization capabilities (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.SS7" title="III-G Visual results on real data ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">III-G</span></span></a>). We implemented the neuron model on GPU and published the code <span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span><a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/ndotr/Spiking-Neural-Resonator-Network" title="">https://github.com/ndotr/Spiking-Neural-Resonator-Network</a></span></span></span>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Neuron model and network architecture</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.7">Typically, automotive radar systems rely on multiple-input-multiple-output FMCW radar sensors with multiple transmitting as well as receiving antennas. This enables a high range and angular resolution as well as unambiguous velocity detection. Various multiplexing methods <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib30" title="">30</a>]</cite> enable simultaneous transmission from multiple antennas while maintaining orthogonality between the signals. The combination and mixing of multiple transmit and receive antennas leads to <span class="ltx_text" id="S2.p1.1.1"><math alttext="N_{\text{vx}}=N_{\text{tx}}\times N_{\text{rx}}" class="ltx_Math" display="inline" id="S2.p1.1.1.m1.1"><semantics id="S2.p1.1.1.m1.1a"><mrow id="S2.p1.1.1.m1.1.1" xref="S2.p1.1.1.m1.1.1.cmml"><msub id="S2.p1.1.1.m1.1.1.2" xref="S2.p1.1.1.m1.1.1.2.cmml"><mi id="S2.p1.1.1.m1.1.1.2.2" xref="S2.p1.1.1.m1.1.1.2.2.cmml">N</mi><mtext id="S2.p1.1.1.m1.1.1.2.3" xref="S2.p1.1.1.m1.1.1.2.3a.cmml">vx</mtext></msub><mo id="S2.p1.1.1.m1.1.1.1" xref="S2.p1.1.1.m1.1.1.1.cmml">=</mo><mrow id="S2.p1.1.1.m1.1.1.3" xref="S2.p1.1.1.m1.1.1.3.cmml"><msub id="S2.p1.1.1.m1.1.1.3.2" xref="S2.p1.1.1.m1.1.1.3.2.cmml"><mi id="S2.p1.1.1.m1.1.1.3.2.2" xref="S2.p1.1.1.m1.1.1.3.2.2.cmml">N</mi><mtext id="S2.p1.1.1.m1.1.1.3.2.3" xref="S2.p1.1.1.m1.1.1.3.2.3a.cmml">tx</mtext></msub><mo id="S2.p1.1.1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p1.1.1.m1.1.1.3.1.cmml">×</mo><msub id="S2.p1.1.1.m1.1.1.3.3" xref="S2.p1.1.1.m1.1.1.3.3.cmml"><mi id="S2.p1.1.1.m1.1.1.3.3.2" xref="S2.p1.1.1.m1.1.1.3.3.2.cmml">N</mi><mtext id="S2.p1.1.1.m1.1.1.3.3.3" xref="S2.p1.1.1.m1.1.1.3.3.3a.cmml">rx</mtext></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.1.m1.1b"><apply id="S2.p1.1.1.m1.1.1.cmml" xref="S2.p1.1.1.m1.1.1"><eq id="S2.p1.1.1.m1.1.1.1.cmml" xref="S2.p1.1.1.m1.1.1.1"></eq><apply id="S2.p1.1.1.m1.1.1.2.cmml" xref="S2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.1.1.m1.1.1.2.1.cmml" xref="S2.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S2.p1.1.1.m1.1.1.2.2.cmml" xref="S2.p1.1.1.m1.1.1.2.2">𝑁</ci><ci id="S2.p1.1.1.m1.1.1.2.3a.cmml" xref="S2.p1.1.1.m1.1.1.2.3"><mtext id="S2.p1.1.1.m1.1.1.2.3.cmml" mathsize="70%" xref="S2.p1.1.1.m1.1.1.2.3">vx</mtext></ci></apply><apply id="S2.p1.1.1.m1.1.1.3.cmml" xref="S2.p1.1.1.m1.1.1.3"><times id="S2.p1.1.1.m1.1.1.3.1.cmml" xref="S2.p1.1.1.m1.1.1.3.1"></times><apply id="S2.p1.1.1.m1.1.1.3.2.cmml" xref="S2.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.p1.1.1.m1.1.1.3.2.1.cmml" xref="S2.p1.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S2.p1.1.1.m1.1.1.3.2.2.cmml" xref="S2.p1.1.1.m1.1.1.3.2.2">𝑁</ci><ci id="S2.p1.1.1.m1.1.1.3.2.3a.cmml" xref="S2.p1.1.1.m1.1.1.3.2.3"><mtext id="S2.p1.1.1.m1.1.1.3.2.3.cmml" mathsize="70%" xref="S2.p1.1.1.m1.1.1.3.2.3">tx</mtext></ci></apply><apply id="S2.p1.1.1.m1.1.1.3.3.cmml" xref="S2.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S2.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S2.p1.1.1.m1.1.1.3.3.2">𝑁</ci><ci id="S2.p1.1.1.m1.1.1.3.3.3a.cmml" xref="S2.p1.1.1.m1.1.1.3.3.3"><mtext id="S2.p1.1.1.m1.1.1.3.3.3.cmml" mathsize="70%" xref="S2.p1.1.1.m1.1.1.3.3.3">rx</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.1.m1.1c">N_{\text{vx}}=N_{\text{tx}}\times N_{\text{rx}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.1.m1.1d">italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT tx end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT rx end_POSTSUBSCRIPT</annotation></semantics></math></span> virtual antennas. The <math alttext="m" class="ltx_Math" display="inline" id="S2.p1.2.m1.1"><semantics id="S2.p1.2.m1.1a"><mi id="S2.p1.2.m1.1.1" xref="S2.p1.2.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.1b"><ci id="S2.p1.2.m1.1.1.cmml" xref="S2.p1.2.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.1d">italic_m</annotation></semantics></math>-th virtual antenna returns an intermediate-frequency (IF) signal <math alttext="x_{m}(t)" class="ltx_Math" display="inline" id="S2.p1.3.m2.1"><semantics id="S2.p1.3.m2.1a"><mrow id="S2.p1.3.m2.1.2" xref="S2.p1.3.m2.1.2.cmml"><msub id="S2.p1.3.m2.1.2.2" xref="S2.p1.3.m2.1.2.2.cmml"><mi id="S2.p1.3.m2.1.2.2.2" xref="S2.p1.3.m2.1.2.2.2.cmml">x</mi><mi id="S2.p1.3.m2.1.2.2.3" xref="S2.p1.3.m2.1.2.2.3.cmml">m</mi></msub><mo id="S2.p1.3.m2.1.2.1" xref="S2.p1.3.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p1.3.m2.1.2.3.2" 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end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, leading to a continuous data vector <span class="ltx_text" id="S2.p1.4.2"><math alttext="\vec{x}(t)\in\mathbb{C}^{N_{\text{vx}}}" class="ltx_Math" display="inline" id="S2.p1.4.2.m1.1"><semantics id="S2.p1.4.2.m1.1a"><mrow id="S2.p1.4.2.m1.1.2" xref="S2.p1.4.2.m1.1.2.cmml"><mrow id="S2.p1.4.2.m1.1.2.2" xref="S2.p1.4.2.m1.1.2.2.cmml"><mover accent="true" id="S2.p1.4.2.m1.1.2.2.2" xref="S2.p1.4.2.m1.1.2.2.2.cmml"><mi id="S2.p1.4.2.m1.1.2.2.2.2" xref="S2.p1.4.2.m1.1.2.2.2.2.cmml">x</mi><mo id="S2.p1.4.2.m1.1.2.2.2.1" stretchy="false" xref="S2.p1.4.2.m1.1.2.2.2.1.cmml">→</mo></mover><mo id="S2.p1.4.2.m1.1.2.2.1" xref="S2.p1.4.2.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.4.2.m1.1.2.2.3.2" xref="S2.p1.4.2.m1.1.2.2.cmml"><mo id="S2.p1.4.2.m1.1.2.2.3.2.1" stretchy="false" xref="S2.p1.4.2.m1.1.2.2.cmml">(</mo><mi id="S2.p1.4.2.m1.1.1" xref="S2.p1.4.2.m1.1.1.cmml">t</mi><mo id="S2.p1.4.2.m1.1.2.2.3.2.2" stretchy="false" xref="S2.p1.4.2.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.4.2.m1.1.2.1" xref="S2.p1.4.2.m1.1.2.1.cmml">∈</mo><msup id="S2.p1.4.2.m1.1.2.3" xref="S2.p1.4.2.m1.1.2.3.cmml"><mi id="S2.p1.4.2.m1.1.2.3.2" xref="S2.p1.4.2.m1.1.2.3.2.cmml">ℂ</mi><msub id="S2.p1.4.2.m1.1.2.3.3" xref="S2.p1.4.2.m1.1.2.3.3.cmml"><mi id="S2.p1.4.2.m1.1.2.3.3.2" xref="S2.p1.4.2.m1.1.2.3.3.2.cmml">N</mi><mtext id="S2.p1.4.2.m1.1.2.3.3.3" xref="S2.p1.4.2.m1.1.2.3.3.3a.cmml">vx</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.2.m1.1b"><apply id="S2.p1.4.2.m1.1.2.cmml" xref="S2.p1.4.2.m1.1.2"><in id="S2.p1.4.2.m1.1.2.1.cmml" xref="S2.p1.4.2.m1.1.2.1"></in><apply id="S2.p1.4.2.m1.1.2.2.cmml" xref="S2.p1.4.2.m1.1.2.2"><times id="S2.p1.4.2.m1.1.2.2.1.cmml" xref="S2.p1.4.2.m1.1.2.2.1"></times><apply id="S2.p1.4.2.m1.1.2.2.2.cmml" xref="S2.p1.4.2.m1.1.2.2.2"><ci id="S2.p1.4.2.m1.1.2.2.2.1.cmml" xref="S2.p1.4.2.m1.1.2.2.2.1">→</ci><ci id="S2.p1.4.2.m1.1.2.2.2.2.cmml" xref="S2.p1.4.2.m1.1.2.2.2.2">𝑥</ci></apply><ci id="S2.p1.4.2.m1.1.1.cmml" xref="S2.p1.4.2.m1.1.1">𝑡</ci></apply><apply id="S2.p1.4.2.m1.1.2.3.cmml" xref="S2.p1.4.2.m1.1.2.3"><csymbol cd="ambiguous" id="S2.p1.4.2.m1.1.2.3.1.cmml" xref="S2.p1.4.2.m1.1.2.3">superscript</csymbol><ci id="S2.p1.4.2.m1.1.2.3.2.cmml" xref="S2.p1.4.2.m1.1.2.3.2">ℂ</ci><apply id="S2.p1.4.2.m1.1.2.3.3.cmml" xref="S2.p1.4.2.m1.1.2.3.3"><csymbol cd="ambiguous" id="S2.p1.4.2.m1.1.2.3.3.1.cmml" xref="S2.p1.4.2.m1.1.2.3.3">subscript</csymbol><ci id="S2.p1.4.2.m1.1.2.3.3.2.cmml" xref="S2.p1.4.2.m1.1.2.3.3.2">𝑁</ci><ci id="S2.p1.4.2.m1.1.2.3.3.3a.cmml" xref="S2.p1.4.2.m1.1.2.3.3.3"><mtext id="S2.p1.4.2.m1.1.2.3.3.3.cmml" mathsize="50%" xref="S2.p1.4.2.m1.1.2.3.3.3">vx</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.2.m1.1c">\vec{x}(t)\in\mathbb{C}^{N_{\text{vx}}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.2.m1.1d">over→ start_ARG italic_x end_ARG ( italic_t ) ∈ blackboard_C start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math></span> for the entire radar sensor (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F1" title="Figure 1 ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">1</span></a>). Throughout the paper, we assume sawtooth frequency modulation and a simplified antenna layout with one transmitting antenna and <math alttext="N_{\text{rx}}=N_{\text{vx}}" class="ltx_Math" display="inline" id="S2.p1.5.m3.1"><semantics id="S2.p1.5.m3.1a"><mrow id="S2.p1.5.m3.1.1" xref="S2.p1.5.m3.1.1.cmml"><msub id="S2.p1.5.m3.1.1.2" xref="S2.p1.5.m3.1.1.2.cmml"><mi id="S2.p1.5.m3.1.1.2.2" xref="S2.p1.5.m3.1.1.2.2.cmml">N</mi><mtext id="S2.p1.5.m3.1.1.2.3" xref="S2.p1.5.m3.1.1.2.3a.cmml">rx</mtext></msub><mo id="S2.p1.5.m3.1.1.1" xref="S2.p1.5.m3.1.1.1.cmml">=</mo><msub id="S2.p1.5.m3.1.1.3" xref="S2.p1.5.m3.1.1.3.cmml"><mi id="S2.p1.5.m3.1.1.3.2" xref="S2.p1.5.m3.1.1.3.2.cmml">N</mi><mtext id="S2.p1.5.m3.1.1.3.3" xref="S2.p1.5.m3.1.1.3.3a.cmml">vx</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m3.1b"><apply id="S2.p1.5.m3.1.1.cmml" xref="S2.p1.5.m3.1.1"><eq id="S2.p1.5.m3.1.1.1.cmml" xref="S2.p1.5.m3.1.1.1"></eq><apply id="S2.p1.5.m3.1.1.2.cmml" xref="S2.p1.5.m3.1.1.2"><csymbol cd="ambiguous" id="S2.p1.5.m3.1.1.2.1.cmml" xref="S2.p1.5.m3.1.1.2">subscript</csymbol><ci id="S2.p1.5.m3.1.1.2.2.cmml" xref="S2.p1.5.m3.1.1.2.2">𝑁</ci><ci id="S2.p1.5.m3.1.1.2.3a.cmml" xref="S2.p1.5.m3.1.1.2.3"><mtext id="S2.p1.5.m3.1.1.2.3.cmml" mathsize="70%" xref="S2.p1.5.m3.1.1.2.3">rx</mtext></ci></apply><apply id="S2.p1.5.m3.1.1.3.cmml" xref="S2.p1.5.m3.1.1.3"><csymbol cd="ambiguous" id="S2.p1.5.m3.1.1.3.1.cmml" xref="S2.p1.5.m3.1.1.3">subscript</csymbol><ci id="S2.p1.5.m3.1.1.3.2.cmml" xref="S2.p1.5.m3.1.1.3.2">𝑁</ci><ci id="S2.p1.5.m3.1.1.3.3a.cmml" xref="S2.p1.5.m3.1.1.3.3"><mtext id="S2.p1.5.m3.1.1.3.3.cmml" mathsize="70%" xref="S2.p1.5.m3.1.1.3.3">vx</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m3.1c">N_{\text{rx}}=N_{\text{vx}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m3.1d">italic_N start_POSTSUBSCRIPT rx end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT</annotation></semantics></math> receiving antennas in one line parallel to the ground. Reflections of <math alttext="K" class="ltx_Math" display="inline" id="S2.p1.6.m4.1"><semantics id="S2.p1.6.m4.1a"><mi id="S2.p1.6.m4.1.1" xref="S2.p1.6.m4.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m4.1b"><ci id="S2.p1.6.m4.1.1.cmml" xref="S2.p1.6.m4.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m4.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m4.1d">italic_K</annotation></semantics></math> objects lead to an IF signal that can be modeled for one chirp <span class="ltx_text" id="S2.p1.7.3"><math alttext="t\in[0,T_{c}]" class="ltx_Math" display="inline" id="S2.p1.7.3.m1.2"><semantics id="S2.p1.7.3.m1.2a"><mrow id="S2.p1.7.3.m1.2.2" xref="S2.p1.7.3.m1.2.2.cmml"><mi id="S2.p1.7.3.m1.2.2.3" xref="S2.p1.7.3.m1.2.2.3.cmml">t</mi><mo id="S2.p1.7.3.m1.2.2.2" xref="S2.p1.7.3.m1.2.2.2.cmml">∈</mo><mrow id="S2.p1.7.3.m1.2.2.1.1" xref="S2.p1.7.3.m1.2.2.1.2.cmml"><mo id="S2.p1.7.3.m1.2.2.1.1.2" stretchy="false" xref="S2.p1.7.3.m1.2.2.1.2.cmml">[</mo><mn id="S2.p1.7.3.m1.1.1" xref="S2.p1.7.3.m1.1.1.cmml">0</mn><mo id="S2.p1.7.3.m1.2.2.1.1.3" xref="S2.p1.7.3.m1.2.2.1.2.cmml">,</mo><msub id="S2.p1.7.3.m1.2.2.1.1.1" xref="S2.p1.7.3.m1.2.2.1.1.1.cmml"><mi id="S2.p1.7.3.m1.2.2.1.1.1.2" xref="S2.p1.7.3.m1.2.2.1.1.1.2.cmml">T</mi><mi id="S2.p1.7.3.m1.2.2.1.1.1.3" xref="S2.p1.7.3.m1.2.2.1.1.1.3.cmml">c</mi></msub><mo id="S2.p1.7.3.m1.2.2.1.1.4" stretchy="false" xref="S2.p1.7.3.m1.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.3.m1.2b"><apply id="S2.p1.7.3.m1.2.2.cmml" xref="S2.p1.7.3.m1.2.2"><in id="S2.p1.7.3.m1.2.2.2.cmml" xref="S2.p1.7.3.m1.2.2.2"></in><ci id="S2.p1.7.3.m1.2.2.3.cmml" xref="S2.p1.7.3.m1.2.2.3">𝑡</ci><interval closure="closed" id="S2.p1.7.3.m1.2.2.1.2.cmml" xref="S2.p1.7.3.m1.2.2.1.1"><cn id="S2.p1.7.3.m1.1.1.cmml" type="integer" xref="S2.p1.7.3.m1.1.1">0</cn><apply id="S2.p1.7.3.m1.2.2.1.1.1.cmml" 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xref="S2.E1.m1.1.2.2.3">𝑚</ci></apply><ci id="S2.E1.m1.1.1.cmml" xref="S2.E1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.1c">\displaystyle x_{m}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum^{K}_{k}a_{k}e^{im\phi_{k}}e^{i\omega_{k}t}." class="ltx_Math" display="inline" id="S2.E1.m2.1"><semantics id="S2.E1.m2.1a"><mrow id="S2.E1.m2.1.1.1" xref="S2.E1.m2.1.1.1.1.cmml"><mrow id="S2.E1.m2.1.1.1.1" xref="S2.E1.m2.1.1.1.1.cmml"><mi id="S2.E1.m2.1.1.1.1.2" xref="S2.E1.m2.1.1.1.1.2.cmml"></mi><mo id="S2.E1.m2.1.1.1.1.1" xref="S2.E1.m2.1.1.1.1.1.cmml">=</mo><mrow id="S2.E1.m2.1.1.1.1.3" xref="S2.E1.m2.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S2.E1.m2.1.1.1.1.3.1" xref="S2.E1.m2.1.1.1.1.3.1.cmml"><munderover 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xref="S2.E1.m2.1.1.1.1.3.2.3.3.4.3">𝑘</ci></apply></apply></apply><apply id="S2.E1.m2.1.1.1.1.3.2.4.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4"><csymbol cd="ambiguous" id="S2.E1.m2.1.1.1.1.3.2.4.1.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4">superscript</csymbol><ci id="S2.E1.m2.1.1.1.1.3.2.4.2.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.2">𝑒</ci><apply id="S2.E1.m2.1.1.1.1.3.2.4.3.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3"><times id="S2.E1.m2.1.1.1.1.3.2.4.3.1.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.1"></times><ci id="S2.E1.m2.1.1.1.1.3.2.4.3.2.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.2">𝑖</ci><apply id="S2.E1.m2.1.1.1.1.3.2.4.3.3.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.3"><csymbol cd="ambiguous" id="S2.E1.m2.1.1.1.1.3.2.4.3.3.1.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.3">subscript</csymbol><ci id="S2.E1.m2.1.1.1.1.3.2.4.3.3.2.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.3.2">𝜔</ci><ci id="S2.E1.m2.1.1.1.1.3.2.4.3.3.3.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.3.3">𝑘</ci></apply><ci id="S2.E1.m2.1.1.1.1.3.2.4.3.4.cmml" xref="S2.E1.m2.1.1.1.1.3.2.4.3.4">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m2.1c">\displaystyle=\sum^{K}_{k}a_{k}e^{im\phi_{k}}e^{i\omega_{k}t}.</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m2.1d">= ∑ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_m italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.15">The frequency <math alttext="\omega_{k}" class="ltx_Math" display="inline" id="S2.p1.8.m1.1"><semantics id="S2.p1.8.m1.1a"><msub id="S2.p1.8.m1.1.1" xref="S2.p1.8.m1.1.1.cmml"><mi id="S2.p1.8.m1.1.1.2" xref="S2.p1.8.m1.1.1.2.cmml">ω</mi><mi id="S2.p1.8.m1.1.1.3" xref="S2.p1.8.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.8.m1.1b"><apply id="S2.p1.8.m1.1.1.cmml" xref="S2.p1.8.m1.1.1"><csymbol cd="ambiguous" id="S2.p1.8.m1.1.1.1.cmml" xref="S2.p1.8.m1.1.1">subscript</csymbol><ci id="S2.p1.8.m1.1.1.2.cmml" xref="S2.p1.8.m1.1.1.2">𝜔</ci><ci id="S2.p1.8.m1.1.1.3.cmml" xref="S2.p1.8.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m1.1c">\omega_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m1.1d">italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is directly proportional to the radial range <math alttext="r_{k}" class="ltx_Math" display="inline" id="S2.p1.9.m2.1"><semantics id="S2.p1.9.m2.1a"><msub id="S2.p1.9.m2.1.1" xref="S2.p1.9.m2.1.1.cmml"><mi id="S2.p1.9.m2.1.1.2" xref="S2.p1.9.m2.1.1.2.cmml">r</mi><mi id="S2.p1.9.m2.1.1.3" xref="S2.p1.9.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.9.m2.1b"><apply id="S2.p1.9.m2.1.1.cmml" xref="S2.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S2.p1.9.m2.1.1.1.cmml" xref="S2.p1.9.m2.1.1">subscript</csymbol><ci id="S2.p1.9.m2.1.1.2.cmml" xref="S2.p1.9.m2.1.1.2">𝑟</ci><ci id="S2.p1.9.m2.1.1.3.cmml" xref="S2.p1.9.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m2.1c">r_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m2.1d">italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> between the object <math alttext="k" class="ltx_Math" display="inline" id="S2.p1.10.m3.1"><semantics id="S2.p1.10.m3.1a"><mi id="S2.p1.10.m3.1.1" xref="S2.p1.10.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.10.m3.1b"><ci id="S2.p1.10.m3.1.1.cmml" xref="S2.p1.10.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m3.1d">italic_k</annotation></semantics></math> and the sensor. The phase shift <math alttext="m\phi_{k}" class="ltx_Math" display="inline" id="S2.p1.11.m4.1"><semantics id="S2.p1.11.m4.1a"><mrow id="S2.p1.11.m4.1.1" xref="S2.p1.11.m4.1.1.cmml"><mi id="S2.p1.11.m4.1.1.2" xref="S2.p1.11.m4.1.1.2.cmml">m</mi><mo id="S2.p1.11.m4.1.1.1" xref="S2.p1.11.m4.1.1.1.cmml">⁢</mo><msub id="S2.p1.11.m4.1.1.3" xref="S2.p1.11.m4.1.1.3.cmml"><mi id="S2.p1.11.m4.1.1.3.2" xref="S2.p1.11.m4.1.1.3.2.cmml">ϕ</mi><mi id="S2.p1.11.m4.1.1.3.3" xref="S2.p1.11.m4.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.11.m4.1b"><apply id="S2.p1.11.m4.1.1.cmml" xref="S2.p1.11.m4.1.1"><times id="S2.p1.11.m4.1.1.1.cmml" xref="S2.p1.11.m4.1.1.1"></times><ci id="S2.p1.11.m4.1.1.2.cmml" xref="S2.p1.11.m4.1.1.2">𝑚</ci><apply id="S2.p1.11.m4.1.1.3.cmml" xref="S2.p1.11.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p1.11.m4.1.1.3.1.cmml" xref="S2.p1.11.m4.1.1.3">subscript</csymbol><ci id="S2.p1.11.m4.1.1.3.2.cmml" xref="S2.p1.11.m4.1.1.3.2">italic-ϕ</ci><ci id="S2.p1.11.m4.1.1.3.3.cmml" xref="S2.p1.11.m4.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m4.1c">m\phi_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m4.1d">italic_m italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> depends on the angle between object <math alttext="k" class="ltx_Math" display="inline" id="S2.p1.12.m5.1"><semantics id="S2.p1.12.m5.1a"><mi id="S2.p1.12.m5.1.1" xref="S2.p1.12.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.12.m5.1b"><ci id="S2.p1.12.m5.1.1.cmml" xref="S2.p1.12.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m5.1d">italic_k</annotation></semantics></math> and the sensor, and the position of the <math alttext="m" class="ltx_Math" display="inline" id="S2.p1.13.m6.1"><semantics id="S2.p1.13.m6.1a"><mi id="S2.p1.13.m6.1.1" xref="S2.p1.13.m6.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p1.13.m6.1b"><ci id="S2.p1.13.m6.1.1.cmml" xref="S2.p1.13.m6.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m6.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m6.1d">italic_m</annotation></semantics></math>-th antenna within the antenna layout. The amplitude <math alttext="a_{k}" class="ltx_Math" display="inline" id="S2.p1.14.m7.1"><semantics id="S2.p1.14.m7.1a"><msub id="S2.p1.14.m7.1.1" xref="S2.p1.14.m7.1.1.cmml"><mi id="S2.p1.14.m7.1.1.2" xref="S2.p1.14.m7.1.1.2.cmml">a</mi><mi id="S2.p1.14.m7.1.1.3" xref="S2.p1.14.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.14.m7.1b"><apply id="S2.p1.14.m7.1.1.cmml" xref="S2.p1.14.m7.1.1"><csymbol cd="ambiguous" id="S2.p1.14.m7.1.1.1.cmml" xref="S2.p1.14.m7.1.1">subscript</csymbol><ci id="S2.p1.14.m7.1.1.2.cmml" xref="S2.p1.14.m7.1.1.2">𝑎</ci><ci id="S2.p1.14.m7.1.1.3.cmml" xref="S2.p1.14.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m7.1c">a_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m7.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> depends on various factors, such as the radar cross-section of the object, the distance between the radar and object <math alttext="d_{k}" class="ltx_Math" display="inline" id="S2.p1.15.m8.1"><semantics id="S2.p1.15.m8.1a"><msub id="S2.p1.15.m8.1.1" xref="S2.p1.15.m8.1.1.cmml"><mi id="S2.p1.15.m8.1.1.2" xref="S2.p1.15.m8.1.1.2.cmml">d</mi><mi id="S2.p1.15.m8.1.1.3" xref="S2.p1.15.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.15.m8.1b"><apply id="S2.p1.15.m8.1.1.cmml" xref="S2.p1.15.m8.1.1"><csymbol cd="ambiguous" id="S2.p1.15.m8.1.1.1.cmml" xref="S2.p1.15.m8.1.1">subscript</csymbol><ci id="S2.p1.15.m8.1.1.2.cmml" xref="S2.p1.15.m8.1.1.2">𝑑</ci><ci id="S2.p1.15.m8.1.1.3.cmml" xref="S2.p1.15.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m8.1c">d_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m8.1d">italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the dynamics of the sensor <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib8" title="">8</a>]</cite>, and properties of the medium. More details on the fundamentals on radar signal processing can be found in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib11" title="">11</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">A single spiking neural resonator receives and continuously processes the radar signal vector <math alttext="\vec{x}(t)" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.2" xref="S2.p2.1.m1.1.2.cmml"><mover accent="true" id="S2.p2.1.m1.1.2.2" xref="S2.p2.1.m1.1.2.2.cmml"><mi id="S2.p2.1.m1.1.2.2.2" xref="S2.p2.1.m1.1.2.2.2.cmml">x</mi><mo id="S2.p2.1.m1.1.2.2.1" stretchy="false" xref="S2.p2.1.m1.1.2.2.1.cmml">→</mo></mover><mo id="S2.p2.1.m1.1.2.1" xref="S2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p2.1.m1.1.2.3.2" xref="S2.p2.1.m1.1.2.cmml"><mo id="S2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p2.1.m1.1.2.cmml">(</mo><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">t</mi><mo id="S2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.2.cmml" xref="S2.p2.1.m1.1.2"><times id="S2.p2.1.m1.1.2.1.cmml" xref="S2.p2.1.m1.1.2.1"></times><apply id="S2.p2.1.m1.1.2.2.cmml" xref="S2.p2.1.m1.1.2.2"><ci id="S2.p2.1.m1.1.2.2.1.cmml" xref="S2.p2.1.m1.1.2.2.1">→</ci><ci id="S2.p2.1.m1.1.2.2.2.cmml" xref="S2.p2.1.m1.1.2.2.2">𝑥</ci></apply><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\vec{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">over→ start_ARG italic_x end_ARG ( italic_t )</annotation></semantics></math> (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F1" title="Figure 1 ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">1</span></a>). We describe the different processing steps of the neuron model in the following. First, a complex weight matrix multiplication extract the angle information of the sensor data. Second, the resonator dynamics provide information about the distance. Third, neuron dynamics analyze the temporal behavior of the resonator to produce informative spikes. We compare three spiking functions, relying on rate-coded and time-coded approaches, respectively.</p> </div> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="339" id="S2.F1.g1" src="x1.png" width="829"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span> Geometrical visualization of the radar signals. On the left, antenna layout with <math alttext="N_{\text{vx}}" class="ltx_Math" display="inline" id="S2.F1.8.m1.1"><semantics id="S2.F1.8.m1.1b"><msub id="S2.F1.8.m1.1.1" xref="S2.F1.8.m1.1.1.cmml"><mi id="S2.F1.8.m1.1.1.2" xref="S2.F1.8.m1.1.1.2.cmml">N</mi><mtext id="S2.F1.8.m1.1.1.3" xref="S2.F1.8.m1.1.1.3a.cmml">vx</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.F1.8.m1.1c"><apply id="S2.F1.8.m1.1.1.cmml" xref="S2.F1.8.m1.1.1"><csymbol cd="ambiguous" id="S2.F1.8.m1.1.1.1.cmml" xref="S2.F1.8.m1.1.1">subscript</csymbol><ci id="S2.F1.8.m1.1.1.2.cmml" xref="S2.F1.8.m1.1.1.2">𝑁</ci><ci id="S2.F1.8.m1.1.1.3a.cmml" xref="S2.F1.8.m1.1.1.3"><mtext id="S2.F1.8.m1.1.1.3.cmml" mathsize="70%" xref="S2.F1.8.m1.1.1.3">vx</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.8.m1.1d">N_{\text{vx}}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.8.m1.1e">italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT</annotation></semantics></math> virtual antennas in one line and a spacing <math alttext="b" class="ltx_Math" display="inline" id="S2.F1.9.m2.1"><semantics id="S2.F1.9.m2.1b"><mi id="S2.F1.9.m2.1.1" xref="S2.F1.9.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.F1.9.m2.1c"><ci id="S2.F1.9.m2.1.1.cmml" xref="S2.F1.9.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.9.m2.1d">b</annotation><annotation encoding="application/x-llamapun" id="S2.F1.9.m2.1e">italic_b</annotation></semantics></math> between consecutive antennas. Transmitted and reflected signals are indicated with arrows, and the direction-of-arrival (DoA) is given as angle <math alttext="\theta" class="ltx_Math" display="inline" id="S2.F1.10.m3.1"><semantics id="S2.F1.10.m3.1b"><mi id="S2.F1.10.m3.1.1" xref="S2.F1.10.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.F1.10.m3.1c"><ci id="S2.F1.10.m3.1.1.cmml" xref="S2.F1.10.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.10.m3.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.F1.10.m3.1e">italic_θ</annotation></semantics></math>. In the middle, schematic view of temporal dynamics of the IF signals <math alttext="x_{m}(t)" class="ltx_Math" display="inline" id="S2.F1.11.m4.1"><semantics id="S2.F1.11.m4.1b"><mrow id="S2.F1.11.m4.1.2" xref="S2.F1.11.m4.1.2.cmml"><msub id="S2.F1.11.m4.1.2.2" xref="S2.F1.11.m4.1.2.2.cmml"><mi id="S2.F1.11.m4.1.2.2.2" xref="S2.F1.11.m4.1.2.2.2.cmml">x</mi><mi id="S2.F1.11.m4.1.2.2.3" xref="S2.F1.11.m4.1.2.2.3.cmml">m</mi></msub><mo id="S2.F1.11.m4.1.2.1" xref="S2.F1.11.m4.1.2.1.cmml">⁢</mo><mrow id="S2.F1.11.m4.1.2.3.2" xref="S2.F1.11.m4.1.2.cmml"><mo id="S2.F1.11.m4.1.2.3.2.1" stretchy="false" xref="S2.F1.11.m4.1.2.cmml">(</mo><mi id="S2.F1.11.m4.1.1" xref="S2.F1.11.m4.1.1.cmml">t</mi><mo id="S2.F1.11.m4.1.2.3.2.2" stretchy="false" xref="S2.F1.11.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.11.m4.1c"><apply id="S2.F1.11.m4.1.2.cmml" xref="S2.F1.11.m4.1.2"><times id="S2.F1.11.m4.1.2.1.cmml" xref="S2.F1.11.m4.1.2.1"></times><apply id="S2.F1.11.m4.1.2.2.cmml" xref="S2.F1.11.m4.1.2.2"><csymbol cd="ambiguous" id="S2.F1.11.m4.1.2.2.1.cmml" xref="S2.F1.11.m4.1.2.2">subscript</csymbol><ci id="S2.F1.11.m4.1.2.2.2.cmml" xref="S2.F1.11.m4.1.2.2.2">𝑥</ci><ci id="S2.F1.11.m4.1.2.2.3.cmml" xref="S2.F1.11.m4.1.2.2.3">𝑚</ci></apply><ci id="S2.F1.11.m4.1.1.cmml" xref="S2.F1.11.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.11.m4.1d">x_{m}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.11.m4.1e">italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> with frequency <math alttext="\omega" class="ltx_Math" display="inline" id="S2.F1.12.m5.1"><semantics id="S2.F1.12.m5.1b"><mi id="S2.F1.12.m5.1.1" xref="S2.F1.12.m5.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S2.F1.12.m5.1c"><ci id="S2.F1.12.m5.1.1.cmml" xref="S2.F1.12.m5.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.12.m5.1d">\omega</annotation><annotation encoding="application/x-llamapun" id="S2.F1.12.m5.1e">italic_ω</annotation></semantics></math>, where <math alttext="m" class="ltx_Math" display="inline" id="S2.F1.13.m6.1"><semantics id="S2.F1.13.m6.1b"><mi id="S2.F1.13.m6.1.1" xref="S2.F1.13.m6.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.F1.13.m6.1c"><ci id="S2.F1.13.m6.1.1.cmml" xref="S2.F1.13.m6.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.13.m6.1d">m</annotation><annotation encoding="application/x-llamapun" id="S2.F1.13.m6.1e">italic_m</annotation></semantics></math> is the antenna index. Frequency analysis along the temporal dimension provides information on the range of an object. On the right, schematic view of the complex value <math alttext="\exp(im\phi)" class="ltx_Math" display="inline" id="S2.F1.14.m7.2"><semantics id="S2.F1.14.m7.2b"><mrow id="S2.F1.14.m7.2.2.1" xref="S2.F1.14.m7.2.2.2.cmml"><mi id="S2.F1.14.m7.1.1" xref="S2.F1.14.m7.1.1.cmml">exp</mi><mo id="S2.F1.14.m7.2.2.1b" xref="S2.F1.14.m7.2.2.2.cmml">⁡</mo><mrow id="S2.F1.14.m7.2.2.1.1" xref="S2.F1.14.m7.2.2.2.cmml"><mo id="S2.F1.14.m7.2.2.1.1.2" stretchy="false" xref="S2.F1.14.m7.2.2.2.cmml">(</mo><mrow id="S2.F1.14.m7.2.2.1.1.1" xref="S2.F1.14.m7.2.2.1.1.1.cmml"><mi id="S2.F1.14.m7.2.2.1.1.1.2" xref="S2.F1.14.m7.2.2.1.1.1.2.cmml">i</mi><mo id="S2.F1.14.m7.2.2.1.1.1.1" xref="S2.F1.14.m7.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.F1.14.m7.2.2.1.1.1.3" xref="S2.F1.14.m7.2.2.1.1.1.3.cmml">m</mi><mo id="S2.F1.14.m7.2.2.1.1.1.1b" xref="S2.F1.14.m7.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.F1.14.m7.2.2.1.1.1.4" xref="S2.F1.14.m7.2.2.1.1.1.4.cmml">ϕ</mi></mrow><mo id="S2.F1.14.m7.2.2.1.1.3" stretchy="false" xref="S2.F1.14.m7.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.14.m7.2c"><apply id="S2.F1.14.m7.2.2.2.cmml" xref="S2.F1.14.m7.2.2.1"><exp id="S2.F1.14.m7.1.1.cmml" xref="S2.F1.14.m7.1.1"></exp><apply id="S2.F1.14.m7.2.2.1.1.1.cmml" xref="S2.F1.14.m7.2.2.1.1.1"><times id="S2.F1.14.m7.2.2.1.1.1.1.cmml" xref="S2.F1.14.m7.2.2.1.1.1.1"></times><ci id="S2.F1.14.m7.2.2.1.1.1.2.cmml" xref="S2.F1.14.m7.2.2.1.1.1.2">𝑖</ci><ci id="S2.F1.14.m7.2.2.1.1.1.3.cmml" xref="S2.F1.14.m7.2.2.1.1.1.3">𝑚</ci><ci id="S2.F1.14.m7.2.2.1.1.1.4.cmml" xref="S2.F1.14.m7.2.2.1.1.1.4">italic-ϕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.14.m7.2d">\exp(im\phi)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.14.m7.2e">roman_exp ( italic_i italic_m italic_ϕ )</annotation></semantics></math> over virtual antennas. Frequency analysis along the antenna dimension provides information on the DoA. </figcaption> </figure> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS1.5.1.1">II-A</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS1.6.2">Angle estimation - Dendritic vector multiplication</span> </h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.15">The relative phase shifts between the antenna signals of a multi-array antenna system (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F1" title="Figure 1 ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">1</span></a>, red) contain information on the direction-of-arrival (DoA) of the objects. Due to a much larger range <math alttext="r_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><msub id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">r</mi><mi id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">𝑟</ci><ci id="S2.SS1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">r_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> compared to the distance <math alttext="b" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_b</annotation></semantics></math> between consecutive antennas (<math alttext="r_{k}\gg b" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mrow id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><msub id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2.2" xref="S2.SS1.p1.3.m3.1.1.2.2.cmml">r</mi><mi id="S2.SS1.p1.3.m3.1.1.2.3" xref="S2.SS1.p1.3.m3.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.p1.3.m3.1.1.1" xref="S2.SS1.p1.3.m3.1.1.1.cmml">≫</mo><mi id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><csymbol cd="latexml" id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1.1">much-greater-than</csymbol><apply id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.2.1.cmml" xref="S2.SS1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.2.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2.2">𝑟</ci><ci id="S2.SS1.p1.3.m3.1.1.2.3.cmml" xref="S2.SS1.p1.3.m3.1.1.2.3">𝑘</ci></apply><ci id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">r_{k}\gg b</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≫ italic_b</annotation></semantics></math>), we can assume that the incoming signals are aligned parallel on all antennas. Due to the geometric setup of the antenna array, the range <math alttext="r_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><msub id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">r</mi><mi id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑟</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">r_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> between object <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mi id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><ci id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_k</annotation></semantics></math> and receiving antenna changes slightly for each antenna (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F1" title="Figure 1 ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">1</span></a>). The relative displacements of the range between the <math alttext="0" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mn id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><cn id="S2.SS1.p1.6.m6.1.1.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1">0</cn></annotation-xml></semantics></math>-th and <math alttext="m" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mi id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><ci id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_m</annotation></semantics></math>-th antenna <span class="ltx_text" id="S2.SS1.p1.8.1"><math alttext="\Delta d_{k}=mb\sin(\theta_{k})" class="ltx_Math" display="inline" id="S2.SS1.p1.8.1.m1.2"><semantics id="S2.SS1.p1.8.1.m1.2a"><mrow id="S2.SS1.p1.8.1.m1.2.2" xref="S2.SS1.p1.8.1.m1.2.2.cmml"><mrow id="S2.SS1.p1.8.1.m1.2.2.3" xref="S2.SS1.p1.8.1.m1.2.2.3.cmml"><mi id="S2.SS1.p1.8.1.m1.2.2.3.2" mathvariant="normal" xref="S2.SS1.p1.8.1.m1.2.2.3.2.cmml">Δ</mi><mo id="S2.SS1.p1.8.1.m1.2.2.3.1" xref="S2.SS1.p1.8.1.m1.2.2.3.1.cmml">⁢</mo><msub id="S2.SS1.p1.8.1.m1.2.2.3.3" xref="S2.SS1.p1.8.1.m1.2.2.3.3.cmml"><mi id="S2.SS1.p1.8.1.m1.2.2.3.3.2" xref="S2.SS1.p1.8.1.m1.2.2.3.3.2.cmml">d</mi><mi id="S2.SS1.p1.8.1.m1.2.2.3.3.3" xref="S2.SS1.p1.8.1.m1.2.2.3.3.3.cmml">k</mi></msub></mrow><mo id="S2.SS1.p1.8.1.m1.2.2.2" xref="S2.SS1.p1.8.1.m1.2.2.2.cmml">=</mo><mrow id="S2.SS1.p1.8.1.m1.2.2.1" xref="S2.SS1.p1.8.1.m1.2.2.1.cmml"><mi id="S2.SS1.p1.8.1.m1.2.2.1.3" xref="S2.SS1.p1.8.1.m1.2.2.1.3.cmml">m</mi><mo id="S2.SS1.p1.8.1.m1.2.2.1.2" xref="S2.SS1.p1.8.1.m1.2.2.1.2.cmml">⁢</mo><mi id="S2.SS1.p1.8.1.m1.2.2.1.4" xref="S2.SS1.p1.8.1.m1.2.2.1.4.cmml">b</mi><mo id="S2.SS1.p1.8.1.m1.2.2.1.2a" lspace="0.167em" xref="S2.SS1.p1.8.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.8.1.m1.2.2.1.1.1" xref="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml"><mi id="S2.SS1.p1.8.1.m1.1.1" xref="S2.SS1.p1.8.1.m1.1.1.cmml">sin</mi><mo id="S2.SS1.p1.8.1.m1.2.2.1.1.1a" xref="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml">⁡</mo><mrow id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1" xref="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml"><mo id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.2" stretchy="false" xref="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml">(</mo><msub id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.2" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.2.cmml">θ</mi><mi id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.3" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.3" stretchy="false" xref="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.1.m1.2b"><apply id="S2.SS1.p1.8.1.m1.2.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2"><eq id="S2.SS1.p1.8.1.m1.2.2.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.2"></eq><apply id="S2.SS1.p1.8.1.m1.2.2.3.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3"><times id="S2.SS1.p1.8.1.m1.2.2.3.1.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.1"></times><ci id="S2.SS1.p1.8.1.m1.2.2.3.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.2">Δ</ci><apply id="S2.SS1.p1.8.1.m1.2.2.3.3.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.8.1.m1.2.2.3.3.1.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.3">subscript</csymbol><ci id="S2.SS1.p1.8.1.m1.2.2.3.3.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.3.2">𝑑</ci><ci id="S2.SS1.p1.8.1.m1.2.2.3.3.3.cmml" xref="S2.SS1.p1.8.1.m1.2.2.3.3.3">𝑘</ci></apply></apply><apply id="S2.SS1.p1.8.1.m1.2.2.1.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1"><times id="S2.SS1.p1.8.1.m1.2.2.1.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.2"></times><ci id="S2.SS1.p1.8.1.m1.2.2.1.3.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.3">𝑚</ci><ci id="S2.SS1.p1.8.1.m1.2.2.1.4.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.4">𝑏</ci><apply id="S2.SS1.p1.8.1.m1.2.2.1.1.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1"><sin id="S2.SS1.p1.8.1.m1.1.1.cmml" xref="S2.SS1.p1.8.1.m1.1.1"></sin><apply id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.2">𝜃</ci><ci id="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.SS1.p1.8.1.m1.2.2.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.1.m1.2c">\Delta d_{k}=mb\sin(\theta_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.1.m1.2d">roman_Δ italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_m italic_b roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></span> depends on the angle <math alttext="\theta_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m8.1"><semantics id="S2.SS1.p1.9.m8.1a"><msub id="S2.SS1.p1.9.m8.1.1" xref="S2.SS1.p1.9.m8.1.1.cmml"><mi id="S2.SS1.p1.9.m8.1.1.2" xref="S2.SS1.p1.9.m8.1.1.2.cmml">θ</mi><mi id="S2.SS1.p1.9.m8.1.1.3" xref="S2.SS1.p1.9.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m8.1b"><apply id="S2.SS1.p1.9.m8.1.1.cmml" xref="S2.SS1.p1.9.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m8.1.1.1.cmml" xref="S2.SS1.p1.9.m8.1.1">subscript</csymbol><ci id="S2.SS1.p1.9.m8.1.1.2.cmml" xref="S2.SS1.p1.9.m8.1.1.2">𝜃</ci><ci id="S2.SS1.p1.9.m8.1.1.3.cmml" xref="S2.SS1.p1.9.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m8.1c">\theta_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m8.1d">italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> between target <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m9.1"><semantics id="S2.SS1.p1.10.m9.1a"><mi id="S2.SS1.p1.10.m9.1.1" xref="S2.SS1.p1.10.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m9.1b"><ci id="S2.SS1.p1.10.m9.1.1.cmml" xref="S2.SS1.p1.10.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m9.1d">italic_k</annotation></semantics></math> and sensor, and on the antenna distance <math alttext="mb" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m10.1"><semantics id="S2.SS1.p1.11.m10.1a"><mrow id="S2.SS1.p1.11.m10.1.1" xref="S2.SS1.p1.11.m10.1.1.cmml"><mi id="S2.SS1.p1.11.m10.1.1.2" xref="S2.SS1.p1.11.m10.1.1.2.cmml">m</mi><mo id="S2.SS1.p1.11.m10.1.1.1" xref="S2.SS1.p1.11.m10.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p1.11.m10.1.1.3" xref="S2.SS1.p1.11.m10.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m10.1b"><apply id="S2.SS1.p1.11.m10.1.1.cmml" xref="S2.SS1.p1.11.m10.1.1"><times id="S2.SS1.p1.11.m10.1.1.1.cmml" xref="S2.SS1.p1.11.m10.1.1.1"></times><ci id="S2.SS1.p1.11.m10.1.1.2.cmml" xref="S2.SS1.p1.11.m10.1.1.2">𝑚</ci><ci id="S2.SS1.p1.11.m10.1.1.3.cmml" xref="S2.SS1.p1.11.m10.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m10.1c">mb</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m10.1d">italic_m italic_b</annotation></semantics></math>. This displacement results in a phase shift between the IF signals of two antennas. By assuming the antenna distance to match multiples of half of the wavelength of the radar sensor <span class="ltx_text" id="S2.SS1.p1.12.2"><math alttext="b=\lambda/2" class="ltx_Math" display="inline" id="S2.SS1.p1.12.2.m1.1"><semantics id="S2.SS1.p1.12.2.m1.1a"><mrow id="S2.SS1.p1.12.2.m1.1.1" xref="S2.SS1.p1.12.2.m1.1.1.cmml"><mi id="S2.SS1.p1.12.2.m1.1.1.2" xref="S2.SS1.p1.12.2.m1.1.1.2.cmml">b</mi><mo id="S2.SS1.p1.12.2.m1.1.1.1" xref="S2.SS1.p1.12.2.m1.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.12.2.m1.1.1.3" xref="S2.SS1.p1.12.2.m1.1.1.3.cmml"><mi id="S2.SS1.p1.12.2.m1.1.1.3.2" xref="S2.SS1.p1.12.2.m1.1.1.3.2.cmml">λ</mi><mo id="S2.SS1.p1.12.2.m1.1.1.3.1" xref="S2.SS1.p1.12.2.m1.1.1.3.1.cmml">/</mo><mn id="S2.SS1.p1.12.2.m1.1.1.3.3" xref="S2.SS1.p1.12.2.m1.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.12.2.m1.1b"><apply id="S2.SS1.p1.12.2.m1.1.1.cmml" xref="S2.SS1.p1.12.2.m1.1.1"><eq id="S2.SS1.p1.12.2.m1.1.1.1.cmml" xref="S2.SS1.p1.12.2.m1.1.1.1"></eq><ci id="S2.SS1.p1.12.2.m1.1.1.2.cmml" xref="S2.SS1.p1.12.2.m1.1.1.2">𝑏</ci><apply id="S2.SS1.p1.12.2.m1.1.1.3.cmml" xref="S2.SS1.p1.12.2.m1.1.1.3"><divide id="S2.SS1.p1.12.2.m1.1.1.3.1.cmml" xref="S2.SS1.p1.12.2.m1.1.1.3.1"></divide><ci id="S2.SS1.p1.12.2.m1.1.1.3.2.cmml" xref="S2.SS1.p1.12.2.m1.1.1.3.2">𝜆</ci><cn id="S2.SS1.p1.12.2.m1.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.12.2.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.12.2.m1.1c">b=\lambda/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.12.2.m1.1d">italic_b = italic_λ / 2</annotation></semantics></math></span>, the relative phase shift between antenna signals <math alttext="x_{0}(t)" class="ltx_Math" display="inline" id="S2.SS1.p1.13.m11.1"><semantics id="S2.SS1.p1.13.m11.1a"><mrow id="S2.SS1.p1.13.m11.1.2" xref="S2.SS1.p1.13.m11.1.2.cmml"><msub id="S2.SS1.p1.13.m11.1.2.2" xref="S2.SS1.p1.13.m11.1.2.2.cmml"><mi id="S2.SS1.p1.13.m11.1.2.2.2" xref="S2.SS1.p1.13.m11.1.2.2.2.cmml">x</mi><mn id="S2.SS1.p1.13.m11.1.2.2.3" xref="S2.SS1.p1.13.m11.1.2.2.3.cmml">0</mn></msub><mo id="S2.SS1.p1.13.m11.1.2.1" xref="S2.SS1.p1.13.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.13.m11.1.2.3.2" xref="S2.SS1.p1.13.m11.1.2.cmml"><mo id="S2.SS1.p1.13.m11.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.13.m11.1.2.cmml">(</mo><mi id="S2.SS1.p1.13.m11.1.1" xref="S2.SS1.p1.13.m11.1.1.cmml">t</mi><mo id="S2.SS1.p1.13.m11.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.13.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.13.m11.1b"><apply id="S2.SS1.p1.13.m11.1.2.cmml" xref="S2.SS1.p1.13.m11.1.2"><times id="S2.SS1.p1.13.m11.1.2.1.cmml" xref="S2.SS1.p1.13.m11.1.2.1"></times><apply id="S2.SS1.p1.13.m11.1.2.2.cmml" xref="S2.SS1.p1.13.m11.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.13.m11.1.2.2.1.cmml" xref="S2.SS1.p1.13.m11.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.13.m11.1.2.2.2.cmml" xref="S2.SS1.p1.13.m11.1.2.2.2">𝑥</ci><cn id="S2.SS1.p1.13.m11.1.2.2.3.cmml" type="integer" xref="S2.SS1.p1.13.m11.1.2.2.3">0</cn></apply><ci id="S2.SS1.p1.13.m11.1.1.cmml" xref="S2.SS1.p1.13.m11.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.13.m11.1c">x_{0}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.13.m11.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> and <math alttext="x_{m}(t)" class="ltx_Math" display="inline" id="S2.SS1.p1.14.m12.1"><semantics id="S2.SS1.p1.14.m12.1a"><mrow id="S2.SS1.p1.14.m12.1.2" xref="S2.SS1.p1.14.m12.1.2.cmml"><msub id="S2.SS1.p1.14.m12.1.2.2" xref="S2.SS1.p1.14.m12.1.2.2.cmml"><mi id="S2.SS1.p1.14.m12.1.2.2.2" xref="S2.SS1.p1.14.m12.1.2.2.2.cmml">x</mi><mi id="S2.SS1.p1.14.m12.1.2.2.3" xref="S2.SS1.p1.14.m12.1.2.2.3.cmml">m</mi></msub><mo id="S2.SS1.p1.14.m12.1.2.1" xref="S2.SS1.p1.14.m12.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.14.m12.1.2.3.2" xref="S2.SS1.p1.14.m12.1.2.cmml"><mo id="S2.SS1.p1.14.m12.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.14.m12.1.2.cmml">(</mo><mi id="S2.SS1.p1.14.m12.1.1" xref="S2.SS1.p1.14.m12.1.1.cmml">t</mi><mo id="S2.SS1.p1.14.m12.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.14.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.14.m12.1b"><apply id="S2.SS1.p1.14.m12.1.2.cmml" xref="S2.SS1.p1.14.m12.1.2"><times id="S2.SS1.p1.14.m12.1.2.1.cmml" xref="S2.SS1.p1.14.m12.1.2.1"></times><apply id="S2.SS1.p1.14.m12.1.2.2.cmml" xref="S2.SS1.p1.14.m12.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.14.m12.1.2.2.1.cmml" xref="S2.SS1.p1.14.m12.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.14.m12.1.2.2.2.cmml" xref="S2.SS1.p1.14.m12.1.2.2.2">𝑥</ci><ci id="S2.SS1.p1.14.m12.1.2.2.3.cmml" xref="S2.SS1.p1.14.m12.1.2.2.3">𝑚</ci></apply><ci id="S2.SS1.p1.14.m12.1.1.cmml" xref="S2.SS1.p1.14.m12.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.14.m12.1c">x_{m}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.14.m12.1d">italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> can be stated as <span class="ltx_text" id="S2.SS1.p1.15.3"><math alttext="m\cdot\phi(\theta_{k})=m\cdot\pi\sin(\theta_{k})" class="ltx_Math" display="inline" id="S2.SS1.p1.15.3.m1.3"><semantics id="S2.SS1.p1.15.3.m1.3a"><mrow id="S2.SS1.p1.15.3.m1.3.3" xref="S2.SS1.p1.15.3.m1.3.3.cmml"><mrow id="S2.SS1.p1.15.3.m1.2.2.1" xref="S2.SS1.p1.15.3.m1.2.2.1.cmml"><mrow id="S2.SS1.p1.15.3.m1.2.2.1.3" xref="S2.SS1.p1.15.3.m1.2.2.1.3.cmml"><mi id="S2.SS1.p1.15.3.m1.2.2.1.3.2" xref="S2.SS1.p1.15.3.m1.2.2.1.3.2.cmml">m</mi><mo id="S2.SS1.p1.15.3.m1.2.2.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.15.3.m1.2.2.1.3.1.cmml">⋅</mo><mi id="S2.SS1.p1.15.3.m1.2.2.1.3.3" xref="S2.SS1.p1.15.3.m1.2.2.1.3.3.cmml">ϕ</mi></mrow><mo id="S2.SS1.p1.15.3.m1.2.2.1.2" xref="S2.SS1.p1.15.3.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.15.3.m1.2.2.1.1.1" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p1.15.3.m1.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.cmml"><mi id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.2" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.2.cmml">θ</mi><mi id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.3" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.p1.15.3.m1.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.15.3.m1.3.3.3" xref="S2.SS1.p1.15.3.m1.3.3.3.cmml">=</mo><mrow id="S2.SS1.p1.15.3.m1.3.3.2" xref="S2.SS1.p1.15.3.m1.3.3.2.cmml"><mrow id="S2.SS1.p1.15.3.m1.3.3.2.3" xref="S2.SS1.p1.15.3.m1.3.3.2.3.cmml"><mi id="S2.SS1.p1.15.3.m1.3.3.2.3.2" xref="S2.SS1.p1.15.3.m1.3.3.2.3.2.cmml">m</mi><mo id="S2.SS1.p1.15.3.m1.3.3.2.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.15.3.m1.3.3.2.3.1.cmml">⋅</mo><mi id="S2.SS1.p1.15.3.m1.3.3.2.3.3" xref="S2.SS1.p1.15.3.m1.3.3.2.3.3.cmml">π</mi></mrow><mo id="S2.SS1.p1.15.3.m1.3.3.2.2" lspace="0.167em" xref="S2.SS1.p1.15.3.m1.3.3.2.2.cmml">⁢</mo><mrow id="S2.SS1.p1.15.3.m1.3.3.2.1.1" xref="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml"><mi id="S2.SS1.p1.15.3.m1.1.1" xref="S2.SS1.p1.15.3.m1.1.1.cmml">sin</mi><mo id="S2.SS1.p1.15.3.m1.3.3.2.1.1a" xref="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml">⁡</mo><mrow id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1" xref="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml"><mo id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.2" stretchy="false" xref="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml">(</mo><msub id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.cmml"><mi id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.2" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.2.cmml">θ</mi><mi id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.3" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.3" stretchy="false" xref="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.15.3.m1.3b"><apply id="S2.SS1.p1.15.3.m1.3.3.cmml" xref="S2.SS1.p1.15.3.m1.3.3"><eq id="S2.SS1.p1.15.3.m1.3.3.3.cmml" xref="S2.SS1.p1.15.3.m1.3.3.3"></eq><apply id="S2.SS1.p1.15.3.m1.2.2.1.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1"><times id="S2.SS1.p1.15.3.m1.2.2.1.2.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.2"></times><apply id="S2.SS1.p1.15.3.m1.2.2.1.3.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.3"><ci id="S2.SS1.p1.15.3.m1.2.2.1.3.1.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.3.1">⋅</ci><ci id="S2.SS1.p1.15.3.m1.2.2.1.3.2.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.3.2">𝑚</ci><ci id="S2.SS1.p1.15.3.m1.2.2.1.3.3.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.3.3">italic-ϕ</ci></apply><apply id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.2">𝜃</ci><ci id="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p1.15.3.m1.2.2.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S2.SS1.p1.15.3.m1.3.3.2.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2"><times id="S2.SS1.p1.15.3.m1.3.3.2.2.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.2"></times><apply id="S2.SS1.p1.15.3.m1.3.3.2.3.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.3"><ci id="S2.SS1.p1.15.3.m1.3.3.2.3.1.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.3.1">⋅</ci><ci id="S2.SS1.p1.15.3.m1.3.3.2.3.2.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.3.2">𝑚</ci><ci id="S2.SS1.p1.15.3.m1.3.3.2.3.3.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.3.3">𝜋</ci></apply><apply id="S2.SS1.p1.15.3.m1.3.3.2.1.2.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1"><sin id="S2.SS1.p1.15.3.m1.1.1.cmml" xref="S2.SS1.p1.15.3.m1.1.1"></sin><apply id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.1.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.2.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.2">𝜃</ci><ci id="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.3.cmml" xref="S2.SS1.p1.15.3.m1.3.3.2.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.15.3.m1.3c">m\cdot\phi(\theta_{k})=m\cdot\pi\sin(\theta_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.15.3.m1.3d">italic_m ⋅ italic_ϕ ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_m ⋅ italic_π roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></span>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.5">Frequency spectrum analysis, such as the discrete FT (DFT), estimates the angle information in the signal of a multi-array antenna system. The DFT is a matrix multiplication of a complex matrix <math alttext="W" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_W</annotation></semantics></math> with the IF signal vector <math alttext="\vec{x}(t)" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mover accent="true" id="S2.SS1.p2.2.m2.1.2.2" xref="S2.SS1.p2.2.m2.1.2.2.cmml"><mi id="S2.SS1.p2.2.m2.1.2.2.2" xref="S2.SS1.p2.2.m2.1.2.2.2.cmml">x</mi><mo id="S2.SS1.p2.2.m2.1.2.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.2.1.cmml">→</mo></mover><mo id="S2.SS1.p2.2.m2.1.2.1" xref="S2.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.2.m2.1.2.3.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mo id="S2.SS1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">t</mi><mo id="S2.SS1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.2.cmml" xref="S2.SS1.p2.2.m2.1.2"><times id="S2.SS1.p2.2.m2.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.1"></times><apply id="S2.SS1.p2.2.m2.1.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2"><ci id="S2.SS1.p2.2.m2.1.2.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.2.1">→</ci><ci id="S2.SS1.p2.2.m2.1.2.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2.2">𝑥</ci></apply><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\vec{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">over→ start_ARG italic_x end_ARG ( italic_t )</annotation></semantics></math>. The matrix <math alttext="W" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mi id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_W</annotation></semantics></math> consists of matrix elements <span class="ltx_text" id="S2.SS1.p2.4.1"><math alttext="W_{lm}=e^{-im\phi_{l}}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.1.m1.1"><semantics id="S2.SS1.p2.4.1.m1.1a"><mrow id="S2.SS1.p2.4.1.m1.1.1" xref="S2.SS1.p2.4.1.m1.1.1.cmml"><msub id="S2.SS1.p2.4.1.m1.1.1.2" xref="S2.SS1.p2.4.1.m1.1.1.2.cmml"><mi id="S2.SS1.p2.4.1.m1.1.1.2.2" xref="S2.SS1.p2.4.1.m1.1.1.2.2.cmml">W</mi><mrow id="S2.SS1.p2.4.1.m1.1.1.2.3" xref="S2.SS1.p2.4.1.m1.1.1.2.3.cmml"><mi id="S2.SS1.p2.4.1.m1.1.1.2.3.2" xref="S2.SS1.p2.4.1.m1.1.1.2.3.2.cmml">l</mi><mo id="S2.SS1.p2.4.1.m1.1.1.2.3.1" xref="S2.SS1.p2.4.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.4.1.m1.1.1.2.3.3" xref="S2.SS1.p2.4.1.m1.1.1.2.3.3.cmml">m</mi></mrow></msub><mo id="S2.SS1.p2.4.1.m1.1.1.1" xref="S2.SS1.p2.4.1.m1.1.1.1.cmml">=</mo><msup id="S2.SS1.p2.4.1.m1.1.1.3" xref="S2.SS1.p2.4.1.m1.1.1.3.cmml"><mi id="S2.SS1.p2.4.1.m1.1.1.3.2" xref="S2.SS1.p2.4.1.m1.1.1.3.2.cmml">e</mi><mrow id="S2.SS1.p2.4.1.m1.1.1.3.3" xref="S2.SS1.p2.4.1.m1.1.1.3.3.cmml"><mo id="S2.SS1.p2.4.1.m1.1.1.3.3a" xref="S2.SS1.p2.4.1.m1.1.1.3.3.cmml">−</mo><mrow id="S2.SS1.p2.4.1.m1.1.1.3.3.2" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.cmml"><mi id="S2.SS1.p2.4.1.m1.1.1.3.3.2.2" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.2.cmml">i</mi><mo id="S2.SS1.p2.4.1.m1.1.1.3.3.2.1" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.1.cmml">⁢</mo><mi id="S2.SS1.p2.4.1.m1.1.1.3.3.2.3" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.3.cmml">m</mi><mo id="S2.SS1.p2.4.1.m1.1.1.3.3.2.1a" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.cmml"><mi id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.2" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.2.cmml">ϕ</mi><mi id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.3" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.3.cmml">l</mi></msub></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.1.m1.1b"><apply id="S2.SS1.p2.4.1.m1.1.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1"><eq id="S2.SS1.p2.4.1.m1.1.1.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.1"></eq><apply id="S2.SS1.p2.4.1.m1.1.1.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.1.m1.1.1.2.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.4.1.m1.1.1.2.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2.2">𝑊</ci><apply id="S2.SS1.p2.4.1.m1.1.1.2.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2.3"><times id="S2.SS1.p2.4.1.m1.1.1.2.3.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2.3.1"></times><ci id="S2.SS1.p2.4.1.m1.1.1.2.3.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2.3.2">𝑙</ci><ci id="S2.SS1.p2.4.1.m1.1.1.2.3.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.2.3.3">𝑚</ci></apply></apply><apply id="S2.SS1.p2.4.1.m1.1.1.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p2.4.1.m1.1.1.3.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p2.4.1.m1.1.1.3.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.2">𝑒</ci><apply id="S2.SS1.p2.4.1.m1.1.1.3.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3"><minus id="S2.SS1.p2.4.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3"></minus><apply id="S2.SS1.p2.4.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2"><times id="S2.SS1.p2.4.1.m1.1.1.3.3.2.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.1"></times><ci id="S2.SS1.p2.4.1.m1.1.1.3.3.2.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.2">𝑖</ci><ci id="S2.SS1.p2.4.1.m1.1.1.3.3.2.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.3">𝑚</ci><apply id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4"><csymbol cd="ambiguous" id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.1.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4">subscript</csymbol><ci id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.2.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.2">italic-ϕ</ci><ci id="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.3.cmml" xref="S2.SS1.p2.4.1.m1.1.1.3.3.2.4.3">𝑙</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.1.m1.1c">W_{lm}=e^{-im\phi_{l}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_l italic_m end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_i italic_m italic_ϕ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math></span>, where <span class="ltx_text" id="S2.SS1.p2.5.2"><math alttext="l\in[0,N_{\text{vx}}]" class="ltx_Math" display="inline" id="S2.SS1.p2.5.2.m1.2"><semantics id="S2.SS1.p2.5.2.m1.2a"><mrow id="S2.SS1.p2.5.2.m1.2.2" xref="S2.SS1.p2.5.2.m1.2.2.cmml"><mi id="S2.SS1.p2.5.2.m1.2.2.3" xref="S2.SS1.p2.5.2.m1.2.2.3.cmml">l</mi><mo id="S2.SS1.p2.5.2.m1.2.2.2" xref="S2.SS1.p2.5.2.m1.2.2.2.cmml">∈</mo><mrow id="S2.SS1.p2.5.2.m1.2.2.1.1" xref="S2.SS1.p2.5.2.m1.2.2.1.2.cmml"><mo id="S2.SS1.p2.5.2.m1.2.2.1.1.2" stretchy="false" xref="S2.SS1.p2.5.2.m1.2.2.1.2.cmml">[</mo><mn id="S2.SS1.p2.5.2.m1.1.1" xref="S2.SS1.p2.5.2.m1.1.1.cmml">0</mn><mo id="S2.SS1.p2.5.2.m1.2.2.1.1.3" xref="S2.SS1.p2.5.2.m1.2.2.1.2.cmml">,</mo><msub id="S2.SS1.p2.5.2.m1.2.2.1.1.1" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.5.2.m1.2.2.1.1.1.2" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.2.cmml">N</mi><mtext id="S2.SS1.p2.5.2.m1.2.2.1.1.1.3" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.3a.cmml">vx</mtext></msub><mo id="S2.SS1.p2.5.2.m1.2.2.1.1.4" stretchy="false" xref="S2.SS1.p2.5.2.m1.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.2.m1.2b"><apply id="S2.SS1.p2.5.2.m1.2.2.cmml" xref="S2.SS1.p2.5.2.m1.2.2"><in id="S2.SS1.p2.5.2.m1.2.2.2.cmml" xref="S2.SS1.p2.5.2.m1.2.2.2"></in><ci id="S2.SS1.p2.5.2.m1.2.2.3.cmml" xref="S2.SS1.p2.5.2.m1.2.2.3">𝑙</ci><interval closure="closed" id="S2.SS1.p2.5.2.m1.2.2.1.2.cmml" xref="S2.SS1.p2.5.2.m1.2.2.1.1"><cn id="S2.SS1.p2.5.2.m1.1.1.cmml" type="integer" xref="S2.SS1.p2.5.2.m1.1.1">0</cn><apply id="S2.SS1.p2.5.2.m1.2.2.1.1.1.cmml" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.5.2.m1.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.5.2.m1.2.2.1.1.1.2.cmml" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.2">𝑁</ci><ci id="S2.SS1.p2.5.2.m1.2.2.1.1.1.3a.cmml" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.3"><mtext id="S2.SS1.p2.5.2.m1.2.2.1.1.1.3.cmml" mathsize="70%" xref="S2.SS1.p2.5.2.m1.2.2.1.1.1.3">vx</mtext></ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.2.m1.2c">l\in[0,N_{\text{vx}}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.2.m1.2d">italic_l ∈ [ 0 , italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT ]</annotation></semantics></math></span> indicates phase shifts depending on the resolution of the DFT. Accordingly, we rewrite the DFT as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx2"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle y_{l}(t)" class="ltx_Math" display="inline" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.2" xref="S2.E2.m1.1.2.cmml"><msub id="S2.E2.m1.1.2.2" xref="S2.E2.m1.1.2.2.cmml"><mi id="S2.E2.m1.1.2.2.2" xref="S2.E2.m1.1.2.2.2.cmml">y</mi><mi id="S2.E2.m1.1.2.2.3" xref="S2.E2.m1.1.2.2.3.cmml">l</mi></msub><mo id="S2.E2.m1.1.2.1" xref="S2.E2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.1.2.3.2" xref="S2.E2.m1.1.2.cmml"><mo id="S2.E2.m1.1.2.3.2.1" stretchy="false" xref="S2.E2.m1.1.2.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">t</mi><mo id="S2.E2.m1.1.2.3.2.2" stretchy="false" xref="S2.E2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><apply id="S2.E2.m1.1.2.cmml" xref="S2.E2.m1.1.2"><times id="S2.E2.m1.1.2.1.cmml" xref="S2.E2.m1.1.2.1"></times><apply id="S2.E2.m1.1.2.2.cmml" xref="S2.E2.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.1.2.2.1.cmml" xref="S2.E2.m1.1.2.2">subscript</csymbol><ci id="S2.E2.m1.1.2.2.2.cmml" xref="S2.E2.m1.1.2.2.2">𝑦</ci><ci id="S2.E2.m1.1.2.2.3.cmml" xref="S2.E2.m1.1.2.2.3">𝑙</ci></apply><ci id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\displaystyle y_{l}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum^{N_{\text{vx}}}_{m}W_{lm}x_{m}(t)" class="ltx_Math" display="inline" id="S2.E2.m2.1"><semantics id="S2.E2.m2.1a"><mrow id="S2.E2.m2.1.2" 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xref="S2.E3.m1.1.1.3.2.3.2.4.3.2">𝑖</ci><apply id="S2.E3.m1.1.1.3.2.3.2.4.3.3.cmml" xref="S2.E3.m1.1.1.3.2.3.2.4.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.3.2.3.2.4.3.3.1.cmml" xref="S2.E3.m1.1.1.3.2.3.2.4.3.3">subscript</csymbol><ci id="S2.E3.m1.1.1.3.2.3.2.4.3.3.2.cmml" xref="S2.E3.m1.1.1.3.2.3.2.4.3.3.2">𝜔</ci><ci id="S2.E3.m1.1.1.3.2.3.2.4.3.3.3.cmml" xref="S2.E3.m1.1.1.3.2.3.2.4.3.3.3">𝑘</ci></apply><ci id="S2.E3.m1.1.1.3.2.3.2.4.3.4.cmml" xref="S2.E3.m1.1.1.3.2.3.2.4.3.4">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\displaystyle=\sum^{N_{\text{vx}}}_{n}e^{-im\phi_{k}}\sum^{K}_{k}a_{k}e^{im% \phi_{k}}e^{i\omega_{k}t}</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">= ∑ start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i italic_m italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_m italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> <tbody id="S2.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S2.E4.m1.1.2.3.2.4.3.3.1.cmml" xref="S2.E4.m1.1.2.3.2.4.3.3">subscript</csymbol><ci id="S2.E4.m1.1.2.3.2.4.3.3.2.cmml" xref="S2.E4.m1.1.2.3.2.4.3.3.2">𝜔</ci><ci id="S2.E4.m1.1.2.3.2.4.3.3.3.cmml" xref="S2.E4.m1.1.2.3.2.4.3.3.3">𝑘</ci></apply><ci id="S2.E4.m1.1.2.3.2.4.3.4.cmml" xref="S2.E4.m1.1.2.3.2.4.3.4">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">\displaystyle=\sum^{K}_{k}a_{k}\underbrace{\sum_{m}e^{im(\phi_{k}-\phi_{l})}}_% {\beta_{kl}}e^{i\omega_{k}t}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">= ∑ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT under⏟ start_ARG ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_m ( italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> <tbody id="S2.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum^{K}_{k}a_{k}\cdot\beta_{kl}\cdot e^{i\omega_{k}t}." class="ltx_Math" display="inline" id="S2.E5.m1.1"><semantics id="S2.E5.m1.1a"><mrow id="S2.E5.m1.1.1.1" xref="S2.E5.m1.1.1.1.1.cmml"><mrow id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.1.2.cmml"></mi><mo id="S2.E5.m1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S2.E5.m1.1.1.1.1.3.1" xref="S2.E5.m1.1.1.1.1.3.1.cmml"><munderover id="S2.E5.m1.1.1.1.1.3.1a" xref="S2.E5.m1.1.1.1.1.3.1.cmml"><mo id="S2.E5.m1.1.1.1.1.3.1.2.2" movablelimits="false" xref="S2.E5.m1.1.1.1.1.3.1.2.2.cmml">∑</mo><mi id="S2.E5.m1.1.1.1.1.3.1.3" xref="S2.E5.m1.1.1.1.1.3.1.3.cmml">k</mi><mi id="S2.E5.m1.1.1.1.1.3.1.2.3" xref="S2.E5.m1.1.1.1.1.3.1.2.3.cmml">K</mi></munderover></mstyle><mrow id="S2.E5.m1.1.1.1.1.3.2" xref="S2.E5.m1.1.1.1.1.3.2.cmml"><msub id="S2.E5.m1.1.1.1.1.3.2.2" xref="S2.E5.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E5.m1.1.1.1.1.3.2.2.2" xref="S2.E5.m1.1.1.1.1.3.2.2.2.cmml">a</mi><mi id="S2.E5.m1.1.1.1.1.3.2.2.3" xref="S2.E5.m1.1.1.1.1.3.2.2.3.cmml">k</mi></msub><mo id="S2.E5.m1.1.1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E5.m1.1.1.1.1.3.2.1.cmml">⋅</mo><msub id="S2.E5.m1.1.1.1.1.3.2.3" xref="S2.E5.m1.1.1.1.1.3.2.3.cmml"><mi id="S2.E5.m1.1.1.1.1.3.2.3.2" xref="S2.E5.m1.1.1.1.1.3.2.3.2.cmml">β</mi><mrow id="S2.E5.m1.1.1.1.1.3.2.3.3" xref="S2.E5.m1.1.1.1.1.3.2.3.3.cmml"><mi id="S2.E5.m1.1.1.1.1.3.2.3.3.2" xref="S2.E5.m1.1.1.1.1.3.2.3.3.2.cmml">k</mi><mo id="S2.E5.m1.1.1.1.1.3.2.3.3.1" xref="S2.E5.m1.1.1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S2.E5.m1.1.1.1.1.3.2.3.3.3" xref="S2.E5.m1.1.1.1.1.3.2.3.3.3.cmml">l</mi></mrow></msub><mo id="S2.E5.m1.1.1.1.1.3.2.1a" lspace="0.222em" rspace="0.222em" xref="S2.E5.m1.1.1.1.1.3.2.1.cmml">⋅</mo><msup id="S2.E5.m1.1.1.1.1.3.2.4" xref="S2.E5.m1.1.1.1.1.3.2.4.cmml"><mi id="S2.E5.m1.1.1.1.1.3.2.4.2" xref="S2.E5.m1.1.1.1.1.3.2.4.2.cmml">e</mi><mrow id="S2.E5.m1.1.1.1.1.3.2.4.3" xref="S2.E5.m1.1.1.1.1.3.2.4.3.cmml"><mi id="S2.E5.m1.1.1.1.1.3.2.4.3.2" xref="S2.E5.m1.1.1.1.1.3.2.4.3.2.cmml">i</mi><mo id="S2.E5.m1.1.1.1.1.3.2.4.3.1" 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xref="S2.E5.m1.1.1.1.1.3.2.4.3.1"></times><ci id="S2.E5.m1.1.1.1.1.3.2.4.3.2.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.2">𝑖</ci><apply id="S2.E5.m1.1.1.1.1.3.2.4.3.3.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.3"><csymbol cd="ambiguous" id="S2.E5.m1.1.1.1.1.3.2.4.3.3.1.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.3">subscript</csymbol><ci id="S2.E5.m1.1.1.1.1.3.2.4.3.3.2.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.3.2">𝜔</ci><ci id="S2.E5.m1.1.1.1.1.3.2.4.3.3.3.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.3.3">𝑘</ci></apply><ci id="S2.E5.m1.1.1.1.1.3.2.4.3.4.cmml" xref="S2.E5.m1.1.1.1.1.3.2.4.3.4">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.1c">\displaystyle=\sum^{K}_{k}a_{k}\cdot\beta_{kl}\cdot e^{i\omega_{k}t}.</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.1d">= ∑ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋅ italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ⋅ italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.7">The case of matching phase shifts for a single target (<math alttext="l=k" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">l</mi><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">=</mo><mi id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><eq id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></eq><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝑙</ci><ci id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">l=k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_l = italic_k</annotation></semantics></math>) results in <math alttext="\beta_{kl}=1" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mrow id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><msub id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2.2" xref="S2.SS1.p3.2.m2.1.1.2.2.cmml">β</mi><mrow id="S2.SS1.p3.2.m2.1.1.2.3" xref="S2.SS1.p3.2.m2.1.1.2.3.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2.3.2" xref="S2.SS1.p3.2.m2.1.1.2.3.2.cmml">k</mi><mo id="S2.SS1.p3.2.m2.1.1.2.3.1" xref="S2.SS1.p3.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.2.m2.1.1.2.3.3" xref="S2.SS1.p3.2.m2.1.1.2.3.3.cmml">l</mi></mrow></msub><mo id="S2.SS1.p3.2.m2.1.1.1" xref="S2.SS1.p3.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><apply id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1"><eq id="S2.SS1.p3.2.m2.1.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1.1"></eq><apply id="S2.SS1.p3.2.m2.1.1.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.1.2.1.cmml" xref="S2.SS1.p3.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.2.m2.1.1.2.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2.2">𝛽</ci><apply id="S2.SS1.p3.2.m2.1.1.2.3.cmml" xref="S2.SS1.p3.2.m2.1.1.2.3"><times id="S2.SS1.p3.2.m2.1.1.2.3.1.cmml" xref="S2.SS1.p3.2.m2.1.1.2.3.1"></times><ci id="S2.SS1.p3.2.m2.1.1.2.3.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2.3.2">𝑘</ci><ci id="S2.SS1.p3.2.m2.1.1.2.3.3.cmml" xref="S2.SS1.p3.2.m2.1.1.2.3.3">𝑙</ci></apply></apply><cn id="S2.SS1.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\beta_{kl}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT = 1</annotation></semantics></math> and hence in the highest amplification of the signal <math alttext="y_{k}=N_{\text{vx}}a_{k}e^{i\omega_{k}t}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><mrow id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml"><msub id="S2.SS1.p3.3.m3.1.1.2" xref="S2.SS1.p3.3.m3.1.1.2.cmml"><mi id="S2.SS1.p3.3.m3.1.1.2.2" xref="S2.SS1.p3.3.m3.1.1.2.2.cmml">y</mi><mi id="S2.SS1.p3.3.m3.1.1.2.3" xref="S2.SS1.p3.3.m3.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.p3.3.m3.1.1.1" xref="S2.SS1.p3.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.3.m3.1.1.3" xref="S2.SS1.p3.3.m3.1.1.3.cmml"><msub id="S2.SS1.p3.3.m3.1.1.3.2" xref="S2.SS1.p3.3.m3.1.1.3.2.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.2.2" xref="S2.SS1.p3.3.m3.1.1.3.2.2.cmml">N</mi><mtext id="S2.SS1.p3.3.m3.1.1.3.2.3" xref="S2.SS1.p3.3.m3.1.1.3.2.3a.cmml">vx</mtext></msub><mo id="S2.SS1.p3.3.m3.1.1.3.1" xref="S2.SS1.p3.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p3.3.m3.1.1.3.3" xref="S2.SS1.p3.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.3.2" xref="S2.SS1.p3.3.m3.1.1.3.3.2.cmml">a</mi><mi id="S2.SS1.p3.3.m3.1.1.3.3.3" xref="S2.SS1.p3.3.m3.1.1.3.3.3.cmml">k</mi></msub><mo id="S2.SS1.p3.3.m3.1.1.3.1a" xref="S2.SS1.p3.3.m3.1.1.3.1.cmml">⁢</mo><msup id="S2.SS1.p3.3.m3.1.1.3.4" xref="S2.SS1.p3.3.m3.1.1.3.4.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.4.2" xref="S2.SS1.p3.3.m3.1.1.3.4.2.cmml">e</mi><mrow id="S2.SS1.p3.3.m3.1.1.3.4.3" xref="S2.SS1.p3.3.m3.1.1.3.4.3.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.4.3.2" xref="S2.SS1.p3.3.m3.1.1.3.4.3.2.cmml">i</mi><mo id="S2.SS1.p3.3.m3.1.1.3.4.3.1" xref="S2.SS1.p3.3.m3.1.1.3.4.3.1.cmml">⁢</mo><msub id="S2.SS1.p3.3.m3.1.1.3.4.3.3" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.4.3.3.2" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3.2.cmml">ω</mi><mi id="S2.SS1.p3.3.m3.1.1.3.4.3.3.3" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3.3.cmml">k</mi></msub><mo id="S2.SS1.p3.3.m3.1.1.3.4.3.1a" xref="S2.SS1.p3.3.m3.1.1.3.4.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.3.m3.1.1.3.4.3.4" xref="S2.SS1.p3.3.m3.1.1.3.4.3.4.cmml">t</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1"><eq id="S2.SS1.p3.3.m3.1.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1.1"></eq><apply id="S2.SS1.p3.3.m3.1.1.2.cmml" xref="S2.SS1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.2.1.cmml" xref="S2.SS1.p3.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.2.2.cmml" xref="S2.SS1.p3.3.m3.1.1.2.2">𝑦</ci><ci id="S2.SS1.p3.3.m3.1.1.2.3.cmml" xref="S2.SS1.p3.3.m3.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.p3.3.m3.1.1.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3"><times id="S2.SS1.p3.3.m3.1.1.3.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.1"></times><apply id="S2.SS1.p3.3.m3.1.1.3.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2.2">𝑁</ci><ci id="S2.SS1.p3.3.m3.1.1.3.2.3a.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2.3"><mtext id="S2.SS1.p3.3.m3.1.1.3.2.3.cmml" mathsize="70%" xref="S2.SS1.p3.3.m3.1.1.3.2.3">vx</mtext></ci></apply><apply id="S2.SS1.p3.3.m3.1.1.3.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.3.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.3.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.3.2">𝑎</ci><ci id="S2.SS1.p3.3.m3.1.1.3.3.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.3.3">𝑘</ci></apply><apply id="S2.SS1.p3.3.m3.1.1.3.4.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.4.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4">superscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.4.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.2">𝑒</ci><apply id="S2.SS1.p3.3.m3.1.1.3.4.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3"><times id="S2.SS1.p3.3.m3.1.1.3.4.3.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.1"></times><ci id="S2.SS1.p3.3.m3.1.1.3.4.3.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.2">𝑖</ci><apply id="S2.SS1.p3.3.m3.1.1.3.4.3.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.4.3.3.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.4.3.3.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3.2">𝜔</ci><ci id="S2.SS1.p3.3.m3.1.1.3.4.3.3.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.3.3">𝑘</ci></apply><ci id="S2.SS1.p3.3.m3.1.1.3.4.3.4.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3.4">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">y_{k}=N_{\text{vx}}a_{k}e^{i\omega_{k}t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, whereas non-matching phase shifts (<math alttext="l\neq k" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><mrow id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><mi id="S2.SS1.p3.4.m4.1.1.2" xref="S2.SS1.p3.4.m4.1.1.2.cmml">l</mi><mo id="S2.SS1.p3.4.m4.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.cmml">≠</mo><mi id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><apply id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1"><neq id="S2.SS1.p3.4.m4.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1"></neq><ci id="S2.SS1.p3.4.m4.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.2">𝑙</ci><ci id="S2.SS1.p3.4.m4.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">l\neq k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">italic_l ≠ italic_k</annotation></semantics></math>) result in complex factors <math alttext="\beta_{kl}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><msub id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml"><mi id="S2.SS1.p3.5.m5.1.1.2" xref="S2.SS1.p3.5.m5.1.1.2.cmml">β</mi><mrow id="S2.SS1.p3.5.m5.1.1.3" xref="S2.SS1.p3.5.m5.1.1.3.cmml"><mi id="S2.SS1.p3.5.m5.1.1.3.2" xref="S2.SS1.p3.5.m5.1.1.3.2.cmml">k</mi><mo id="S2.SS1.p3.5.m5.1.1.3.1" xref="S2.SS1.p3.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.5.m5.1.1.3.3" xref="S2.SS1.p3.5.m5.1.1.3.3.cmml">l</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><apply id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.1.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p3.5.m5.1.1.2.cmml" xref="S2.SS1.p3.5.m5.1.1.2">𝛽</ci><apply id="S2.SS1.p3.5.m5.1.1.3.cmml" xref="S2.SS1.p3.5.m5.1.1.3"><times id="S2.SS1.p3.5.m5.1.1.3.1.cmml" xref="S2.SS1.p3.5.m5.1.1.3.1"></times><ci id="S2.SS1.p3.5.m5.1.1.3.2.cmml" xref="S2.SS1.p3.5.m5.1.1.3.2">𝑘</ci><ci id="S2.SS1.p3.5.m5.1.1.3.3.cmml" xref="S2.SS1.p3.5.m5.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\beta_{kl}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT</annotation></semantics></math> with <span class="ltx_text" id="S2.SS1.p3.6.1"><math alttext="|\beta_{kl}|<1" class="ltx_Math" display="inline" id="S2.SS1.p3.6.1.m1.1"><semantics id="S2.SS1.p3.6.1.m1.1a"><mrow id="S2.SS1.p3.6.1.m1.1.1" xref="S2.SS1.p3.6.1.m1.1.1.cmml"><mrow id="S2.SS1.p3.6.1.m1.1.1.1.1" xref="S2.SS1.p3.6.1.m1.1.1.1.2.cmml"><mo id="S2.SS1.p3.6.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.6.1.m1.1.1.1.2.1.cmml">|</mo><msub id="S2.SS1.p3.6.1.m1.1.1.1.1.1" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.6.1.m1.1.1.1.1.1.2" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.2.cmml">β</mi><mrow id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.2" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.1" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.3" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.3.cmml">l</mi></mrow></msub><mo id="S2.SS1.p3.6.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.6.1.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.SS1.p3.6.1.m1.1.1.2" xref="S2.SS1.p3.6.1.m1.1.1.2.cmml"><</mo><mn id="S2.SS1.p3.6.1.m1.1.1.3" xref="S2.SS1.p3.6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.1.m1.1b"><apply id="S2.SS1.p3.6.1.m1.1.1.cmml" xref="S2.SS1.p3.6.1.m1.1.1"><lt id="S2.SS1.p3.6.1.m1.1.1.2.cmml" xref="S2.SS1.p3.6.1.m1.1.1.2"></lt><apply id="S2.SS1.p3.6.1.m1.1.1.1.2.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1"><abs id="S2.SS1.p3.6.1.m1.1.1.1.2.1.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.2"></abs><apply id="S2.SS1.p3.6.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.6.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.6.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.2">𝛽</ci><apply id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3"><times id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.1"></times><ci id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.2">𝑘</ci><ci id="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.3.cmml" xref="S2.SS1.p3.6.1.m1.1.1.1.1.1.3.3">𝑙</ci></apply></apply></apply><cn id="S2.SS1.p3.6.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.1.m1.1c">|\beta_{kl}|<1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.1.m1.1d">| italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT | < 1</annotation></semantics></math></span>. After matrix multiplication, high amplifications indicate an object at a given angle <math alttext="\theta_{l}" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m6.1"><semantics id="S2.SS1.p3.7.m6.1a"><msub id="S2.SS1.p3.7.m6.1.1" xref="S2.SS1.p3.7.m6.1.1.cmml"><mi id="S2.SS1.p3.7.m6.1.1.2" xref="S2.SS1.p3.7.m6.1.1.2.cmml">θ</mi><mi id="S2.SS1.p3.7.m6.1.1.3" xref="S2.SS1.p3.7.m6.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m6.1b"><apply id="S2.SS1.p3.7.m6.1.1.cmml" xref="S2.SS1.p3.7.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m6.1.1.1.cmml" xref="S2.SS1.p3.7.m6.1.1">subscript</csymbol><ci id="S2.SS1.p3.7.m6.1.1.2.cmml" xref="S2.SS1.p3.7.m6.1.1.2">𝜃</ci><ci id="S2.SS1.p3.7.m6.1.1.3.cmml" xref="S2.SS1.p3.7.m6.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m6.1c">\theta_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m6.1d">italic_θ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.6">In our proposed network, a single neuron <math alttext="l" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mi id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">l</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_l</annotation></semantics></math> performs a vector multiplication <span class="ltx_text" id="S2.SS1.p4.2.1"><math alttext="y_{l}(t)=\vec{w}_{l}\cdot\vec{x}(t)" class="ltx_Math" display="inline" id="S2.SS1.p4.2.1.m1.2"><semantics id="S2.SS1.p4.2.1.m1.2a"><mrow id="S2.SS1.p4.2.1.m1.2.3" xref="S2.SS1.p4.2.1.m1.2.3.cmml"><mrow 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xref="S2.SS1.p4.2.1.m1.2.3.3.2.3"><ci id="S2.SS1.p4.2.1.m1.2.3.3.2.3.1.cmml" xref="S2.SS1.p4.2.1.m1.2.3.3.2.3.1">→</ci><ci id="S2.SS1.p4.2.1.m1.2.3.3.2.3.2.cmml" xref="S2.SS1.p4.2.1.m1.2.3.3.2.3.2">𝑥</ci></apply></apply><ci id="S2.SS1.p4.2.1.m1.2.2.cmml" xref="S2.SS1.p4.2.1.m1.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.1.m1.2c">y_{l}(t)=\vec{w}_{l}\cdot\vec{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.1.m1.2d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t ) = over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_x end_ARG ( italic_t )</annotation></semantics></math></span> of the weight vector <span class="ltx_text" id="S2.SS1.p4.3.2"><math alttext="\vec{w}_{l}=(W_{l0},...,W_{lm})" class="ltx_Math" display="inline" id="S2.SS1.p4.3.2.m1.3"><semantics id="S2.SS1.p4.3.2.m1.3a"><mrow id="S2.SS1.p4.3.2.m1.3.3" 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xref="S2.SS1.p4.3.2.m1.3.3.2.2.2.2">𝑊</ci><apply id="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.cmml" xref="S2.SS1.p4.3.2.m1.3.3.2.2.2.3"><times id="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.1.cmml" xref="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.1"></times><ci id="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.2.cmml" xref="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.2">𝑙</ci><ci id="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.3.cmml" xref="S2.SS1.p4.3.2.m1.3.3.2.2.2.3.3">𝑚</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.2.m1.3c">\vec{w}_{l}=(W_{l0},...,W_{lm})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.2.m1.3d">over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT = ( italic_W start_POSTSUBSCRIPT italic_l 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT italic_l italic_m end_POSTSUBSCRIPT )</annotation></semantics></math></span> and the radar signal vector <math alttext="\vec{x}(t)" class="ltx_Math" display="inline" 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The resulting signal <math alttext="y_{l}(t)" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m4.1"><semantics id="S2.SS1.p4.6.m4.1a"><mrow id="S2.SS1.p4.6.m4.1.2" xref="S2.SS1.p4.6.m4.1.2.cmml"><msub id="S2.SS1.p4.6.m4.1.2.2" xref="S2.SS1.p4.6.m4.1.2.2.cmml"><mi id="S2.SS1.p4.6.m4.1.2.2.2" xref="S2.SS1.p4.6.m4.1.2.2.2.cmml">y</mi><mi id="S2.SS1.p4.6.m4.1.2.2.3" xref="S2.SS1.p4.6.m4.1.2.2.3.cmml">l</mi></msub><mo id="S2.SS1.p4.6.m4.1.2.1" xref="S2.SS1.p4.6.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p4.6.m4.1.2.3.2" xref="S2.SS1.p4.6.m4.1.2.cmml"><mo id="S2.SS1.p4.6.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p4.6.m4.1.2.cmml">(</mo><mi id="S2.SS1.p4.6.m4.1.1" xref="S2.SS1.p4.6.m4.1.1.cmml">t</mi><mo id="S2.SS1.p4.6.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p4.6.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m4.1b"><apply id="S2.SS1.p4.6.m4.1.2.cmml" xref="S2.SS1.p4.6.m4.1.2"><times id="S2.SS1.p4.6.m4.1.2.1.cmml" xref="S2.SS1.p4.6.m4.1.2.1"></times><apply id="S2.SS1.p4.6.m4.1.2.2.cmml" xref="S2.SS1.p4.6.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p4.6.m4.1.2.2.1.cmml" xref="S2.SS1.p4.6.m4.1.2.2">subscript</csymbol><ci id="S2.SS1.p4.6.m4.1.2.2.2.cmml" xref="S2.SS1.p4.6.m4.1.2.2.2">𝑦</ci><ci id="S2.SS1.p4.6.m4.1.2.2.3.cmml" xref="S2.SS1.p4.6.m4.1.2.2.3">𝑙</ci></apply><ci id="S2.SS1.p4.6.m4.1.1.cmml" xref="S2.SS1.p4.6.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m4.1c">y_{l}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m4.1d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is further processed in the neuron.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS2.5.1.1">II-B</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS2.6.2">Distance estimation - Neural resonators</span> </h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.2">A spectrum analysis along the temporal dimension extracts the range from a single chirp of the IF signal <math alttext="x_{m}(t)" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mrow id="S2.SS2.p1.1.m1.1.2" xref="S2.SS2.p1.1.m1.1.2.cmml"><msub id="S2.SS2.p1.1.m1.1.2.2" xref="S2.SS2.p1.1.m1.1.2.2.cmml"><mi id="S2.SS2.p1.1.m1.1.2.2.2" xref="S2.SS2.p1.1.m1.1.2.2.2.cmml">x</mi><mi id="S2.SS2.p1.1.m1.1.2.2.3" xref="S2.SS2.p1.1.m1.1.2.2.3.cmml">m</mi></msub><mo id="S2.SS2.p1.1.m1.1.2.1" xref="S2.SS2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p1.1.m1.1.2.3.2" xref="S2.SS2.p1.1.m1.1.2.cmml"><mo id="S2.SS2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p1.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">t</mi><mo id="S2.SS2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.2.cmml" xref="S2.SS2.p1.1.m1.1.2"><times id="S2.SS2.p1.1.m1.1.2.1.cmml" xref="S2.SS2.p1.1.m1.1.2.1"></times><apply id="S2.SS2.p1.1.m1.1.2.2.cmml" xref="S2.SS2.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.1.m1.1.2.2.1.cmml" xref="S2.SS2.p1.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.1.m1.1.2.2.2.cmml" xref="S2.SS2.p1.1.m1.1.2.2.2">𝑥</ci><ci id="S2.SS2.p1.1.m1.1.2.2.3.cmml" xref="S2.SS2.p1.1.m1.1.2.2.3">𝑚</ci></apply><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">x_{m}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> or the weighted IF signal <math alttext="y_{l}(t)" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mrow id="S2.SS2.p1.2.m2.1.2" xref="S2.SS2.p1.2.m2.1.2.cmml"><msub id="S2.SS2.p1.2.m2.1.2.2" xref="S2.SS2.p1.2.m2.1.2.2.cmml"><mi id="S2.SS2.p1.2.m2.1.2.2.2" xref="S2.SS2.p1.2.m2.1.2.2.2.cmml">y</mi><mi id="S2.SS2.p1.2.m2.1.2.2.3" xref="S2.SS2.p1.2.m2.1.2.2.3.cmml">l</mi></msub><mo id="S2.SS2.p1.2.m2.1.2.1" xref="S2.SS2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p1.2.m2.1.2.3.2" xref="S2.SS2.p1.2.m2.1.2.cmml"><mo id="S2.SS2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS2.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml">t</mi><mo id="S2.SS2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><apply id="S2.SS2.p1.2.m2.1.2.cmml" xref="S2.SS2.p1.2.m2.1.2"><times id="S2.SS2.p1.2.m2.1.2.1.cmml" xref="S2.SS2.p1.2.m2.1.2.1"></times><apply id="S2.SS2.p1.2.m2.1.2.2.cmml" xref="S2.SS2.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.2.m2.1.2.2.1.cmml" xref="S2.SS2.p1.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.2.m2.1.2.2.2.cmml" xref="S2.SS2.p1.2.m2.1.2.2.2">𝑦</ci><ci id="S2.SS2.p1.2.m2.1.2.2.3.cmml" xref="S2.SS2.p1.2.m2.1.2.2.3">𝑙</ci></apply><ci id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">y_{l}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> of an FMCW radar as the frequency is directly proportional to the range of an object to the sensor. 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ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.4">with <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_j</annotation></semantics></math> indexing the eigenfrequencies that represent different distances. Typically, we set the initial value of <math alttext="s" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mi id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><ci id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_s</annotation></semantics></math> at <math alttext="s(0)=0" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><mrow id="S2.SS2.p2.3.m3.1.2" xref="S2.SS2.p2.3.m3.1.2.cmml"><mrow id="S2.SS2.p2.3.m3.1.2.2" xref="S2.SS2.p2.3.m3.1.2.2.cmml"><mi id="S2.SS2.p2.3.m3.1.2.2.2" xref="S2.SS2.p2.3.m3.1.2.2.2.cmml">s</mi><mo id="S2.SS2.p2.3.m3.1.2.2.1" xref="S2.SS2.p2.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.3.m3.1.2.2.3.2" xref="S2.SS2.p2.3.m3.1.2.2.cmml"><mo id="S2.SS2.p2.3.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p2.3.m3.1.2.2.cmml">(</mo><mn id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">0</mn><mo id="S2.SS2.p2.3.m3.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p2.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.3.m3.1.2.1" xref="S2.SS2.p2.3.m3.1.2.1.cmml">=</mo><mn id="S2.SS2.p2.3.m3.1.2.3" xref="S2.SS2.p2.3.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.1b"><apply id="S2.SS2.p2.3.m3.1.2.cmml" xref="S2.SS2.p2.3.m3.1.2"><eq id="S2.SS2.p2.3.m3.1.2.1.cmml" xref="S2.SS2.p2.3.m3.1.2.1"></eq><apply id="S2.SS2.p2.3.m3.1.2.2.cmml" xref="S2.SS2.p2.3.m3.1.2.2"><times id="S2.SS2.p2.3.m3.1.2.2.1.cmml" xref="S2.SS2.p2.3.m3.1.2.2.1"></times><ci id="S2.SS2.p2.3.m3.1.2.2.2.cmml" xref="S2.SS2.p2.3.m3.1.2.2.2">𝑠</ci><cn id="S2.SS2.p2.3.m3.1.1.cmml" type="integer" xref="S2.SS2.p2.3.m3.1.1">0</cn></apply><cn id="S2.SS2.p2.3.m3.1.2.3.cmml" type="integer" xref="S2.SS2.p2.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.1c">s(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.1d">italic_s ( 0 ) = 0</annotation></semantics></math> for each chirp. 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id="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.3.2.cmml" xref="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.3.2">𝜔</ci><ci id="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.3.3.cmml" xref="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.3.3">𝑘</ci></apply><ci id="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.4.cmml" xref="S2.SS2.p2.4.1.m1.1.2.3.2.4.3.4">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.1.m1.1c">y_{l}(t)=\sum_{k}^{K}a_{k}\beta_{kl}e^{i\omega_{k}t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math></span>, we can determine the analytic solution to the differential equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E6" title="In II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">6</span></a>),</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx4"> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle s_{jl}(t)=\sum_{k}^{K}\frac{ia_{k}\beta_{kl}e^{iw_{j}t}}{w_{j}-w% _{k}}(e^{-i(w_{j}-w_{k})t}-1)." class="ltx_Math" display="inline" id="S2.E7.m1.3"><semantics id="S2.E7.m1.3a"><mrow id="S2.E7.m1.3.3.1" xref="S2.E7.m1.3.3.1.1.cmml"><mrow id="S2.E7.m1.3.3.1.1" xref="S2.E7.m1.3.3.1.1.cmml"><mrow 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s_{jl}(t)=\sum_{k}^{K}\frac{ia_{k}\beta_{kl}e^{iw_{j}t}}{w_{j}-w% _{k}}(e^{-i(w_{j}-w_{k})t}-1).</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.3d">italic_s start_POSTSUBSCRIPT italic_j italic_l end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT divide start_ARG italic_i italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT end_ARG start_ARG italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ( italic_e start_POSTSUPERSCRIPT - italic_i ( italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_t end_POSTSUPERSCRIPT - 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.1">To reduce computational costs, we use the discrete version of the neuron model in the following work and no decay by setting <math alttext="a_{kl}=1\,\forall k" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><msub id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2.2" xref="S2.SS2.p3.1.m1.1.1.2.2.cmml">a</mi><mrow id="S2.SS2.p3.1.m1.1.1.2.3" xref="S2.SS2.p3.1.m1.1.1.2.3.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2.3.2" xref="S2.SS2.p3.1.m1.1.1.2.3.2.cmml">k</mi><mo id="S2.SS2.p3.1.m1.1.1.2.3.1" xref="S2.SS2.p3.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p3.1.m1.1.1.2.3.3" 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k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT = 1 ∀ italic_k</annotation></semantics></math>. Because an analog-to-digital converter (ADC) typically samples the radar data, the discrete version of the neuron model is sufficient.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx5"> <tbody id="S2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle s_{jl,t+1}=e^{i\Delta\omega_{j}}\cdot s_{jl,t}+y_{l,t}." class="ltx_Math" display="inline" id="S2.E8.m1.7"><semantics id="S2.E8.m1.7a"><mrow id="S2.E8.m1.7.7.1" xref="S2.E8.m1.7.7.1.1.cmml"><mrow id="S2.E8.m1.7.7.1.1" xref="S2.E8.m1.7.7.1.1.cmml"><msub id="S2.E8.m1.7.7.1.1.2" xref="S2.E8.m1.7.7.1.1.2.cmml"><mi id="S2.E8.m1.7.7.1.1.2.2" xref="S2.E8.m1.7.7.1.1.2.2.cmml">s</mi><mrow id="S2.E8.m1.2.2.2.2" xref="S2.E8.m1.2.2.2.3.cmml"><mrow id="S2.E8.m1.1.1.1.1.1" xref="S2.E8.m1.1.1.1.1.1.cmml"><mi id="S2.E8.m1.1.1.1.1.1.2" 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class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.12">The complex state <math alttext="s" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mi id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><ci id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">italic_s</annotation></semantics></math> of a neuron rotates around an angle <math alttext="\Delta\omega_{j}" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mrow id="S2.SS2.p4.2.m2.1.1" 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xref="S2.SS2.p4.2.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">\Delta\omega_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">roman_Δ italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> per sampling time <math alttext="\Delta t" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mrow id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><mi id="S2.SS2.p4.3.m3.1.1.2" mathvariant="normal" xref="S2.SS2.p4.3.m3.1.1.2.cmml">Δ</mi><mo id="S2.SS2.p4.3.m3.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p4.3.m3.1.1.3" xref="S2.SS2.p4.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1"><times id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1"></times><ci id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2">Δ</ci><ci id="S2.SS2.p4.3.m3.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">roman_Δ italic_t</annotation></semantics></math> leading to an angular velocity <span class="ltx_text" id="S2.SS2.p4.4.1"><math alttext="\omega_{j}=\frac{\Delta\omega_{j}}{\Delta t}" class="ltx_Math" display="inline" id="S2.SS2.p4.4.1.m1.1"><semantics id="S2.SS2.p4.4.1.m1.1a"><mrow id="S2.SS2.p4.4.1.m1.1.1" xref="S2.SS2.p4.4.1.m1.1.1.cmml"><msub id="S2.SS2.p4.4.1.m1.1.1.2" xref="S2.SS2.p4.4.1.m1.1.1.2.cmml"><mi id="S2.SS2.p4.4.1.m1.1.1.2.2" xref="S2.SS2.p4.4.1.m1.1.1.2.2.cmml">ω</mi><mi id="S2.SS2.p4.4.1.m1.1.1.2.3" xref="S2.SS2.p4.4.1.m1.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS2.p4.4.1.m1.1.1.1" xref="S2.SS2.p4.4.1.m1.1.1.1.cmml">=</mo><mfrac id="S2.SS2.p4.4.1.m1.1.1.3" xref="S2.SS2.p4.4.1.m1.1.1.3.cmml"><mrow id="S2.SS2.p4.4.1.m1.1.1.3.2" xref="S2.SS2.p4.4.1.m1.1.1.3.2.cmml"><mi id="S2.SS2.p4.4.1.m1.1.1.3.2.2" mathvariant="normal" xref="S2.SS2.p4.4.1.m1.1.1.3.2.2.cmml">Δ</mi><mo id="S2.SS2.p4.4.1.m1.1.1.3.2.1" xref="S2.SS2.p4.4.1.m1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS2.p4.4.1.m1.1.1.3.2.3" xref="S2.SS2.p4.4.1.m1.1.1.3.2.3.cmml"><mi id="S2.SS2.p4.4.1.m1.1.1.3.2.3.2" xref="S2.SS2.p4.4.1.m1.1.1.3.2.3.2.cmml">ω</mi><mi id="S2.SS2.p4.4.1.m1.1.1.3.2.3.3" xref="S2.SS2.p4.4.1.m1.1.1.3.2.3.3.cmml">j</mi></msub></mrow><mrow id="S2.SS2.p4.4.1.m1.1.1.3.3" xref="S2.SS2.p4.4.1.m1.1.1.3.3.cmml"><mi id="S2.SS2.p4.4.1.m1.1.1.3.3.2" mathvariant="normal" xref="S2.SS2.p4.4.1.m1.1.1.3.3.2.cmml">Δ</mi><mo id="S2.SS2.p4.4.1.m1.1.1.3.3.1" xref="S2.SS2.p4.4.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p4.4.1.m1.1.1.3.3.3" xref="S2.SS2.p4.4.1.m1.1.1.3.3.3.cmml">t</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.1.m1.1b"><apply id="S2.SS2.p4.4.1.m1.1.1.cmml" xref="S2.SS2.p4.4.1.m1.1.1"><eq id="S2.SS2.p4.4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.4.1.m1.1.1.1"></eq><apply id="S2.SS2.p4.4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.4.1.m1.1.1.2.1.cmml" xref="S2.SS2.p4.4.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.4.1.m1.1.1.2.2.cmml" xref="S2.SS2.p4.4.1.m1.1.1.2.2">𝜔</ci><ci id="S2.SS2.p4.4.1.m1.1.1.2.3.cmml" xref="S2.SS2.p4.4.1.m1.1.1.2.3">𝑗</ci></apply><apply id="S2.SS2.p4.4.1.m1.1.1.3.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3"><divide id="S2.SS2.p4.4.1.m1.1.1.3.1.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3"></divide><apply id="S2.SS2.p4.4.1.m1.1.1.3.2.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3.2"><times id="S2.SS2.p4.4.1.m1.1.1.3.2.1.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3.2.1"></times><ci id="S2.SS2.p4.4.1.m1.1.1.3.2.2.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3.2.2">Δ</ci><apply id="S2.SS2.p4.4.1.m1.1.1.3.2.3.cmml" xref="S2.SS2.p4.4.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" 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italic_t end_ARG</annotation></semantics></math></span>. The sampling time <math alttext="\Delta t" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m4.1"><semantics id="S2.SS2.p4.5.m4.1a"><mrow id="S2.SS2.p4.5.m4.1.1" xref="S2.SS2.p4.5.m4.1.1.cmml"><mi id="S2.SS2.p4.5.m4.1.1.2" mathvariant="normal" xref="S2.SS2.p4.5.m4.1.1.2.cmml">Δ</mi><mo id="S2.SS2.p4.5.m4.1.1.1" xref="S2.SS2.p4.5.m4.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p4.5.m4.1.1.3" xref="S2.SS2.p4.5.m4.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m4.1b"><apply id="S2.SS2.p4.5.m4.1.1.cmml" xref="S2.SS2.p4.5.m4.1.1"><times id="S2.SS2.p4.5.m4.1.1.1.cmml" xref="S2.SS2.p4.5.m4.1.1.1"></times><ci id="S2.SS2.p4.5.m4.1.1.2.cmml" xref="S2.SS2.p4.5.m4.1.1.2">Δ</ci><ci id="S2.SS2.p4.5.m4.1.1.3.cmml" xref="S2.SS2.p4.5.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m4.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m4.1d">roman_Δ italic_t</annotation></semantics></math> and the chirp time <math alttext="T_{c}" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m5.1"><semantics id="S2.SS2.p4.6.m5.1a"><msub id="S2.SS2.p4.6.m5.1.1" xref="S2.SS2.p4.6.m5.1.1.cmml"><mi id="S2.SS2.p4.6.m5.1.1.2" xref="S2.SS2.p4.6.m5.1.1.2.cmml">T</mi><mi id="S2.SS2.p4.6.m5.1.1.3" xref="S2.SS2.p4.6.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m5.1b"><apply id="S2.SS2.p4.6.m5.1.1.cmml" xref="S2.SS2.p4.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m5.1.1.1.cmml" xref="S2.SS2.p4.6.m5.1.1">subscript</csymbol><ci id="S2.SS2.p4.6.m5.1.1.2.cmml" xref="S2.SS2.p4.6.m5.1.1.2">𝑇</ci><ci id="S2.SS2.p4.6.m5.1.1.3.cmml" xref="S2.SS2.p4.6.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m5.1c">T_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m5.1d">italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> define the number of samples <math alttext="N" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m6.1"><semantics id="S2.SS2.p4.7.m6.1a"><mi id="S2.SS2.p4.7.m6.1.1" xref="S2.SS2.p4.7.m6.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m6.1b"><ci id="S2.SS2.p4.7.m6.1.1.cmml" xref="S2.SS2.p4.7.m6.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m6.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m6.1d">italic_N</annotation></semantics></math> for one chirp. By iteratively applying Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E8" title="In II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">8</span></a>) it can be shown, that the neuron state <math alttext="s(T_{c})" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m7.1"><semantics id="S2.SS2.p4.8.m7.1a"><mrow id="S2.SS2.p4.8.m7.1.1" xref="S2.SS2.p4.8.m7.1.1.cmml"><mi id="S2.SS2.p4.8.m7.1.1.3" xref="S2.SS2.p4.8.m7.1.1.3.cmml">s</mi><mo id="S2.SS2.p4.8.m7.1.1.2" xref="S2.SS2.p4.8.m7.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.8.m7.1.1.1.1" xref="S2.SS2.p4.8.m7.1.1.1.1.1.cmml"><mo id="S2.SS2.p4.8.m7.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.8.m7.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p4.8.m7.1.1.1.1.1" xref="S2.SS2.p4.8.m7.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.8.m7.1.1.1.1.1.2" xref="S2.SS2.p4.8.m7.1.1.1.1.1.2.cmml">T</mi><mi id="S2.SS2.p4.8.m7.1.1.1.1.1.3" xref="S2.SS2.p4.8.m7.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S2.SS2.p4.8.m7.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.8.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m7.1b"><apply id="S2.SS2.p4.8.m7.1.1.cmml" xref="S2.SS2.p4.8.m7.1.1"><times id="S2.SS2.p4.8.m7.1.1.2.cmml" xref="S2.SS2.p4.8.m7.1.1.2"></times><ci id="S2.SS2.p4.8.m7.1.1.3.cmml" xref="S2.SS2.p4.8.m7.1.1.3">𝑠</ci><apply id="S2.SS2.p4.8.m7.1.1.1.1.1.cmml" xref="S2.SS2.p4.8.m7.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.8.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.8.m7.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p4.8.m7.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.8.m7.1.1.1.1.1.2">𝑇</ci><ci id="S2.SS2.p4.8.m7.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.8.m7.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m7.1c">s(T_{c})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m7.1d">italic_s ( italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> at time <math alttext="T_{c}" class="ltx_Math" display="inline" id="S2.SS2.p4.9.m8.1"><semantics id="S2.SS2.p4.9.m8.1a"><msub id="S2.SS2.p4.9.m8.1.1" xref="S2.SS2.p4.9.m8.1.1.cmml"><mi id="S2.SS2.p4.9.m8.1.1.2" xref="S2.SS2.p4.9.m8.1.1.2.cmml">T</mi><mi id="S2.SS2.p4.9.m8.1.1.3" xref="S2.SS2.p4.9.m8.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m8.1b"><apply id="S2.SS2.p4.9.m8.1.1.cmml" xref="S2.SS2.p4.9.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m8.1.1.1.cmml" xref="S2.SS2.p4.9.m8.1.1">subscript</csymbol><ci id="S2.SS2.p4.9.m8.1.1.2.cmml" xref="S2.SS2.p4.9.m8.1.1.2">𝑇</ci><ci id="S2.SS2.p4.9.m8.1.1.3.cmml" xref="S2.SS2.p4.9.m8.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m8.1c">T_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m8.1d">italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="N" class="ltx_Math" display="inline" id="S2.SS2.p4.10.m9.1"><semantics id="S2.SS2.p4.10.m9.1a"><mi id="S2.SS2.p4.10.m9.1.1" xref="S2.SS2.p4.10.m9.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.10.m9.1b"><ci id="S2.SS2.p4.10.m9.1.1.cmml" xref="S2.SS2.p4.10.m9.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.10.m9.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.10.m9.1d">italic_N</annotation></semantics></math> samples resembles the result of a DFT of the data vector <span class="ltx_text" id="S2.SS2.p4.11.2"><math alttext="y_{l}=(y_{0},...,y_{N-1})^{T}" class="ltx_Math" display="inline" id="S2.SS2.p4.11.2.m1.3"><semantics id="S2.SS2.p4.11.2.m1.3a"><mrow id="S2.SS2.p4.11.2.m1.3.3" xref="S2.SS2.p4.11.2.m1.3.3.cmml"><msub id="S2.SS2.p4.11.2.m1.3.3.4" xref="S2.SS2.p4.11.2.m1.3.3.4.cmml"><mi id="S2.SS2.p4.11.2.m1.3.3.4.2" 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xref="S2.SS2.p4.11.2.m1.2.2.1.1.1.1.3">0</cn></apply><ci id="S2.SS2.p4.11.2.m1.1.1.cmml" xref="S2.SS2.p4.11.2.m1.1.1">…</ci><apply id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.1.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.2.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.2">𝑦</ci><apply id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3"><minus id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.1.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.1"></minus><ci id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.2.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.2">𝑁</ci><cn id="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS2.p4.11.2.m1.3.3.2.2.2.2.3.3">1</cn></apply></apply></vector><ci id="S2.SS2.p4.11.2.m1.3.3.2.4.cmml" xref="S2.SS2.p4.11.2.m1.3.3.2.4">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.11.2.m1.3c">y_{l}=(y_{0},...,y_{N-1})^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.11.2.m1.3d">italic_y start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT = ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math></span> by setting <math alttext="\Delta\omega_{j}=\frac{2\pi jn}{N}" class="ltx_Math" display="inline" id="S2.SS2.p4.12.m10.1"><semantics id="S2.SS2.p4.12.m10.1a"><mrow id="S2.SS2.p4.12.m10.1.1" xref="S2.SS2.p4.12.m10.1.1.cmml"><mrow id="S2.SS2.p4.12.m10.1.1.2" xref="S2.SS2.p4.12.m10.1.1.2.cmml"><mi id="S2.SS2.p4.12.m10.1.1.2.2" mathvariant="normal" xref="S2.SS2.p4.12.m10.1.1.2.2.cmml">Δ</mi><mo id="S2.SS2.p4.12.m10.1.1.2.1" xref="S2.SS2.p4.12.m10.1.1.2.1.cmml">⁢</mo><msub id="S2.SS2.p4.12.m10.1.1.2.3" xref="S2.SS2.p4.12.m10.1.1.2.3.cmml"><mi id="S2.SS2.p4.12.m10.1.1.2.3.2" xref="S2.SS2.p4.12.m10.1.1.2.3.2.cmml">ω</mi><mi id="S2.SS2.p4.12.m10.1.1.2.3.3" xref="S2.SS2.p4.12.m10.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S2.SS2.p4.12.m10.1.1.1" xref="S2.SS2.p4.12.m10.1.1.1.cmml">=</mo><mfrac id="S2.SS2.p4.12.m10.1.1.3" xref="S2.SS2.p4.12.m10.1.1.3.cmml"><mrow id="S2.SS2.p4.12.m10.1.1.3.2" xref="S2.SS2.p4.12.m10.1.1.3.2.cmml"><mn id="S2.SS2.p4.12.m10.1.1.3.2.2" xref="S2.SS2.p4.12.m10.1.1.3.2.2.cmml">2</mn><mo id="S2.SS2.p4.12.m10.1.1.3.2.1" xref="S2.SS2.p4.12.m10.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p4.12.m10.1.1.3.2.3" xref="S2.SS2.p4.12.m10.1.1.3.2.3.cmml">π</mi><mo id="S2.SS2.p4.12.m10.1.1.3.2.1a" xref="S2.SS2.p4.12.m10.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p4.12.m10.1.1.3.2.4" xref="S2.SS2.p4.12.m10.1.1.3.2.4.cmml">j</mi><mo id="S2.SS2.p4.12.m10.1.1.3.2.1b" xref="S2.SS2.p4.12.m10.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p4.12.m10.1.1.3.2.5" xref="S2.SS2.p4.12.m10.1.1.3.2.5.cmml">n</mi></mrow><mi id="S2.SS2.p4.12.m10.1.1.3.3" xref="S2.SS2.p4.12.m10.1.1.3.3.cmml">N</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.12.m10.1b"><apply id="S2.SS2.p4.12.m10.1.1.cmml" xref="S2.SS2.p4.12.m10.1.1"><eq id="S2.SS2.p4.12.m10.1.1.1.cmml" xref="S2.SS2.p4.12.m10.1.1.1"></eq><apply id="S2.SS2.p4.12.m10.1.1.2.cmml" xref="S2.SS2.p4.12.m10.1.1.2"><times id="S2.SS2.p4.12.m10.1.1.2.1.cmml" xref="S2.SS2.p4.12.m10.1.1.2.1"></times><ci id="S2.SS2.p4.12.m10.1.1.2.2.cmml" xref="S2.SS2.p4.12.m10.1.1.2.2">Δ</ci><apply id="S2.SS2.p4.12.m10.1.1.2.3.cmml" xref="S2.SS2.p4.12.m10.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p4.12.m10.1.1.2.3.1.cmml" xref="S2.SS2.p4.12.m10.1.1.2.3">subscript</csymbol><ci id="S2.SS2.p4.12.m10.1.1.2.3.2.cmml" xref="S2.SS2.p4.12.m10.1.1.2.3.2">𝜔</ci><ci id="S2.SS2.p4.12.m10.1.1.2.3.3.cmml" xref="S2.SS2.p4.12.m10.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S2.SS2.p4.12.m10.1.1.3.cmml" xref="S2.SS2.p4.12.m10.1.1.3"><divide id="S2.SS2.p4.12.m10.1.1.3.1.cmml" xref="S2.SS2.p4.12.m10.1.1.3"></divide><apply id="S2.SS2.p4.12.m10.1.1.3.2.cmml" xref="S2.SS2.p4.12.m10.1.1.3.2"><times id="S2.SS2.p4.12.m10.1.1.3.2.1.cmml" xref="S2.SS2.p4.12.m10.1.1.3.2.1"></times><cn id="S2.SS2.p4.12.m10.1.1.3.2.2.cmml" type="integer" xref="S2.SS2.p4.12.m10.1.1.3.2.2">2</cn><ci id="S2.SS2.p4.12.m10.1.1.3.2.3.cmml" xref="S2.SS2.p4.12.m10.1.1.3.2.3">𝜋</ci><ci id="S2.SS2.p4.12.m10.1.1.3.2.4.cmml" xref="S2.SS2.p4.12.m10.1.1.3.2.4">𝑗</ci><ci id="S2.SS2.p4.12.m10.1.1.3.2.5.cmml" xref="S2.SS2.p4.12.m10.1.1.3.2.5">𝑛</ci></apply><ci id="S2.SS2.p4.12.m10.1.1.3.3.cmml" xref="S2.SS2.p4.12.m10.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.12.m10.1c">\Delta\omega_{j}=\frac{2\pi jn}{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.12.m10.1d">roman_Δ italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 2 italic_π italic_j italic_n end_ARG start_ARG italic_N end_ARG</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx6"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle s_{jl}(T_{c})" class="ltx_Math" display="inline" id="S2.Ex1.m1.1"><semantics id="S2.Ex1.m1.1a"><mrow id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><msub id="S2.Ex1.m1.1.1.3" xref="S2.Ex1.m1.1.1.3.cmml"><mi id="S2.Ex1.m1.1.1.3.2" xref="S2.Ex1.m1.1.1.3.2.cmml">s</mi><mrow id="S2.Ex1.m1.1.1.3.3" xref="S2.Ex1.m1.1.1.3.3.cmml"><mi id="S2.Ex1.m1.1.1.3.3.2" xref="S2.Ex1.m1.1.1.3.3.2.cmml">j</mi><mo id="S2.Ex1.m1.1.1.3.3.1" xref="S2.Ex1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.Ex1.m1.1.1.3.3.3" xref="S2.Ex1.m1.1.1.3.3.3.cmml">l</mi></mrow></msub><mo id="S2.Ex1.m1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.2.cmml">T</mi><mi id="S2.Ex1.m1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S2.Ex1.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.1b"><apply id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1"><times id="S2.Ex1.m1.1.1.2.cmml" xref="S2.Ex1.m1.1.1.2"></times><apply id="S2.Ex1.m1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.3.1.cmml" xref="S2.Ex1.m1.1.1.3">subscript</csymbol><ci id="S2.Ex1.m1.1.1.3.2.cmml" xref="S2.Ex1.m1.1.1.3.2">𝑠</ci><apply id="S2.Ex1.m1.1.1.3.3.cmml" xref="S2.Ex1.m1.1.1.3.3"><times id="S2.Ex1.m1.1.1.3.3.1.cmml" xref="S2.Ex1.m1.1.1.3.3.1"></times><ci id="S2.Ex1.m1.1.1.3.3.2.cmml" xref="S2.Ex1.m1.1.1.3.3.2">𝑗</ci><ci id="S2.Ex1.m1.1.1.3.3.3.cmml" xref="S2.Ex1.m1.1.1.3.3.3">𝑙</ci></apply></apply><apply id="S2.Ex1.m1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.1.1.1.1">subscript</csymbol><ci id="S2.Ex1.m1.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.1.1.1.1.1.2">𝑇</ci><ci id="S2.Ex1.m1.1.1.1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.1c">\displaystyle s_{jl}(T_{c})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.1d">italic_s start_POSTSUBSCRIPT italic_j italic_l end_POSTSUBSCRIPT ( italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=e^{iN\Delta\omega_{j}}y_{0}+e^{i(N-1)\Delta\omega_{j}}y_{l,1}+..% .+e^{i\Delta\omega_{j}}y_{l,N-1}" class="ltx_Math" display="inline" id="S2.Ex1.m2.5"><semantics 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start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT italic_i ( italic_N - 1 ) roman_Δ italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_l , 1 end_POSTSUBSCRIPT + … + italic_e start_POSTSUPERSCRIPT italic_i roman_Δ italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_l , italic_N - 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{n=0}^{N-1}e^{-i2\pi jn/N}y_{l,n}." class="ltx_Math" display="inline" id="S2.E9.m1.3"><semantics id="S2.E9.m1.3a"><mrow 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xref="S2.E9.m1.3.3.1.1.3.2.3.2.cmml">y</mi><mrow id="S2.E9.m1.2.2.2.4" xref="S2.E9.m1.2.2.2.3.cmml"><mi id="S2.E9.m1.1.1.1.1" xref="S2.E9.m1.1.1.1.1.cmml">l</mi><mo id="S2.E9.m1.2.2.2.4.1" xref="S2.E9.m1.2.2.2.3.cmml">,</mo><mi id="S2.E9.m1.2.2.2.2" xref="S2.E9.m1.2.2.2.2.cmml">n</mi></mrow></msub></mrow></mrow></mrow><mo id="S2.E9.m1.3.3.1.2" lspace="0em" xref="S2.E9.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.3b"><apply id="S2.E9.m1.3.3.1.1.cmml" xref="S2.E9.m1.3.3.1"><eq id="S2.E9.m1.3.3.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.1"></eq><csymbol cd="latexml" id="S2.E9.m1.3.3.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.2">absent</csymbol><apply id="S2.E9.m1.3.3.1.1.3.cmml" xref="S2.E9.m1.3.3.1.1.3"><apply id="S2.E9.m1.3.3.1.1.3.1.cmml" xref="S2.E9.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S2.E9.m1.3.3.1.1.3.1.1.cmml" xref="S2.E9.m1.3.3.1.1.3.1">superscript</csymbol><apply id="S2.E9.m1.3.3.1.1.3.1.2.cmml" xref="S2.E9.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" 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xref="S2.E9.m1.3.3.1.1.3.2.2.3.2.2.4">𝜋</ci><ci id="S2.E9.m1.3.3.1.1.3.2.2.3.2.2.5.cmml" xref="S2.E9.m1.3.3.1.1.3.2.2.3.2.2.5">𝑗</ci><ci id="S2.E9.m1.3.3.1.1.3.2.2.3.2.2.6.cmml" xref="S2.E9.m1.3.3.1.1.3.2.2.3.2.2.6">𝑛</ci></apply><ci id="S2.E9.m1.3.3.1.1.3.2.2.3.2.3.cmml" xref="S2.E9.m1.3.3.1.1.3.2.2.3.2.3">𝑁</ci></apply></apply></apply><apply id="S2.E9.m1.3.3.1.1.3.2.3.cmml" xref="S2.E9.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.E9.m1.3.3.1.1.3.2.3.1.cmml" xref="S2.E9.m1.3.3.1.1.3.2.3">subscript</csymbol><ci id="S2.E9.m1.3.3.1.1.3.2.3.2.cmml" xref="S2.E9.m1.3.3.1.1.3.2.3.2">𝑦</ci><list id="S2.E9.m1.2.2.2.3.cmml" xref="S2.E9.m1.2.2.2.4"><ci id="S2.E9.m1.1.1.1.1.cmml" xref="S2.E9.m1.1.1.1.1">𝑙</ci><ci id="S2.E9.m1.2.2.2.2.cmml" xref="S2.E9.m1.2.2.2.2">𝑛</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.3c">\displaystyle=\sum_{n=0}^{N-1}e^{-i2\pi jn/N}y_{l,n}.</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.3d">= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i 2 italic_π italic_j italic_n / italic_N end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_l , italic_n end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.22">Each neuron <math alttext="s_{jl}" class="ltx_Math" display="inline" id="S2.SS2.p4.13.m1.1"><semantics id="S2.SS2.p4.13.m1.1a"><msub id="S2.SS2.p4.13.m1.1.1" xref="S2.SS2.p4.13.m1.1.1.cmml"><mi id="S2.SS2.p4.13.m1.1.1.2" xref="S2.SS2.p4.13.m1.1.1.2.cmml">s</mi><mrow id="S2.SS2.p4.13.m1.1.1.3" xref="S2.SS2.p4.13.m1.1.1.3.cmml"><mi id="S2.SS2.p4.13.m1.1.1.3.2" xref="S2.SS2.p4.13.m1.1.1.3.2.cmml">j</mi><mo id="S2.SS2.p4.13.m1.1.1.3.1" xref="S2.SS2.p4.13.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p4.13.m1.1.1.3.3" xref="S2.SS2.p4.13.m1.1.1.3.3.cmml">l</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.13.m1.1b"><apply id="S2.SS2.p4.13.m1.1.1.cmml" xref="S2.SS2.p4.13.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.13.m1.1.1.1.cmml" xref="S2.SS2.p4.13.m1.1.1">subscript</csymbol><ci id="S2.SS2.p4.13.m1.1.1.2.cmml" xref="S2.SS2.p4.13.m1.1.1.2">𝑠</ci><apply id="S2.SS2.p4.13.m1.1.1.3.cmml" xref="S2.SS2.p4.13.m1.1.1.3"><times id="S2.SS2.p4.13.m1.1.1.3.1.cmml" xref="S2.SS2.p4.13.m1.1.1.3.1"></times><ci id="S2.SS2.p4.13.m1.1.1.3.2.cmml" xref="S2.SS2.p4.13.m1.1.1.3.2">𝑗</ci><ci id="S2.SS2.p4.13.m1.1.1.3.3.cmml" xref="S2.SS2.p4.13.m1.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.13.m1.1c">s_{jl}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.13.m1.1d">italic_s start_POSTSUBSCRIPT italic_j italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is indexed by <math alttext="l" class="ltx_Math" display="inline" id="S2.SS2.p4.14.m2.1"><semantics id="S2.SS2.p4.14.m2.1a"><mi id="S2.SS2.p4.14.m2.1.1" xref="S2.SS2.p4.14.m2.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.14.m2.1b"><ci id="S2.SS2.p4.14.m2.1.1.cmml" xref="S2.SS2.p4.14.m2.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.14.m2.1c">l</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.14.m2.1d">italic_l</annotation></semantics></math> arising from the DFT along the antenna dimension. The phase shift <math alttext="\phi_{l}" class="ltx_Math" display="inline" id="S2.SS2.p4.15.m3.1"><semantics id="S2.SS2.p4.15.m3.1a"><msub id="S2.SS2.p4.15.m3.1.1" xref="S2.SS2.p4.15.m3.1.1.cmml"><mi id="S2.SS2.p4.15.m3.1.1.2" xref="S2.SS2.p4.15.m3.1.1.2.cmml">ϕ</mi><mi id="S2.SS2.p4.15.m3.1.1.3" xref="S2.SS2.p4.15.m3.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.15.m3.1b"><apply id="S2.SS2.p4.15.m3.1.1.cmml" xref="S2.SS2.p4.15.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.15.m3.1.1.1.cmml" xref="S2.SS2.p4.15.m3.1.1">subscript</csymbol><ci id="S2.SS2.p4.15.m3.1.1.2.cmml" xref="S2.SS2.p4.15.m3.1.1.2">italic-ϕ</ci><ci id="S2.SS2.p4.15.m3.1.1.3.cmml" xref="S2.SS2.p4.15.m3.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.15.m3.1c">\phi_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.15.m3.1d">italic_ϕ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> represents an angle <math alttext="\theta_{l}" class="ltx_Math" display="inline" id="S2.SS2.p4.16.m4.1"><semantics id="S2.SS2.p4.16.m4.1a"><msub id="S2.SS2.p4.16.m4.1.1" xref="S2.SS2.p4.16.m4.1.1.cmml"><mi id="S2.SS2.p4.16.m4.1.1.2" xref="S2.SS2.p4.16.m4.1.1.2.cmml">θ</mi><mi id="S2.SS2.p4.16.m4.1.1.3" xref="S2.SS2.p4.16.m4.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.16.m4.1b"><apply id="S2.SS2.p4.16.m4.1.1.cmml" xref="S2.SS2.p4.16.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.16.m4.1.1.1.cmml" xref="S2.SS2.p4.16.m4.1.1">subscript</csymbol><ci id="S2.SS2.p4.16.m4.1.1.2.cmml" xref="S2.SS2.p4.16.m4.1.1.2">𝜃</ci><ci id="S2.SS2.p4.16.m4.1.1.3.cmml" xref="S2.SS2.p4.16.m4.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.16.m4.1c">\theta_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.16.m4.1d">italic_θ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>. Each neuron is also indexed by <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p4.17.m5.1"><semantics id="S2.SS2.p4.17.m5.1a"><mi id="S2.SS2.p4.17.m5.1.1" xref="S2.SS2.p4.17.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.17.m5.1b"><ci id="S2.SS2.p4.17.m5.1.1.cmml" xref="S2.SS2.p4.17.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.17.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.17.m5.1d">italic_j</annotation></semantics></math> depending on the angular velocity <math alttext="\omega_{j}" class="ltx_Math" display="inline" id="S2.SS2.p4.18.m6.1"><semantics id="S2.SS2.p4.18.m6.1a"><msub id="S2.SS2.p4.18.m6.1.1" xref="S2.SS2.p4.18.m6.1.1.cmml"><mi id="S2.SS2.p4.18.m6.1.1.2" xref="S2.SS2.p4.18.m6.1.1.2.cmml">ω</mi><mi id="S2.SS2.p4.18.m6.1.1.3" xref="S2.SS2.p4.18.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.18.m6.1b"><apply id="S2.SS2.p4.18.m6.1.1.cmml" xref="S2.SS2.p4.18.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.18.m6.1.1.1.cmml" xref="S2.SS2.p4.18.m6.1.1">subscript</csymbol><ci id="S2.SS2.p4.18.m6.1.1.2.cmml" xref="S2.SS2.p4.18.m6.1.1.2">𝜔</ci><ci id="S2.SS2.p4.18.m6.1.1.3.cmml" xref="S2.SS2.p4.18.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.18.m6.1c">\omega_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.18.m6.1d">italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, which represents a range <math alttext="r_{j}" class="ltx_Math" display="inline" id="S2.SS2.p4.19.m7.1"><semantics id="S2.SS2.p4.19.m7.1a"><msub id="S2.SS2.p4.19.m7.1.1" xref="S2.SS2.p4.19.m7.1.1.cmml"><mi id="S2.SS2.p4.19.m7.1.1.2" xref="S2.SS2.p4.19.m7.1.1.2.cmml">r</mi><mi id="S2.SS2.p4.19.m7.1.1.3" xref="S2.SS2.p4.19.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.19.m7.1b"><apply id="S2.SS2.p4.19.m7.1.1.cmml" xref="S2.SS2.p4.19.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.19.m7.1.1.1.cmml" xref="S2.SS2.p4.19.m7.1.1">subscript</csymbol><ci id="S2.SS2.p4.19.m7.1.1.2.cmml" xref="S2.SS2.p4.19.m7.1.1.2">𝑟</ci><ci id="S2.SS2.p4.19.m7.1.1.3.cmml" xref="S2.SS2.p4.19.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.19.m7.1c">r_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.19.m7.1d">italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, each neuron is specific for a DoA angle <math alttext="\theta_{l}" class="ltx_Math" display="inline" id="S2.SS2.p4.20.m8.1"><semantics id="S2.SS2.p4.20.m8.1a"><msub id="S2.SS2.p4.20.m8.1.1" xref="S2.SS2.p4.20.m8.1.1.cmml"><mi id="S2.SS2.p4.20.m8.1.1.2" xref="S2.SS2.p4.20.m8.1.1.2.cmml">θ</mi><mi id="S2.SS2.p4.20.m8.1.1.3" xref="S2.SS2.p4.20.m8.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.20.m8.1b"><apply id="S2.SS2.p4.20.m8.1.1.cmml" xref="S2.SS2.p4.20.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.20.m8.1.1.1.cmml" xref="S2.SS2.p4.20.m8.1.1">subscript</csymbol><ci id="S2.SS2.p4.20.m8.1.1.2.cmml" xref="S2.SS2.p4.20.m8.1.1.2">𝜃</ci><ci id="S2.SS2.p4.20.m8.1.1.3.cmml" xref="S2.SS2.p4.20.m8.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.20.m8.1c">\theta_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.20.m8.1d">italic_θ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and a range <math alttext="r_{j}" class="ltx_Math" display="inline" id="S2.SS2.p4.21.m9.1"><semantics id="S2.SS2.p4.21.m9.1a"><msub id="S2.SS2.p4.21.m9.1.1" xref="S2.SS2.p4.21.m9.1.1.cmml"><mi id="S2.SS2.p4.21.m9.1.1.2" xref="S2.SS2.p4.21.m9.1.1.2.cmml">r</mi><mi id="S2.SS2.p4.21.m9.1.1.3" xref="S2.SS2.p4.21.m9.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.21.m9.1b"><apply id="S2.SS2.p4.21.m9.1.1.cmml" xref="S2.SS2.p4.21.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.21.m9.1.1.1.cmml" xref="S2.SS2.p4.21.m9.1.1">subscript</csymbol><ci id="S2.SS2.p4.21.m9.1.1.2.cmml" xref="S2.SS2.p4.21.m9.1.1.2">𝑟</ci><ci id="S2.SS2.p4.21.m9.1.1.3.cmml" xref="S2.SS2.p4.21.m9.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.21.m9.1c">r_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.21.m9.1d">italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F2" title="Figure 2 ‣ II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">2</span></a> visualizes the layout of the network architecture. As depicted in (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.Ex1" title="II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-B</span></span></a>), the state of the neuron at time <math alttext="T_{c}" class="ltx_Math" display="inline" id="S2.SS2.p4.22.m10.1"><semantics id="S2.SS2.p4.22.m10.1a"><msub id="S2.SS2.p4.22.m10.1.1" xref="S2.SS2.p4.22.m10.1.1.cmml"><mi id="S2.SS2.p4.22.m10.1.1.2" xref="S2.SS2.p4.22.m10.1.1.2.cmml">T</mi><mi id="S2.SS2.p4.22.m10.1.1.3" xref="S2.SS2.p4.22.m10.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.22.m10.1b"><apply id="S2.SS2.p4.22.m10.1.1.cmml" xref="S2.SS2.p4.22.m10.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.22.m10.1.1.1.cmml" xref="S2.SS2.p4.22.m10.1.1">subscript</csymbol><ci id="S2.SS2.p4.22.m10.1.1.2.cmml" xref="S2.SS2.p4.22.m10.1.1.2">𝑇</ci><ci id="S2.SS2.p4.22.m10.1.1.3.cmml" xref="S2.SS2.p4.22.m10.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.22.m10.1c">T_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.22.m10.1d">italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> represents the result of an FT. In the next section, we work out neuron dynamics and spiking functions that emit spike trains before this state is reached while maintaining the accuracy of the FT.</p> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="336" id="S2.F2.g1" src="x2.png" width="831"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span> Network of spiking neural resonators. Reflected signals are indicated as arrows, detected by virtual antennas. Each virtual antenna passes its IF signal <math alttext="x_{m}(t)" class="ltx_Math" display="inline" id="S2.F2.6.m1.1"><semantics id="S2.F2.6.m1.1b"><mrow id="S2.F2.6.m1.1.2" xref="S2.F2.6.m1.1.2.cmml"><msub id="S2.F2.6.m1.1.2.2" xref="S2.F2.6.m1.1.2.2.cmml"><mi id="S2.F2.6.m1.1.2.2.2" xref="S2.F2.6.m1.1.2.2.2.cmml">x</mi><mi id="S2.F2.6.m1.1.2.2.3" xref="S2.F2.6.m1.1.2.2.3.cmml">m</mi></msub><mo id="S2.F2.6.m1.1.2.1" xref="S2.F2.6.m1.1.2.1.cmml">⁢</mo><mrow id="S2.F2.6.m1.1.2.3.2" xref="S2.F2.6.m1.1.2.cmml"><mo id="S2.F2.6.m1.1.2.3.2.1" stretchy="false" xref="S2.F2.6.m1.1.2.cmml">(</mo><mi id="S2.F2.6.m1.1.1" xref="S2.F2.6.m1.1.1.cmml">t</mi><mo id="S2.F2.6.m1.1.2.3.2.2" stretchy="false" xref="S2.F2.6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.6.m1.1c"><apply id="S2.F2.6.m1.1.2.cmml" xref="S2.F2.6.m1.1.2"><times id="S2.F2.6.m1.1.2.1.cmml" xref="S2.F2.6.m1.1.2.1"></times><apply id="S2.F2.6.m1.1.2.2.cmml" xref="S2.F2.6.m1.1.2.2"><csymbol cd="ambiguous" id="S2.F2.6.m1.1.2.2.1.cmml" xref="S2.F2.6.m1.1.2.2">subscript</csymbol><ci id="S2.F2.6.m1.1.2.2.2.cmml" xref="S2.F2.6.m1.1.2.2.2">𝑥</ci><ci id="S2.F2.6.m1.1.2.2.3.cmml" xref="S2.F2.6.m1.1.2.2.3">𝑚</ci></apply><ci id="S2.F2.6.m1.1.1.cmml" xref="S2.F2.6.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.6.m1.1d">x_{m}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.F2.6.m1.1e">italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> to a neural resonator with given eigenfrequency <math alttext="\omega_{j}" class="ltx_Math" display="inline" id="S2.F2.7.m2.1"><semantics id="S2.F2.7.m2.1b"><msub id="S2.F2.7.m2.1.1" xref="S2.F2.7.m2.1.1.cmml"><mi id="S2.F2.7.m2.1.1.2" xref="S2.F2.7.m2.1.1.2.cmml">ω</mi><mi id="S2.F2.7.m2.1.1.3" xref="S2.F2.7.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.F2.7.m2.1c"><apply id="S2.F2.7.m2.1.1.cmml" xref="S2.F2.7.m2.1.1"><csymbol cd="ambiguous" id="S2.F2.7.m2.1.1.1.cmml" xref="S2.F2.7.m2.1.1">subscript</csymbol><ci id="S2.F2.7.m2.1.1.2.cmml" xref="S2.F2.7.m2.1.1.2">𝜔</ci><ci id="S2.F2.7.m2.1.1.3.cmml" xref="S2.F2.7.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.7.m2.1d">\omega_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.7.m2.1e">italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> weighted by the complex weight vector <math alttext="\vec{w}_{l}" class="ltx_Math" display="inline" id="S2.F2.8.m3.1"><semantics id="S2.F2.8.m3.1b"><msub id="S2.F2.8.m3.1.1" xref="S2.F2.8.m3.1.1.cmml"><mover accent="true" id="S2.F2.8.m3.1.1.2" xref="S2.F2.8.m3.1.1.2.cmml"><mi id="S2.F2.8.m3.1.1.2.2" xref="S2.F2.8.m3.1.1.2.2.cmml">w</mi><mo id="S2.F2.8.m3.1.1.2.1" stretchy="false" xref="S2.F2.8.m3.1.1.2.1.cmml">→</mo></mover><mi id="S2.F2.8.m3.1.1.3" xref="S2.F2.8.m3.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.F2.8.m3.1c"><apply id="S2.F2.8.m3.1.1.cmml" xref="S2.F2.8.m3.1.1"><csymbol cd="ambiguous" id="S2.F2.8.m3.1.1.1.cmml" xref="S2.F2.8.m3.1.1">subscript</csymbol><apply id="S2.F2.8.m3.1.1.2.cmml" xref="S2.F2.8.m3.1.1.2"><ci id="S2.F2.8.m3.1.1.2.1.cmml" xref="S2.F2.8.m3.1.1.2.1">→</ci><ci id="S2.F2.8.m3.1.1.2.2.cmml" xref="S2.F2.8.m3.1.1.2.2">𝑤</ci></apply><ci id="S2.F2.8.m3.1.1.3.cmml" xref="S2.F2.8.m3.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.8.m3.1d">\vec{w}_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.8.m3.1e">over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>. Along the vertical dimension, complex weight vectors optimized for a specific angle <math alttext="\theta" class="ltx_Math" display="inline" id="S2.F2.9.m4.1"><semantics id="S2.F2.9.m4.1b"><mi id="S2.F2.9.m4.1.1" xref="S2.F2.9.m4.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.F2.9.m4.1c"><ci id="S2.F2.9.m4.1.1.cmml" xref="S2.F2.9.m4.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.9.m4.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.F2.9.m4.1e">italic_θ</annotation></semantics></math> are visualized (gray, black). Along the horizontal dimension, the eigenfrequency of the neuron changes optimized for a specific range <math alttext="r" class="ltx_Math" display="inline" id="S2.F2.10.m5.1"><semantics id="S2.F2.10.m5.1b"><mi id="S2.F2.10.m5.1.1" xref="S2.F2.10.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.F2.10.m5.1c"><ci id="S2.F2.10.m5.1.1.cmml" xref="S2.F2.10.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.10.m5.1d">r</annotation><annotation encoding="application/x-llamapun" id="S2.F2.10.m5.1e">italic_r</annotation></semantics></math>. The transmitted spikes can be visualized as range-angle map. </figcaption> </figure> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS3.5.1.1">II-C</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS3.6.2">Envelope estimation and gradient estimation</span> </h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.3">To estimate the angle and range of an object, we need to detect neurons that match the phase (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E5" title="In II-A Angle estimation - Dendritic vector multiplication ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">5</span></a>) and the frequency (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E7" title="In II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">7</span></a>) of an object. 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xref="S2.SS3.p1.1.m1.1.1.5">much-less-than</csymbol><share href="https://arxiv.org/html/2503.00898v1#S2.SS3.p1.1.m1.1.1.4.cmml" id="S2.SS3.p1.1.m1.1.1d.cmml" xref="S2.SS3.p1.1.m1.1.1"></share><cn id="S2.SS3.p1.1.m1.1.1.6.cmml" type="integer" xref="S2.SS3.p1.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">\Delta\omega_{jk}=\omega_{j}-\omega_{k}\ll 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">roman_Δ italic_ω start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≪ 1</annotation></semantics></math> of the neuron state, and we apply Taylor-approximation on the magnitude of (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E7" title="In II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">7</span></a>) for a single target <math alttext="k" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.1"><semantics id="S2.SS3.p1.2.m2.1a"><mi id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.1b"><ci id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.1d">italic_k</annotation></semantics></math> around <math alttext="\Delta\omega_{jk}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m3.1"><semantics id="S2.SS3.p1.3.m3.1a"><mrow id="S2.SS3.p1.3.m3.1.1" xref="S2.SS3.p1.3.m3.1.1.cmml"><mi id="S2.SS3.p1.3.m3.1.1.2" mathvariant="normal" xref="S2.SS3.p1.3.m3.1.1.2.cmml">Δ</mi><mo id="S2.SS3.p1.3.m3.1.1.1" xref="S2.SS3.p1.3.m3.1.1.1.cmml">⁢</mo><msub 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id="S2.SS3.p1.3.m3.1.1.3.3.cmml" xref="S2.SS3.p1.3.m3.1.1.3.3"><times id="S2.SS3.p1.3.m3.1.1.3.3.1.cmml" xref="S2.SS3.p1.3.m3.1.1.3.3.1"></times><ci id="S2.SS3.p1.3.m3.1.1.3.3.2.cmml" xref="S2.SS3.p1.3.m3.1.1.3.3.2">𝑗</ci><ci id="S2.SS3.p1.3.m3.1.1.3.3.3.cmml" xref="S2.SS3.p1.3.m3.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m3.1c">\Delta\omega_{jk}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">roman_Δ italic_ω start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx7"> <tbody id="S2.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|s_{jl}(t)|\simeq\beta_{kl}t^{1}+\mathcal{O}(t^{2})." class="ltx_Math" display="inline" 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id="S2.E10.m1.2.2.1.1.2.1.cmml" xref="S2.E10.m1.2.2.1.1.2.1"><times id="S2.E10.m1.2.2.1.1.2.1.2.cmml" xref="S2.E10.m1.2.2.1.1.2.1.2"></times><ci id="S2.E10.m1.2.2.1.1.2.1.3.cmml" xref="S2.E10.m1.2.2.1.1.2.1.3">𝒪</ci><apply id="S2.E10.m1.2.2.1.1.2.1.1.1.1.cmml" xref="S2.E10.m1.2.2.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S2.E10.m1.2.2.1.1.2.1.1.1.1.1.cmml" xref="S2.E10.m1.2.2.1.1.2.1.1.1">superscript</csymbol><ci id="S2.E10.m1.2.2.1.1.2.1.1.1.1.2.cmml" xref="S2.E10.m1.2.2.1.1.2.1.1.1.1.2">𝑡</ci><cn id="S2.E10.m1.2.2.1.1.2.1.1.1.1.3.cmml" type="integer" xref="S2.E10.m1.2.2.1.1.2.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.2c">\displaystyle|s_{jl}(t)|\simeq\beta_{kl}t^{1}+\mathcal{O}(t^{2}).</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.2d">| italic_s start_POSTSUBSCRIPT italic_j italic_l end_POSTSUBSCRIPT ( italic_t ) | ≃ italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT + caligraphic_O ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.6">A matching phase of an object <math alttext="k" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m1.1"><semantics id="S2.SS3.p1.4.m1.1a"><mi id="S2.SS3.p1.4.m1.1.1" xref="S2.SS3.p1.4.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m1.1b"><ci id="S2.SS3.p1.4.m1.1.1.cmml" xref="S2.SS3.p1.4.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m1.1d">italic_k</annotation></semantics></math> leads also to an increased amplitude <math alttext="\beta_{kl}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m2.1"><semantics id="S2.SS3.p1.5.m2.1a"><msub id="S2.SS3.p1.5.m2.1.1" xref="S2.SS3.p1.5.m2.1.1.cmml"><mi id="S2.SS3.p1.5.m2.1.1.2" xref="S2.SS3.p1.5.m2.1.1.2.cmml">β</mi><mrow id="S2.SS3.p1.5.m2.1.1.3" xref="S2.SS3.p1.5.m2.1.1.3.cmml"><mi id="S2.SS3.p1.5.m2.1.1.3.2" xref="S2.SS3.p1.5.m2.1.1.3.2.cmml">k</mi><mo id="S2.SS3.p1.5.m2.1.1.3.1" xref="S2.SS3.p1.5.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.5.m2.1.1.3.3" xref="S2.SS3.p1.5.m2.1.1.3.3.cmml">l</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m2.1b"><apply id="S2.SS3.p1.5.m2.1.1.cmml" xref="S2.SS3.p1.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.5.m2.1.1.1.cmml" xref="S2.SS3.p1.5.m2.1.1">subscript</csymbol><ci id="S2.SS3.p1.5.m2.1.1.2.cmml" xref="S2.SS3.p1.5.m2.1.1.2">𝛽</ci><apply id="S2.SS3.p1.5.m2.1.1.3.cmml" xref="S2.SS3.p1.5.m2.1.1.3"><times id="S2.SS3.p1.5.m2.1.1.3.1.cmml" xref="S2.SS3.p1.5.m2.1.1.3.1"></times><ci id="S2.SS3.p1.5.m2.1.1.3.2.cmml" xref="S2.SS3.p1.5.m2.1.1.3.2">𝑘</ci><ci id="S2.SS3.p1.5.m2.1.1.3.3.cmml" xref="S2.SS3.p1.5.m2.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m2.1c">\beta_{kl}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m2.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT</annotation></semantics></math> of the neuron state (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E7" title="In II-B Distance estimation - Neural resonators ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">7</span></a>), i.e. <math alttext="\beta_{kl}\approx 1" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m3.1"><semantics id="S2.SS3.p1.6.m3.1a"><mrow id="S2.SS3.p1.6.m3.1.1" xref="S2.SS3.p1.6.m3.1.1.cmml"><msub id="S2.SS3.p1.6.m3.1.1.2" xref="S2.SS3.p1.6.m3.1.1.2.cmml"><mi id="S2.SS3.p1.6.m3.1.1.2.2" xref="S2.SS3.p1.6.m3.1.1.2.2.cmml">β</mi><mrow id="S2.SS3.p1.6.m3.1.1.2.3" xref="S2.SS3.p1.6.m3.1.1.2.3.cmml"><mi id="S2.SS3.p1.6.m3.1.1.2.3.2" xref="S2.SS3.p1.6.m3.1.1.2.3.2.cmml">k</mi><mo id="S2.SS3.p1.6.m3.1.1.2.3.1" xref="S2.SS3.p1.6.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.6.m3.1.1.2.3.3" xref="S2.SS3.p1.6.m3.1.1.2.3.3.cmml">l</mi></mrow></msub><mo id="S2.SS3.p1.6.m3.1.1.1" xref="S2.SS3.p1.6.m3.1.1.1.cmml">≈</mo><mn id="S2.SS3.p1.6.m3.1.1.3" xref="S2.SS3.p1.6.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.6.m3.1b"><apply id="S2.SS3.p1.6.m3.1.1.cmml" xref="S2.SS3.p1.6.m3.1.1"><approx id="S2.SS3.p1.6.m3.1.1.1.cmml" xref="S2.SS3.p1.6.m3.1.1.1"></approx><apply id="S2.SS3.p1.6.m3.1.1.2.cmml" xref="S2.SS3.p1.6.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p1.6.m3.1.1.2.1.cmml" xref="S2.SS3.p1.6.m3.1.1.2">subscript</csymbol><ci id="S2.SS3.p1.6.m3.1.1.2.2.cmml" xref="S2.SS3.p1.6.m3.1.1.2.2">𝛽</ci><apply id="S2.SS3.p1.6.m3.1.1.2.3.cmml" xref="S2.SS3.p1.6.m3.1.1.2.3"><times id="S2.SS3.p1.6.m3.1.1.2.3.1.cmml" xref="S2.SS3.p1.6.m3.1.1.2.3.1"></times><ci id="S2.SS3.p1.6.m3.1.1.2.3.2.cmml" xref="S2.SS3.p1.6.m3.1.1.2.3.2">𝑘</ci><ci id="S2.SS3.p1.6.m3.1.1.2.3.3.cmml" xref="S2.SS3.p1.6.m3.1.1.2.3.3">𝑙</ci></apply></apply><cn id="S2.SS3.p1.6.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p1.6.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.6.m3.1c">\beta_{kl}\approx 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.6.m3.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ≈ 1</annotation></semantics></math>. Therefore, the first-order term of the magnitude of the neuron state stores the information of interest. We assume that the remaining parts of the sum (non-resonant) affect only the offset and higher-order oscillations.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.3">Ultimately, two criteria distinguish a neuron that matches the phase and frequency of an object and a non-matching neuron: (a) the magnitude <math alttext="|s_{j}l(t)|" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.2"><semantics id="S2.SS3.p2.1.m1.2a"><mrow id="S2.SS3.p2.1.m1.2.2.1" xref="S2.SS3.p2.1.m1.2.2.2.cmml"><mo id="S2.SS3.p2.1.m1.2.2.1.2" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.2.1.cmml">|</mo><mrow id="S2.SS3.p2.1.m1.2.2.1.1" xref="S2.SS3.p2.1.m1.2.2.1.1.cmml"><msub id="S2.SS3.p2.1.m1.2.2.1.1.2" xref="S2.SS3.p2.1.m1.2.2.1.1.2.cmml"><mi id="S2.SS3.p2.1.m1.2.2.1.1.2.2" xref="S2.SS3.p2.1.m1.2.2.1.1.2.2.cmml">s</mi><mi id="S2.SS3.p2.1.m1.2.2.1.1.2.3" xref="S2.SS3.p2.1.m1.2.2.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS3.p2.1.m1.2.2.1.1.1" xref="S2.SS3.p2.1.m1.2.2.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.1.m1.2.2.1.1.3" xref="S2.SS3.p2.1.m1.2.2.1.1.3.cmml">l</mi><mo id="S2.SS3.p2.1.m1.2.2.1.1.1a" xref="S2.SS3.p2.1.m1.2.2.1.1.1.cmml">⁢</mo><mrow id="S2.SS3.p2.1.m1.2.2.1.1.4.2" xref="S2.SS3.p2.1.m1.2.2.1.1.cmml"><mo id="S2.SS3.p2.1.m1.2.2.1.1.4.2.1" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.1.1.cmml">(</mo><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">t</mi><mo id="S2.SS3.p2.1.m1.2.2.1.1.4.2.2" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p2.1.m1.2.2.1.3" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.2b"><apply id="S2.SS3.p2.1.m1.2.2.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1"><abs id="S2.SS3.p2.1.m1.2.2.2.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.2"></abs><apply id="S2.SS3.p2.1.m1.2.2.1.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1"><times id="S2.SS3.p2.1.m1.2.2.1.1.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1"></times><apply id="S2.SS3.p2.1.m1.2.2.1.1.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.1.m1.2.2.1.1.2.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.1.m1.2.2.1.1.2.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.2.2">𝑠</ci><ci id="S2.SS3.p2.1.m1.2.2.1.1.2.3.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.2.3">𝑗</ci></apply><ci id="S2.SS3.p2.1.m1.2.2.1.1.3.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.3">𝑙</ci><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.2c">|s_{j}l(t)|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.2d">| italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_l ( italic_t ) |</annotation></semantics></math> follows a linearly increasing function <math alttext="\beta_{kl}\cdot t" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mrow id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml"><msub id="S2.SS3.p2.2.m2.1.1.2" xref="S2.SS3.p2.2.m2.1.1.2.cmml"><mi id="S2.SS3.p2.2.m2.1.1.2.2" xref="S2.SS3.p2.2.m2.1.1.2.2.cmml">β</mi><mrow id="S2.SS3.p2.2.m2.1.1.2.3" xref="S2.SS3.p2.2.m2.1.1.2.3.cmml"><mi id="S2.SS3.p2.2.m2.1.1.2.3.2" xref="S2.SS3.p2.2.m2.1.1.2.3.2.cmml">k</mi><mo id="S2.SS3.p2.2.m2.1.1.2.3.1" xref="S2.SS3.p2.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.2.m2.1.1.2.3.3" xref="S2.SS3.p2.2.m2.1.1.2.3.3.cmml">l</mi></mrow></msub><mo id="S2.SS3.p2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p2.2.m2.1.1.1.cmml">⋅</mo><mi id="S2.SS3.p2.2.m2.1.1.3" xref="S2.SS3.p2.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><apply id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1"><ci id="S2.SS3.p2.2.m2.1.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1.1">⋅</ci><apply id="S2.SS3.p2.2.m2.1.1.2.cmml" xref="S2.SS3.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.2.m2.1.1.2.1.cmml" xref="S2.SS3.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.2.m2.1.1.2.2.cmml" xref="S2.SS3.p2.2.m2.1.1.2.2">𝛽</ci><apply id="S2.SS3.p2.2.m2.1.1.2.3.cmml" xref="S2.SS3.p2.2.m2.1.1.2.3"><times id="S2.SS3.p2.2.m2.1.1.2.3.1.cmml" xref="S2.SS3.p2.2.m2.1.1.2.3.1"></times><ci id="S2.SS3.p2.2.m2.1.1.2.3.2.cmml" xref="S2.SS3.p2.2.m2.1.1.2.3.2">𝑘</ci><ci id="S2.SS3.p2.2.m2.1.1.2.3.3.cmml" xref="S2.SS3.p2.2.m2.1.1.2.3.3">𝑙</ci></apply></apply><ci id="S2.SS3.p2.2.m2.1.1.3.cmml" xref="S2.SS3.p2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">\beta_{kl}\cdot t</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ⋅ italic_t</annotation></semantics></math> over time and (b) the gradient of the linear function is large, <math alttext="\beta_{kl}\gg 1" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mrow id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><msub id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2.2" xref="S2.SS3.p2.3.m3.1.1.2.2.cmml">β</mi><mrow id="S2.SS3.p2.3.m3.1.1.2.3" xref="S2.SS3.p2.3.m3.1.1.2.3.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2.3.2" xref="S2.SS3.p2.3.m3.1.1.2.3.2.cmml">k</mi><mo id="S2.SS3.p2.3.m3.1.1.2.3.1" xref="S2.SS3.p2.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.3.m3.1.1.2.3.3" xref="S2.SS3.p2.3.m3.1.1.2.3.3.cmml">l</mi></mrow></msub><mo id="S2.SS3.p2.3.m3.1.1.1" xref="S2.SS3.p2.3.m3.1.1.1.cmml">≫</mo><mn id="S2.SS3.p2.3.m3.1.1.3" xref="S2.SS3.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><csymbol cd="latexml" id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1.1">much-greater-than</csymbol><apply id="S2.SS3.p2.3.m3.1.1.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.3.m3.1.1.2.1.cmml" xref="S2.SS3.p2.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.3.m3.1.1.2.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2.2">𝛽</ci><apply id="S2.SS3.p2.3.m3.1.1.2.3.cmml" xref="S2.SS3.p2.3.m3.1.1.2.3"><times id="S2.SS3.p2.3.m3.1.1.2.3.1.cmml" xref="S2.SS3.p2.3.m3.1.1.2.3.1"></times><ci id="S2.SS3.p2.3.m3.1.1.2.3.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2.3.2">𝑘</ci><ci id="S2.SS3.p2.3.m3.1.1.2.3.3.cmml" xref="S2.SS3.p2.3.m3.1.1.2.3.3">𝑙</ci></apply></apply><cn id="S2.SS3.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">\beta_{kl}\gg 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_β start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ≫ 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.2">Applying this knowledge, we develop a neuron that estimates an envelope <math alttext="\Lambda(t)" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.1"><semantics id="S2.SS3.p3.1.m1.1a"><mrow id="S2.SS3.p3.1.m1.1.2" xref="S2.SS3.p3.1.m1.1.2.cmml"><mi id="S2.SS3.p3.1.m1.1.2.2" mathvariant="normal" xref="S2.SS3.p3.1.m1.1.2.2.cmml">Λ</mi><mo id="S2.SS3.p3.1.m1.1.2.1" xref="S2.SS3.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.1.m1.1.2.3.2" xref="S2.SS3.p3.1.m1.1.2.cmml"><mo id="S2.SS3.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.p3.1.m1.1.1" xref="S2.SS3.p3.1.m1.1.1.cmml">t</mi><mo id="S2.SS3.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.1b"><apply id="S2.SS3.p3.1.m1.1.2.cmml" xref="S2.SS3.p3.1.m1.1.2"><times id="S2.SS3.p3.1.m1.1.2.1.cmml" xref="S2.SS3.p3.1.m1.1.2.1"></times><ci id="S2.SS3.p3.1.m1.1.2.2.cmml" xref="S2.SS3.p3.1.m1.1.2.2">Λ</ci><ci id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.1c">\Lambda(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.1d">roman_Λ ( italic_t )</annotation></semantics></math> of the magnitude of the neuron state by removing uninformative high-order oscillations. We start by estimating the maximum of the neuron’s magnitude <math alttext="s_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><mrow id="S2.SS3.p3.2.m2.1.2" xref="S2.SS3.p3.2.m2.1.2.cmml"><msub id="S2.SS3.p3.2.m2.1.2.2" xref="S2.SS3.p3.2.m2.1.2.2.cmml"><mi id="S2.SS3.p3.2.m2.1.2.2.2" xref="S2.SS3.p3.2.m2.1.2.2.2.cmml">s</mi><mtext id="S2.SS3.p3.2.m2.1.2.2.3" xref="S2.SS3.p3.2.m2.1.2.2.3a.cmml">max</mtext></msub><mo id="S2.SS3.p3.2.m2.1.2.1" xref="S2.SS3.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.2.m2.1.2.3.2" xref="S2.SS3.p3.2.m2.1.2.cmml"><mo id="S2.SS3.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml">t</mi><mo id="S2.SS3.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.1b"><apply id="S2.SS3.p3.2.m2.1.2.cmml" xref="S2.SS3.p3.2.m2.1.2"><times 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class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle s_{\text{max}}(t)=\begin{cases}\|s(t)\|,&\text{ if }\|s(t)\|>s_{% \text{max}}(t)\\ s_{\text{max}}(t),&\text{ else.}\end{cases}" class="ltx_Math" display="inline" id="S2.E11.m1.5"><semantics id="S2.E11.m1.5a"><mrow id="S2.E11.m1.5.6" xref="S2.E11.m1.5.6.cmml"><mrow id="S2.E11.m1.5.6.2" xref="S2.E11.m1.5.6.2.cmml"><msub id="S2.E11.m1.5.6.2.2" xref="S2.E11.m1.5.6.2.2.cmml"><mi id="S2.E11.m1.5.6.2.2.2" xref="S2.E11.m1.5.6.2.2.2.cmml">s</mi><mtext id="S2.E11.m1.5.6.2.2.3" xref="S2.E11.m1.5.6.2.2.3a.cmml">max</mtext></msub><mo id="S2.E11.m1.5.6.2.1" xref="S2.E11.m1.5.6.2.1.cmml">⁢</mo><mrow id="S2.E11.m1.5.6.2.3.2" xref="S2.E11.m1.5.6.2.cmml"><mo id="S2.E11.m1.5.6.2.3.2.1" stretchy="false" xref="S2.E11.m1.5.6.2.cmml">(</mo><mi id="S2.E11.m1.5.5" xref="S2.E11.m1.5.5.cmml">t</mi><mo id="S2.E11.m1.5.6.2.3.2.2" stretchy="false" 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end_CELL start_CELL else. end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.2">By estimating only the upper boundary, we neglect any decays or oscillations in the magnitude, which indicate non-resonating behavior. Therefore, we also estimate the maximum width <math alttext="w_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.1"><semantics id="S2.SS3.p4.1.m1.1a"><mrow id="S2.SS3.p4.1.m1.1.2" xref="S2.SS3.p4.1.m1.1.2.cmml"><msub id="S2.SS3.p4.1.m1.1.2.2" xref="S2.SS3.p4.1.m1.1.2.2.cmml"><mi id="S2.SS3.p4.1.m1.1.2.2.2" xref="S2.SS3.p4.1.m1.1.2.2.2.cmml">w</mi><mtext id="S2.SS3.p4.1.m1.1.2.2.3" xref="S2.SS3.p4.1.m1.1.2.2.3a.cmml">max</mtext></msub><mo id="S2.SS3.p4.1.m1.1.2.1" xref="S2.SS3.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p4.1.m1.1.2.3.2" xref="S2.SS3.p4.1.m1.1.2.cmml"><mo id="S2.SS3.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p4.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml">t</mi><mo id="S2.SS3.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.1.m1.1b"><apply id="S2.SS3.p4.1.m1.1.2.cmml" xref="S2.SS3.p4.1.m1.1.2"><times id="S2.SS3.p4.1.m1.1.2.1.cmml" xref="S2.SS3.p4.1.m1.1.2.1"></times><apply id="S2.SS3.p4.1.m1.1.2.2.cmml" xref="S2.SS3.p4.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p4.1.m1.1.2.2.1.cmml" xref="S2.SS3.p4.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS3.p4.1.m1.1.2.2.2.cmml" xref="S2.SS3.p4.1.m1.1.2.2.2">𝑤</ci><ci id="S2.SS3.p4.1.m1.1.2.2.3a.cmml" xref="S2.SS3.p4.1.m1.1.2.2.3"><mtext id="S2.SS3.p4.1.m1.1.2.2.3.cmml" mathsize="70%" xref="S2.SS3.p4.1.m1.1.2.2.3">max</mtext></ci></apply><ci id="S2.SS3.p4.1.m1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.1.m1.1c">w_{\text{max}}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.1.m1.1d">italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> between the neuron’s magnitude and the upper boundary to extract the additional information on non-resonance. The maximum of the width <math alttext="w_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p4.2.m2.1"><semantics id="S2.SS3.p4.2.m2.1a"><mrow id="S2.SS3.p4.2.m2.1.2" xref="S2.SS3.p4.2.m2.1.2.cmml"><msub id="S2.SS3.p4.2.m2.1.2.2" xref="S2.SS3.p4.2.m2.1.2.2.cmml"><mi id="S2.SS3.p4.2.m2.1.2.2.2" xref="S2.SS3.p4.2.m2.1.2.2.2.cmml">w</mi><mtext id="S2.SS3.p4.2.m2.1.2.2.3" xref="S2.SS3.p4.2.m2.1.2.2.3a.cmml">max</mtext></msub><mo id="S2.SS3.p4.2.m2.1.2.1" xref="S2.SS3.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p4.2.m2.1.2.3.2" xref="S2.SS3.p4.2.m2.1.2.cmml"><mo id="S2.SS3.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.p4.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.p4.2.m2.1.1" xref="S2.SS3.p4.2.m2.1.1.cmml">t</mi><mo id="S2.SS3.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.2.m2.1b"><apply id="S2.SS3.p4.2.m2.1.2.cmml" xref="S2.SS3.p4.2.m2.1.2"><times id="S2.SS3.p4.2.m2.1.2.1.cmml" 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end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p4.3">The difference <math alttext="\Lambda(t)=s_{\text{max}}(t)-w_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p4.3.m1.3"><semantics id="S2.SS3.p4.3.m1.3a"><mrow id="S2.SS3.p4.3.m1.3.4" xref="S2.SS3.p4.3.m1.3.4.cmml"><mrow id="S2.SS3.p4.3.m1.3.4.2" xref="S2.SS3.p4.3.m1.3.4.2.cmml"><mi id="S2.SS3.p4.3.m1.3.4.2.2" mathvariant="normal" xref="S2.SS3.p4.3.m1.3.4.2.2.cmml">Λ</mi><mo id="S2.SS3.p4.3.m1.3.4.2.1" xref="S2.SS3.p4.3.m1.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS3.p4.3.m1.3.4.2.3.2" xref="S2.SS3.p4.3.m1.3.4.2.cmml"><mo id="S2.SS3.p4.3.m1.3.4.2.3.2.1" stretchy="false" xref="S2.SS3.p4.3.m1.3.4.2.cmml">(</mo><mi id="S2.SS3.p4.3.m1.1.1" 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id="S2.SS3.p4.3.m1.3c">\Lambda(t)=s_{\text{max}}(t)-w_{\text{max}}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.3.m1.3d">roman_Λ ( italic_t ) = italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t ) - italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> follows the neuron’s magnitude as a lower boundary and replicates an envelope estimate removing higher-order oscillations.</p> </div> <div class="ltx_para" id="S2.SS3.p5"> <p class="ltx_p" id="S2.SS3.p5.1">We estimate the gradient of <math alttext="\Lambda(t)" class="ltx_Math" display="inline" id="S2.SS3.p5.1.m1.1"><semantics id="S2.SS3.p5.1.m1.1a"><mrow id="S2.SS3.p5.1.m1.1.2" xref="S2.SS3.p5.1.m1.1.2.cmml"><mi id="S2.SS3.p5.1.m1.1.2.2" mathvariant="normal" xref="S2.SS3.p5.1.m1.1.2.2.cmml">Λ</mi><mo id="S2.SS3.p5.1.m1.1.2.1" xref="S2.SS3.p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.1.m1.1.2.3.2" xref="S2.SS3.p5.1.m1.1.2.cmml"><mo 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rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p5.5"><math alttext="\Delta\Lambda" class="ltx_Math" display="inline" id="S2.SS3.p5.2.m1.1"><semantics id="S2.SS3.p5.2.m1.1a"><mrow id="S2.SS3.p5.2.m1.1.1" xref="S2.SS3.p5.2.m1.1.1.cmml"><mi id="S2.SS3.p5.2.m1.1.1.2" mathvariant="normal" xref="S2.SS3.p5.2.m1.1.1.2.cmml">Δ</mi><mo id="S2.SS3.p5.2.m1.1.1.1" xref="S2.SS3.p5.2.m1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p5.2.m1.1.1.3" mathvariant="normal" xref="S2.SS3.p5.2.m1.1.1.3.cmml">Λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.2.m1.1b"><apply id="S2.SS3.p5.2.m1.1.1.cmml" xref="S2.SS3.p5.2.m1.1.1"><times id="S2.SS3.p5.2.m1.1.1.1.cmml" xref="S2.SS3.p5.2.m1.1.1.1"></times><ci id="S2.SS3.p5.2.m1.1.1.2.cmml" xref="S2.SS3.p5.2.m1.1.1.2">Δ</ci><ci id="S2.SS3.p5.2.m1.1.1.3.cmml" xref="S2.SS3.p5.2.m1.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.2.m1.1c">\Delta\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.2.m1.1d">roman_Δ roman_Λ</annotation></semantics></math> being the update step for the envelope which depends on the updates in <math alttext="s_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p5.3.m2.1"><semantics id="S2.SS3.p5.3.m2.1a"><mrow id="S2.SS3.p5.3.m2.1.2" xref="S2.SS3.p5.3.m2.1.2.cmml"><msub id="S2.SS3.p5.3.m2.1.2.2" xref="S2.SS3.p5.3.m2.1.2.2.cmml"><mi id="S2.SS3.p5.3.m2.1.2.2.2" xref="S2.SS3.p5.3.m2.1.2.2.2.cmml">s</mi><mtext id="S2.SS3.p5.3.m2.1.2.2.3" xref="S2.SS3.p5.3.m2.1.2.2.3a.cmml">max</mtext></msub><mo id="S2.SS3.p5.3.m2.1.2.1" xref="S2.SS3.p5.3.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.3.m2.1.2.3.2" xref="S2.SS3.p5.3.m2.1.2.cmml"><mo id="S2.SS3.p5.3.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.p5.3.m2.1.2.cmml">(</mo><mi id="S2.SS3.p5.3.m2.1.1" xref="S2.SS3.p5.3.m2.1.1.cmml">t</mi><mo id="S2.SS3.p5.3.m2.1.2.3.2.2" stretchy="false" 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encoding="application/x-tex" id="S2.SS3.p5.5.1.m1.1c">\Delta\Lambda=\Delta s_{\text{max}}-\Delta w_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.5.1.m1.1d">roman_Δ roman_Λ = roman_Δ italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT - roman_Δ italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math></span>.</p> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.7">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F3" title="Figure 3 ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">3</span></a> shows the estimated gradient <math alttext="g(t)" class="ltx_Math" display="inline" id="S2.SS3.p6.1.m1.1"><semantics id="S2.SS3.p6.1.m1.1a"><mrow id="S2.SS3.p6.1.m1.1.2" xref="S2.SS3.p6.1.m1.1.2.cmml"><mi 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xref="S2.SS3.p6.2.m2.1.2.1"></times><apply id="S2.SS3.p6.2.m2.1.2.2.cmml" xref="S2.SS3.p6.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p6.2.m2.1.2.2.1.cmml" xref="S2.SS3.p6.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS3.p6.2.m2.1.2.2.2.cmml" xref="S2.SS3.p6.2.m2.1.2.2.2">𝑠</ci><ci id="S2.SS3.p6.2.m2.1.2.2.3a.cmml" xref="S2.SS3.p6.2.m2.1.2.2.3"><mtext id="S2.SS3.p6.2.m2.1.2.2.3.cmml" mathsize="70%" xref="S2.SS3.p6.2.m2.1.2.2.3">max</mtext></ci></apply><ci id="S2.SS3.p6.2.m2.1.1.cmml" xref="S2.SS3.p6.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.2.m2.1c">s_{\text{max}}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.2.m2.1d">italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, and <math alttext="w_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.SS3.p6.3.m3.1"><semantics id="S2.SS3.p6.3.m3.1a"><mrow id="S2.SS3.p6.3.m3.1.2" xref="S2.SS3.p6.3.m3.1.2.cmml"><msub 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id="S2.SS3.p6.3.m3.1.2.2.2.cmml" xref="S2.SS3.p6.3.m3.1.2.2.2">𝑤</ci><ci id="S2.SS3.p6.3.m3.1.2.2.3a.cmml" xref="S2.SS3.p6.3.m3.1.2.2.3"><mtext id="S2.SS3.p6.3.m3.1.2.2.3.cmml" mathsize="70%" xref="S2.SS3.p6.3.m3.1.2.2.3">max</mtext></ci></apply><ci id="S2.SS3.p6.3.m3.1.1.cmml" xref="S2.SS3.p6.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.3.m3.1c">w_{\text{max}}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.3.m3.1d">italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.T1" title="TABLE I ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">I</span></a> summarizes all variables in the neuron model and indicates their dimensions, temporal behavior, and a short description. For the first chirp, we set the initial value to <math alttext="g(0)=0" class="ltx_Math" display="inline" id="S2.SS3.p6.4.m4.1"><semantics id="S2.SS3.p6.4.m4.1a"><mrow id="S2.SS3.p6.4.m4.1.2" xref="S2.SS3.p6.4.m4.1.2.cmml"><mrow id="S2.SS3.p6.4.m4.1.2.2" xref="S2.SS3.p6.4.m4.1.2.2.cmml"><mi id="S2.SS3.p6.4.m4.1.2.2.2" xref="S2.SS3.p6.4.m4.1.2.2.2.cmml">g</mi><mo id="S2.SS3.p6.4.m4.1.2.2.1" xref="S2.SS3.p6.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p6.4.m4.1.2.2.3.2" xref="S2.SS3.p6.4.m4.1.2.2.cmml"><mo id="S2.SS3.p6.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p6.4.m4.1.2.2.cmml">(</mo><mn id="S2.SS3.p6.4.m4.1.1" xref="S2.SS3.p6.4.m4.1.1.cmml">0</mn><mo id="S2.SS3.p6.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p6.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p6.4.m4.1.2.1" xref="S2.SS3.p6.4.m4.1.2.1.cmml">=</mo><mn id="S2.SS3.p6.4.m4.1.2.3" xref="S2.SS3.p6.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.4.m4.1b"><apply id="S2.SS3.p6.4.m4.1.2.cmml" xref="S2.SS3.p6.4.m4.1.2"><eq id="S2.SS3.p6.4.m4.1.2.1.cmml" xref="S2.SS3.p6.4.m4.1.2.1"></eq><apply id="S2.SS3.p6.4.m4.1.2.2.cmml" xref="S2.SS3.p6.4.m4.1.2.2"><times id="S2.SS3.p6.4.m4.1.2.2.1.cmml" xref="S2.SS3.p6.4.m4.1.2.2.1"></times><ci id="S2.SS3.p6.4.m4.1.2.2.2.cmml" xref="S2.SS3.p6.4.m4.1.2.2.2">𝑔</ci><cn id="S2.SS3.p6.4.m4.1.1.cmml" type="integer" xref="S2.SS3.p6.4.m4.1.1">0</cn></apply><cn id="S2.SS3.p6.4.m4.1.2.3.cmml" type="integer" xref="S2.SS3.p6.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.4.m4.1c">g(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.4.m4.1d">italic_g ( 0 ) = 0</annotation></semantics></math>. Otherwise, we use the last gradient estimate <math alttext="g(T_{c})" class="ltx_Math" display="inline" id="S2.SS3.p6.5.m5.1"><semantics id="S2.SS3.p6.5.m5.1a"><mrow id="S2.SS3.p6.5.m5.1.1" xref="S2.SS3.p6.5.m5.1.1.cmml"><mi id="S2.SS3.p6.5.m5.1.1.3" xref="S2.SS3.p6.5.m5.1.1.3.cmml">g</mi><mo id="S2.SS3.p6.5.m5.1.1.2" xref="S2.SS3.p6.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.p6.5.m5.1.1.1.1" xref="S2.SS3.p6.5.m5.1.1.1.1.1.cmml"><mo id="S2.SS3.p6.5.m5.1.1.1.1.2" stretchy="false" xref="S2.SS3.p6.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.p6.5.m5.1.1.1.1.1" xref="S2.SS3.p6.5.m5.1.1.1.1.1.cmml"><mi id="S2.SS3.p6.5.m5.1.1.1.1.1.2" xref="S2.SS3.p6.5.m5.1.1.1.1.1.2.cmml">T</mi><mi id="S2.SS3.p6.5.m5.1.1.1.1.1.3" xref="S2.SS3.p6.5.m5.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S2.SS3.p6.5.m5.1.1.1.1.3" stretchy="false" xref="S2.SS3.p6.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.5.m5.1b"><apply id="S2.SS3.p6.5.m5.1.1.cmml" xref="S2.SS3.p6.5.m5.1.1"><times id="S2.SS3.p6.5.m5.1.1.2.cmml" xref="S2.SS3.p6.5.m5.1.1.2"></times><ci id="S2.SS3.p6.5.m5.1.1.3.cmml" xref="S2.SS3.p6.5.m5.1.1.3">𝑔</ci><apply id="S2.SS3.p6.5.m5.1.1.1.1.1.cmml" xref="S2.SS3.p6.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p6.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS3.p6.5.m5.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p6.5.m5.1.1.1.1.1.2.cmml" xref="S2.SS3.p6.5.m5.1.1.1.1.1.2">𝑇</ci><ci id="S2.SS3.p6.5.m5.1.1.1.1.1.3.cmml" xref="S2.SS3.p6.5.m5.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.5.m5.1c">g(T_{c})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.5.m5.1d">italic_g ( italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> as the initial value for the next chirp. Instead of depending on the result of <math alttext="s(T_{c})" class="ltx_Math" display="inline" id="S2.SS3.p6.6.m6.1"><semantics id="S2.SS3.p6.6.m6.1a"><mrow id="S2.SS3.p6.6.m6.1.1" xref="S2.SS3.p6.6.m6.1.1.cmml"><mi id="S2.SS3.p6.6.m6.1.1.3" xref="S2.SS3.p6.6.m6.1.1.3.cmml">s</mi><mo id="S2.SS3.p6.6.m6.1.1.2" xref="S2.SS3.p6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.p6.6.m6.1.1.1.1" xref="S2.SS3.p6.6.m6.1.1.1.1.1.cmml"><mo id="S2.SS3.p6.6.m6.1.1.1.1.2" stretchy="false" xref="S2.SS3.p6.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.p6.6.m6.1.1.1.1.1" xref="S2.SS3.p6.6.m6.1.1.1.1.1.cmml"><mi id="S2.SS3.p6.6.m6.1.1.1.1.1.2" xref="S2.SS3.p6.6.m6.1.1.1.1.1.2.cmml">T</mi><mi id="S2.SS3.p6.6.m6.1.1.1.1.1.3" xref="S2.SS3.p6.6.m6.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S2.SS3.p6.6.m6.1.1.1.1.3" stretchy="false" xref="S2.SS3.p6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.6.m6.1b"><apply id="S2.SS3.p6.6.m6.1.1.cmml" xref="S2.SS3.p6.6.m6.1.1"><times id="S2.SS3.p6.6.m6.1.1.2.cmml" xref="S2.SS3.p6.6.m6.1.1.2"></times><ci id="S2.SS3.p6.6.m6.1.1.3.cmml" xref="S2.SS3.p6.6.m6.1.1.3">𝑠</ci><apply id="S2.SS3.p6.6.m6.1.1.1.1.1.cmml" xref="S2.SS3.p6.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p6.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS3.p6.6.m6.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p6.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS3.p6.6.m6.1.1.1.1.1.2">𝑇</ci><ci id="S2.SS3.p6.6.m6.1.1.1.1.1.3.cmml" xref="S2.SS3.p6.6.m6.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.6.m6.1c">s(T_{c})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.6.m6.1d">italic_s ( italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math>, as it replicates the result of an FT, we utilize information about the gradient estimate <math alttext="g(t)" class="ltx_Math" display="inline" id="S2.SS3.p6.7.m7.1"><semantics id="S2.SS3.p6.7.m7.1a"><mrow id="S2.SS3.p6.7.m7.1.2" xref="S2.SS3.p6.7.m7.1.2.cmml"><mi id="S2.SS3.p6.7.m7.1.2.2" xref="S2.SS3.p6.7.m7.1.2.2.cmml">g</mi><mo id="S2.SS3.p6.7.m7.1.2.1" xref="S2.SS3.p6.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p6.7.m7.1.2.3.2" xref="S2.SS3.p6.7.m7.1.2.cmml"><mo id="S2.SS3.p6.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS3.p6.7.m7.1.2.cmml">(</mo><mi id="S2.SS3.p6.7.m7.1.1" xref="S2.SS3.p6.7.m7.1.1.cmml">t</mi><mo id="S2.SS3.p6.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS3.p6.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.7.m7.1b"><apply id="S2.SS3.p6.7.m7.1.2.cmml" xref="S2.SS3.p6.7.m7.1.2"><times id="S2.SS3.p6.7.m7.1.2.1.cmml" xref="S2.SS3.p6.7.m7.1.2.1"></times><ci id="S2.SS3.p6.7.m7.1.2.2.cmml" xref="S2.SS3.p6.7.m7.1.2.2">𝑔</ci><ci id="S2.SS3.p6.7.m7.1.1.cmml" xref="S2.SS3.p6.7.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.7.m7.1c">g(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.7.m7.1d">italic_g ( italic_t )</annotation></semantics></math> to produce informative spikes. This procedure allows us to already transmit information during the sampling process.</p> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS4.5.1.1">II-D</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS4.6.2">Spiking Functions</span> </h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.1">Neuromorphic applications rely on different spike encoding schemes, such as rate-coding and temporal-coding <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib1" title="">1</a>]</cite>. In the following section, we describe three approaches that offer individual advantages and trade-offs. The adaptive threshold spiking function builds on rate-coding and benefits from fewer computation steps, but transmits the most spikes. The rate-coded LIF reduces the spike number and allows for early spike transmission, but increases the computational complexity. The temporal-coded LIF reduces the spike number even more, but does not benefit from early spike transmission. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F3" title="Figure 3 ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">3</span></a> compares the dynamics and spiking behavior of all spiking functions for a single chirp. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.T2" title="TABLE II ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">II</span></a> summarizes the variables and parameters of all the spiking functions.</p> </div> <section class="ltx_subsubsection" id="S2.SS4.SSSx1"> <h4 class="ltx_title ltx_title_subsubsection">Adaptive Threshold Spiking Function</h4> <div class="ltx_para" id="S2.SS4.SSSx1.p1"> <p class="ltx_p" id="S2.SS4.SSSx1.p1.6">The first spiking function relies on an adaptive threshold proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib2" title="">2</a>]</cite>. Once the neuron’s state reaches a threshold <math alttext="u_{\text{th}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.1.m1.1"><semantics id="S2.SS4.SSSx1.p1.1.m1.1a"><msub id="S2.SS4.SSSx1.p1.1.m1.1.1" xref="S2.SS4.SSSx1.p1.1.m1.1.1.cmml"><mi id="S2.SS4.SSSx1.p1.1.m1.1.1.2" xref="S2.SS4.SSSx1.p1.1.m1.1.1.2.cmml">u</mi><mtext id="S2.SS4.SSSx1.p1.1.m1.1.1.3" xref="S2.SS4.SSSx1.p1.1.m1.1.1.3a.cmml">th</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.1.m1.1b"><apply id="S2.SS4.SSSx1.p1.1.m1.1.1.cmml" xref="S2.SS4.SSSx1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.1.m1.1.1.1.cmml" xref="S2.SS4.SSSx1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.1.m1.1.1.2.cmml" xref="S2.SS4.SSSx1.p1.1.m1.1.1.2">𝑢</ci><ci id="S2.SS4.SSSx1.p1.1.m1.1.1.3a.cmml" xref="S2.SS4.SSSx1.p1.1.m1.1.1.3"><mtext id="S2.SS4.SSSx1.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p1.1.m1.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.1.m1.1c">u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.1.m1.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math>, the neuron spikes and increases this threshold by <math alttext="\Delta u_{\text{th}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.2.m2.1"><semantics id="S2.SS4.SSSx1.p1.2.m2.1a"><mrow id="S2.SS4.SSSx1.p1.2.m2.1.1" xref="S2.SS4.SSSx1.p1.2.m2.1.1.cmml"><mi id="S2.SS4.SSSx1.p1.2.m2.1.1.2" mathvariant="normal" xref="S2.SS4.SSSx1.p1.2.m2.1.1.2.cmml">Δ</mi><mo id="S2.SS4.SSSx1.p1.2.m2.1.1.1" xref="S2.SS4.SSSx1.p1.2.m2.1.1.1.cmml">⁢</mo><msub id="S2.SS4.SSSx1.p1.2.m2.1.1.3" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.cmml"><mi id="S2.SS4.SSSx1.p1.2.m2.1.1.3.2" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.2.cmml">u</mi><mtext id="S2.SS4.SSSx1.p1.2.m2.1.1.3.3" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.3a.cmml">th</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.2.m2.1b"><apply id="S2.SS4.SSSx1.p1.2.m2.1.1.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1"><times id="S2.SS4.SSSx1.p1.2.m2.1.1.1.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.1"></times><ci id="S2.SS4.SSSx1.p1.2.m2.1.1.2.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.2">Δ</ci><apply id="S2.SS4.SSSx1.p1.2.m2.1.1.3.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.2.m2.1.1.3.1.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.2.m2.1.1.3.2.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.2">𝑢</ci><ci id="S2.SS4.SSSx1.p1.2.m2.1.1.3.3a.cmml" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.3"><mtext id="S2.SS4.SSSx1.p1.2.m2.1.1.3.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p1.2.m2.1.1.3.3">th</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.2.m2.1c">\Delta u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.2.m2.1d">roman_Δ italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, the total number of spikes is proportional to the maximum value of the neuron within a time window, independent of the negative gradient within this window. Since the magnitude of the neuron state <math alttext="|s|" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.3.m3.1"><semantics id="S2.SS4.SSSx1.p1.3.m3.1a"><mrow id="S2.SS4.SSSx1.p1.3.m3.1.2.2" xref="S2.SS4.SSSx1.p1.3.m3.1.2.1.cmml"><mo id="S2.SS4.SSSx1.p1.3.m3.1.2.2.1" stretchy="false" xref="S2.SS4.SSSx1.p1.3.m3.1.2.1.1.cmml">|</mo><mi id="S2.SS4.SSSx1.p1.3.m3.1.1" xref="S2.SS4.SSSx1.p1.3.m3.1.1.cmml">s</mi><mo id="S2.SS4.SSSx1.p1.3.m3.1.2.2.2" stretchy="false" xref="S2.SS4.SSSx1.p1.3.m3.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.3.m3.1b"><apply id="S2.SS4.SSSx1.p1.3.m3.1.2.1.cmml" xref="S2.SS4.SSSx1.p1.3.m3.1.2.2"><abs id="S2.SS4.SSSx1.p1.3.m3.1.2.1.1.cmml" xref="S2.SS4.SSSx1.p1.3.m3.1.2.2.1"></abs><ci id="S2.SS4.SSSx1.p1.3.m3.1.1.cmml" xref="S2.SS4.SSSx1.p1.3.m3.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.3.m3.1c">|s|</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.3.m3.1d">| italic_s |</annotation></semantics></math> oscillates in case of no resonance <span class="ltx_text" id="S2.SS4.SSSx1.p1.4.1"><math alttext="\omega_{k}\neq\omega_{j}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.4.1.m1.1"><semantics id="S2.SS4.SSSx1.p1.4.1.m1.1a"><mrow id="S2.SS4.SSSx1.p1.4.1.m1.1.1" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.cmml"><msub id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.cmml"><mi id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.2" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.2.cmml">ω</mi><mi id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.3" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS4.SSSx1.p1.4.1.m1.1.1.1" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.1.cmml">≠</mo><msub id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.cmml"><mi id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.2" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.2.cmml">ω</mi><mi id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.3" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.4.1.m1.1b"><apply id="S2.SS4.SSSx1.p1.4.1.m1.1.1.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1"><neq id="S2.SS4.SSSx1.p1.4.1.m1.1.1.1.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.1"></neq><apply id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.1.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.2.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.2">𝜔</ci><ci id="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.3.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.2.3">𝑘</ci></apply><apply id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.1.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.2.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.2">𝜔</ci><ci id="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.3.cmml" xref="S2.SS4.SSSx1.p1.4.1.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.4.1.m1.1c">\omega_{k}\neq\omega_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.4.1.m1.1d">italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≠ italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></span>, the decrease of the neuron state also contains information. Hence, we propose positive and negative spikes that depend on adaptive thresholds <math alttext="u_{\text{th}}^{s}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.5.m4.1"><semantics id="S2.SS4.SSSx1.p1.5.m4.1a"><msubsup id="S2.SS4.SSSx1.p1.5.m4.1.1" xref="S2.SS4.SSSx1.p1.5.m4.1.1.cmml"><mi id="S2.SS4.SSSx1.p1.5.m4.1.1.2.2" xref="S2.SS4.SSSx1.p1.5.m4.1.1.2.2.cmml">u</mi><mtext id="S2.SS4.SSSx1.p1.5.m4.1.1.2.3" xref="S2.SS4.SSSx1.p1.5.m4.1.1.2.3a.cmml">th</mtext><mi id="S2.SS4.SSSx1.p1.5.m4.1.1.3" xref="S2.SS4.SSSx1.p1.5.m4.1.1.3.cmml">s</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.5.m4.1b"><apply id="S2.SS4.SSSx1.p1.5.m4.1.1.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.5.m4.1.1.1.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1">superscript</csymbol><apply id="S2.SS4.SSSx1.p1.5.m4.1.1.2.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.5.m4.1.1.2.1.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.5.m4.1.1.2.2.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1.2.2">𝑢</ci><ci id="S2.SS4.SSSx1.p1.5.m4.1.1.2.3a.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1.2.3"><mtext id="S2.SS4.SSSx1.p1.5.m4.1.1.2.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p1.5.m4.1.1.2.3">th</mtext></ci></apply><ci id="S2.SS4.SSSx1.p1.5.m4.1.1.3.cmml" xref="S2.SS4.SSSx1.p1.5.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.5.m4.1c">u_{\text{th}}^{s}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.5.m4.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="u_{\text{th}}^{w}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p1.6.m5.1"><semantics id="S2.SS4.SSSx1.p1.6.m5.1a"><msubsup id="S2.SS4.SSSx1.p1.6.m5.1.1" xref="S2.SS4.SSSx1.p1.6.m5.1.1.cmml"><mi id="S2.SS4.SSSx1.p1.6.m5.1.1.2.2" xref="S2.SS4.SSSx1.p1.6.m5.1.1.2.2.cmml">u</mi><mtext id="S2.SS4.SSSx1.p1.6.m5.1.1.2.3" xref="S2.SS4.SSSx1.p1.6.m5.1.1.2.3a.cmml">th</mtext><mi id="S2.SS4.SSSx1.p1.6.m5.1.1.3" xref="S2.SS4.SSSx1.p1.6.m5.1.1.3.cmml">w</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p1.6.m5.1b"><apply id="S2.SS4.SSSx1.p1.6.m5.1.1.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.6.m5.1.1.1.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1">superscript</csymbol><apply id="S2.SS4.SSSx1.p1.6.m5.1.1.2.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p1.6.m5.1.1.2.1.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1">subscript</csymbol><ci id="S2.SS4.SSSx1.p1.6.m5.1.1.2.2.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1.2.2">𝑢</ci><ci id="S2.SS4.SSSx1.p1.6.m5.1.1.2.3a.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1.2.3"><mtext id="S2.SS4.SSSx1.p1.6.m5.1.1.2.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p1.6.m5.1.1.2.3">th</mtext></ci></apply><ci id="S2.SS4.SSSx1.p1.6.m5.1.1.3.cmml" xref="S2.SS4.SSSx1.p1.6.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p1.6.m5.1c">u_{\text{th}}^{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p1.6.m5.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT</annotation></semantics></math>. The neuron follows the spike generation rules,</p> </div> <div class="ltx_para" id="S2.SS4.SSSx1.p2"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx11"> <tbody id="S2.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta^{+}(t)=\begin{cases}1&s_{\text{max}}(t)>u_{\text{th}}^{s}(% t),\\ 0&\text{else,}\end{cases}" class="ltx_Math" display="inline" id="S2.E14.m1.5"><semantics id="S2.E14.m1.5a"><mrow id="S2.E14.m1.5.6" xref="S2.E14.m1.5.6.cmml"><mrow id="S2.E14.m1.5.6.2" xref="S2.E14.m1.5.6.2.cmml"><msup id="S2.E14.m1.5.6.2.2" xref="S2.E14.m1.5.6.2.2.cmml"><mi id="S2.E14.m1.5.6.2.2.2" xref="S2.E14.m1.5.6.2.2.2.cmml">δ</mi><mo id="S2.E14.m1.5.6.2.2.3" xref="S2.E14.m1.5.6.2.2.3.cmml">+</mo></msup><mo id="S2.E14.m1.5.6.2.1" xref="S2.E14.m1.5.6.2.1.cmml">⁢</mo><mrow id="S2.E14.m1.5.6.2.3.2" xref="S2.E14.m1.5.6.2.cmml"><mo 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start_ROW start_CELL 1 end_CELL start_CELL italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t ) > italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL else, end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx1.p2.4">and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx12"> <tbody id="S2.E15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta^{-}(t)=\begin{cases}-1&w_{\text{max}}(t)>u_{\text{th}}^{w}% (t),\\ 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL else, end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx1.p2.3">where <math alttext="s_{\text{max}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p2.1.m1.1"><semantics id="S2.SS4.SSSx1.p2.1.m1.1a"><msub id="S2.SS4.SSSx1.p2.1.m1.1.1" xref="S2.SS4.SSSx1.p2.1.m1.1.1.cmml"><mi id="S2.SS4.SSSx1.p2.1.m1.1.1.2" xref="S2.SS4.SSSx1.p2.1.m1.1.1.2.cmml">s</mi><mtext id="S2.SS4.SSSx1.p2.1.m1.1.1.3" xref="S2.SS4.SSSx1.p2.1.m1.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p2.1.m1.1b"><apply id="S2.SS4.SSSx1.p2.1.m1.1.1.cmml" xref="S2.SS4.SSSx1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p2.1.m1.1.1.1.cmml" xref="S2.SS4.SSSx1.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.SSSx1.p2.1.m1.1.1.2.cmml" xref="S2.SS4.SSSx1.p2.1.m1.1.1.2">𝑠</ci><ci id="S2.SS4.SSSx1.p2.1.m1.1.1.3a.cmml" xref="S2.SS4.SSSx1.p2.1.m1.1.1.3"><mtext id="S2.SS4.SSSx1.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p2.1.m1.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p2.1.m1.1c">s_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p2.1.m1.1d">italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w_{\text{max}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p2.2.m2.1"><semantics id="S2.SS4.SSSx1.p2.2.m2.1a"><msub id="S2.SS4.SSSx1.p2.2.m2.1.1" xref="S2.SS4.SSSx1.p2.2.m2.1.1.cmml"><mi id="S2.SS4.SSSx1.p2.2.m2.1.1.2" xref="S2.SS4.SSSx1.p2.2.m2.1.1.2.cmml">w</mi><mtext id="S2.SS4.SSSx1.p2.2.m2.1.1.3" xref="S2.SS4.SSSx1.p2.2.m2.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p2.2.m2.1b"><apply id="S2.SS4.SSSx1.p2.2.m2.1.1.cmml" xref="S2.SS4.SSSx1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p2.2.m2.1.1.1.cmml" xref="S2.SS4.SSSx1.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS4.SSSx1.p2.2.m2.1.1.2.cmml" xref="S2.SS4.SSSx1.p2.2.m2.1.1.2">𝑤</ci><ci id="S2.SS4.SSSx1.p2.2.m2.1.1.3a.cmml" xref="S2.SS4.SSSx1.p2.2.m2.1.1.3"><mtext id="S2.SS4.SSSx1.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p2.2.m2.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p2.2.m2.1c">w_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p2.2.m2.1d">italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> describe the estimated maxima in section <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.SS3" title="II-C Envelope estimation and gradient estimation ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-C</span></span></a>. The initial thresholds <math alttext="u_{\text{th}}^{s/w}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p2.3.m3.1"><semantics id="S2.SS4.SSSx1.p2.3.m3.1a"><msubsup id="S2.SS4.SSSx1.p2.3.m3.1.1" xref="S2.SS4.SSSx1.p2.3.m3.1.1.cmml"><mi id="S2.SS4.SSSx1.p2.3.m3.1.1.2.2" xref="S2.SS4.SSSx1.p2.3.m3.1.1.2.2.cmml">u</mi><mtext id="S2.SS4.SSSx1.p2.3.m3.1.1.2.3" xref="S2.SS4.SSSx1.p2.3.m3.1.1.2.3a.cmml">th</mtext><mrow id="S2.SS4.SSSx1.p2.3.m3.1.1.3" xref="S2.SS4.SSSx1.p2.3.m3.1.1.3.cmml"><mi id="S2.SS4.SSSx1.p2.3.m3.1.1.3.2" xref="S2.SS4.SSSx1.p2.3.m3.1.1.3.2.cmml">s</mi><mo id="S2.SS4.SSSx1.p2.3.m3.1.1.3.1" xref="S2.SS4.SSSx1.p2.3.m3.1.1.3.1.cmml">/</mo><mi id="S2.SS4.SSSx1.p2.3.m3.1.1.3.3" xref="S2.SS4.SSSx1.p2.3.m3.1.1.3.3.cmml">w</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p2.3.m3.1b"><apply id="S2.SS4.SSSx1.p2.3.m3.1.1.cmml" xref="S2.SS4.SSSx1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx1.p2.3.m3.1.1.1.cmml" 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id="S2.SS4.SSSx1.p2.3.m3.1c">u_{\text{th}}^{s/w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p2.3.m3.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s / italic_w end_POSTSUPERSCRIPT</annotation></semantics></math> are updated according to</p> </div> <div class="ltx_para" id="S2.SS4.SSSx1.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx13"> <tbody id="S2.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle u_{\text{th}}^{s/w}(t)=u_{\text{th}}^{s/w}+\Delta u_{\text{th}}% \int_{0}^{t}\delta^{\pm}(t^{\prime})dt^{\prime}=u_{\text{th}}^{\pm}+\gamma N^{% \pm}_{\text{spikes}}\,." class="ltx_Math" display="inline" id="S2.E16.m1.2"><semantics id="S2.E16.m1.2a"><mrow id="S2.E16.m1.2.2.1" xref="S2.E16.m1.2.2.1.1.cmml"><mrow id="S2.E16.m1.2.2.1.1" 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encoding="application/x-llamapun" id="S2.E16.m1.2d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s / italic_w end_POSTSUPERSCRIPT ( italic_t ) = italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s / italic_w end_POSTSUPERSCRIPT + roman_Δ italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_δ start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_d italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT + italic_γ italic_N start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT start_POSTSUBSCRIPT spikes end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS4.SSSx1.p4"> <p class="ltx_p" id="S2.SS4.SSSx1.p4.3">For the readout, the number of negative spikes is subtracted from the number of positive spikes <math alttext="N^{+}_{\text{spikes}}-N^{-}_{\text{spikes}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p4.1.m1.1"><semantics id="S2.SS4.SSSx1.p4.1.m1.1a"><mrow id="S2.SS4.SSSx1.p4.1.m1.1.1" xref="S2.SS4.SSSx1.p4.1.m1.1.1.cmml"><msubsup id="S2.SS4.SSSx1.p4.1.m1.1.1.2" xref="S2.SS4.SSSx1.p4.1.m1.1.1.2.cmml"><mi id="S2.SS4.SSSx1.p4.1.m1.1.1.2.2.2" xref="S2.SS4.SSSx1.p4.1.m1.1.1.2.2.2.cmml">N</mi><mtext id="S2.SS4.SSSx1.p4.1.m1.1.1.2.3" xref="S2.SS4.SSSx1.p4.1.m1.1.1.2.3a.cmml">spikes</mtext><mo id="S2.SS4.SSSx1.p4.1.m1.1.1.2.2.3" xref="S2.SS4.SSSx1.p4.1.m1.1.1.2.2.3.cmml">+</mo></msubsup><mo id="S2.SS4.SSSx1.p4.1.m1.1.1.1" xref="S2.SS4.SSSx1.p4.1.m1.1.1.1.cmml">−</mo><msubsup 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id="S2.SS4.SSSx1.p4.1.m1.1.1.3.3a.cmml" xref="S2.SS4.SSSx1.p4.1.m1.1.1.3.3"><mtext id="S2.SS4.SSSx1.p4.1.m1.1.1.3.3.cmml" mathsize="70%" xref="S2.SS4.SSSx1.p4.1.m1.1.1.3.3">spikes</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p4.1.m1.1c">N^{+}_{\text{spikes}}-N^{-}_{\text{spikes}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p4.1.m1.1d">italic_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT spikes end_POSTSUBSCRIPT - italic_N start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT spikes end_POSTSUBSCRIPT</annotation></semantics></math>. This result gives an estimate of the envelope gradient <math alttext="\Lambda(t)" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p4.2.m2.1"><semantics id="S2.SS4.SSSx1.p4.2.m2.1a"><mrow id="S2.SS4.SSSx1.p4.2.m2.1.2" xref="S2.SS4.SSSx1.p4.2.m2.1.2.cmml"><mi id="S2.SS4.SSSx1.p4.2.m2.1.2.2" mathvariant="normal" xref="S2.SS4.SSSx1.p4.2.m2.1.2.2.cmml">Λ</mi><mo id="S2.SS4.SSSx1.p4.2.m2.1.2.1" xref="S2.SS4.SSSx1.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx1.p4.2.m2.1.2.3.2" xref="S2.SS4.SSSx1.p4.2.m2.1.2.cmml"><mo id="S2.SS4.SSSx1.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS4.SSSx1.p4.2.m2.1.2.cmml">(</mo><mi id="S2.SS4.SSSx1.p4.2.m2.1.1" xref="S2.SS4.SSSx1.p4.2.m2.1.1.cmml">t</mi><mo id="S2.SS4.SSSx1.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS4.SSSx1.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx1.p4.2.m2.1b"><apply id="S2.SS4.SSSx1.p4.2.m2.1.2.cmml" xref="S2.SS4.SSSx1.p4.2.m2.1.2"><times id="S2.SS4.SSSx1.p4.2.m2.1.2.1.cmml" xref="S2.SS4.SSSx1.p4.2.m2.1.2.1"></times><ci id="S2.SS4.SSSx1.p4.2.m2.1.2.2.cmml" xref="S2.SS4.SSSx1.p4.2.m2.1.2.2">Λ</ci><ci id="S2.SS4.SSSx1.p4.2.m2.1.1.cmml" xref="S2.SS4.SSSx1.p4.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p4.2.m2.1c">\Lambda(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p4.2.m2.1d">roman_Λ ( italic_t )</annotation></semantics></math> without calculating Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.E13" title="In II-C Envelope estimation and gradient estimation ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">13</span></a>) directly. For consecutive chirps, the thresholds are reset to <math alttext="\Gamma^{\pm}(0)=0" class="ltx_Math" display="inline" id="S2.SS4.SSSx1.p4.3.m3.1"><semantics id="S2.SS4.SSSx1.p4.3.m3.1a"><mrow id="S2.SS4.SSSx1.p4.3.m3.1.2" xref="S2.SS4.SSSx1.p4.3.m3.1.2.cmml"><mrow id="S2.SS4.SSSx1.p4.3.m3.1.2.2" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.cmml"><msup id="S2.SS4.SSSx1.p4.3.m3.1.2.2.2" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.cmml"><mi id="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.2" mathvariant="normal" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.2.cmml">Γ</mi><mo id="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.3" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.3.cmml">±</mo></msup><mo id="S2.SS4.SSSx1.p4.3.m3.1.2.2.1" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx1.p4.3.m3.1.2.2.3.2" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.cmml"><mo id="S2.SS4.SSSx1.p4.3.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.cmml">(</mo><mn id="S2.SS4.SSSx1.p4.3.m3.1.1" xref="S2.SS4.SSSx1.p4.3.m3.1.1.cmml">0</mn><mo 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id="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.3.cmml" xref="S2.SS4.SSSx1.p4.3.m3.1.2.2.2.3">plus-or-minus</csymbol></apply><cn id="S2.SS4.SSSx1.p4.3.m3.1.1.cmml" type="integer" xref="S2.SS4.SSSx1.p4.3.m3.1.1">0</cn></apply><cn id="S2.SS4.SSSx1.p4.3.m3.1.2.3.cmml" type="integer" xref="S2.SS4.SSSx1.p4.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx1.p4.3.m3.1c">\Gamma^{\pm}(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx1.p4.3.m3.1d">roman_Γ start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( 0 ) = 0</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS4.SSSx2"> <h4 class="ltx_title ltx_title_subsubsection">Rate-coded LIF Spiking Function</h4> <div class="ltx_para" id="S2.SS4.SSSx2.p1"> <p class="ltx_p" id="S2.SS4.SSSx2.p1.9">Alternatively, we can extend the neuron model using the dynamics of a LIF neuron model,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" 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xref="S2.E17.m1.2.2.1.1.3.4.2">𝑢</ci><ci id="S2.E17.m1.2.2.1.1.3.4.3a.cmml" xref="S2.E17.m1.2.2.1.1.3.4.3"><mtext id="S2.E17.m1.2.2.1.1.3.4.3.cmml" mathsize="70%" xref="S2.E17.m1.2.2.1.1.3.4.3">rest</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E17.m1.2c">\displaystyle\tau\frac{du}{dt}=-u+g(t)+u_{\text{rest}}\,,</annotation><annotation encoding="application/x-llamapun" id="S2.E17.m1.2d">italic_τ divide start_ARG italic_d italic_u end_ARG start_ARG italic_d italic_t end_ARG = - italic_u + italic_g ( italic_t ) + italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx2.p1.5">to produce spikes with a spike rate <math alttext="r(t)\sim g(t)" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.1.m1.2"><semantics id="S2.SS4.SSSx2.p1.1.m1.2a"><mrow id="S2.SS4.SSSx2.p1.1.m1.2.3" xref="S2.SS4.SSSx2.p1.1.m1.2.3.cmml"><mrow id="S2.SS4.SSSx2.p1.1.m1.2.3.2" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.cmml"><mi id="S2.SS4.SSSx2.p1.1.m1.2.3.2.2" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.2.cmml">r</mi><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.2.1" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx2.p1.1.m1.2.3.2.3.2" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.cmml"><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.cmml">(</mo><mi id="S2.SS4.SSSx2.p1.1.m1.1.1" xref="S2.SS4.SSSx2.p1.1.m1.1.1.cmml">t</mi><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.1" xref="S2.SS4.SSSx2.p1.1.m1.2.3.1.cmml">∼</mo><mrow id="S2.SS4.SSSx2.p1.1.m1.2.3.3" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.cmml"><mi id="S2.SS4.SSSx2.p1.1.m1.2.3.3.2" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.2.cmml">g</mi><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.3.1" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx2.p1.1.m1.2.3.3.3.2" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.cmml"><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.cmml">(</mo><mi id="S2.SS4.SSSx2.p1.1.m1.2.2" xref="S2.SS4.SSSx2.p1.1.m1.2.2.cmml">t</mi><mo id="S2.SS4.SSSx2.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.1.m1.2b"><apply id="S2.SS4.SSSx2.p1.1.m1.2.3.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3"><csymbol cd="latexml" id="S2.SS4.SSSx2.p1.1.m1.2.3.1.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.1">similar-to</csymbol><apply id="S2.SS4.SSSx2.p1.1.m1.2.3.2.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2"><times id="S2.SS4.SSSx2.p1.1.m1.2.3.2.1.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.1"></times><ci id="S2.SS4.SSSx2.p1.1.m1.2.3.2.2.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.2.2">𝑟</ci><ci id="S2.SS4.SSSx2.p1.1.m1.1.1.cmml" xref="S2.SS4.SSSx2.p1.1.m1.1.1">𝑡</ci></apply><apply id="S2.SS4.SSSx2.p1.1.m1.2.3.3.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3"><times id="S2.SS4.SSSx2.p1.1.m1.2.3.3.1.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.1"></times><ci id="S2.SS4.SSSx2.p1.1.m1.2.3.3.2.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.3.3.2">𝑔</ci><ci id="S2.SS4.SSSx2.p1.1.m1.2.2.cmml" xref="S2.SS4.SSSx2.p1.1.m1.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.1.m1.2c">r(t)\sim g(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.1.m1.2d">italic_r ( italic_t ) ∼ italic_g ( italic_t )</annotation></semantics></math>. The LIF neuron spikes once the neuron membrane <math alttext="u(t)" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.2.m2.1"><semantics id="S2.SS4.SSSx2.p1.2.m2.1a"><mrow id="S2.SS4.SSSx2.p1.2.m2.1.2" xref="S2.SS4.SSSx2.p1.2.m2.1.2.cmml"><mi id="S2.SS4.SSSx2.p1.2.m2.1.2.2" xref="S2.SS4.SSSx2.p1.2.m2.1.2.2.cmml">u</mi><mo id="S2.SS4.SSSx2.p1.2.m2.1.2.1" xref="S2.SS4.SSSx2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx2.p1.2.m2.1.2.3.2" xref="S2.SS4.SSSx2.p1.2.m2.1.2.cmml"><mo id="S2.SS4.SSSx2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS4.SSSx2.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS4.SSSx2.p1.2.m2.1.1" xref="S2.SS4.SSSx2.p1.2.m2.1.1.cmml">t</mi><mo id="S2.SS4.SSSx2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS4.SSSx2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.2.m2.1b"><apply id="S2.SS4.SSSx2.p1.2.m2.1.2.cmml" xref="S2.SS4.SSSx2.p1.2.m2.1.2"><times id="S2.SS4.SSSx2.p1.2.m2.1.2.1.cmml" xref="S2.SS4.SSSx2.p1.2.m2.1.2.1"></times><ci id="S2.SS4.SSSx2.p1.2.m2.1.2.2.cmml" xref="S2.SS4.SSSx2.p1.2.m2.1.2.2">𝑢</ci><ci id="S2.SS4.SSSx2.p1.2.m2.1.1.cmml" xref="S2.SS4.SSSx2.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.2.m2.1c">u(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.2.m2.1d">italic_u ( italic_t )</annotation></semantics></math> reaches the threshold <math alttext="u_{\text{th}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.3.m3.1"><semantics id="S2.SS4.SSSx2.p1.3.m3.1a"><msub id="S2.SS4.SSSx2.p1.3.m3.1.1" xref="S2.SS4.SSSx2.p1.3.m3.1.1.cmml"><mi id="S2.SS4.SSSx2.p1.3.m3.1.1.2" xref="S2.SS4.SSSx2.p1.3.m3.1.1.2.cmml">u</mi><mtext id="S2.SS4.SSSx2.p1.3.m3.1.1.3" xref="S2.SS4.SSSx2.p1.3.m3.1.1.3a.cmml">th</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.3.m3.1b"><apply id="S2.SS4.SSSx2.p1.3.m3.1.1.cmml" xref="S2.SS4.SSSx2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx2.p1.3.m3.1.1.1.cmml" xref="S2.SS4.SSSx2.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS4.SSSx2.p1.3.m3.1.1.2.cmml" xref="S2.SS4.SSSx2.p1.3.m3.1.1.2">𝑢</ci><ci id="S2.SS4.SSSx2.p1.3.m3.1.1.3a.cmml" xref="S2.SS4.SSSx2.p1.3.m3.1.1.3"><mtext id="S2.SS4.SSSx2.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S2.SS4.SSSx2.p1.3.m3.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.3.m3.1c">u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.3.m3.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math> and resets its state by subtracting <math alttext="u_{\text{th}}" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.4.m4.1"><semantics id="S2.SS4.SSSx2.p1.4.m4.1a"><msub id="S2.SS4.SSSx2.p1.4.m4.1.1" xref="S2.SS4.SSSx2.p1.4.m4.1.1.cmml"><mi id="S2.SS4.SSSx2.p1.4.m4.1.1.2" xref="S2.SS4.SSSx2.p1.4.m4.1.1.2.cmml">u</mi><mtext id="S2.SS4.SSSx2.p1.4.m4.1.1.3" xref="S2.SS4.SSSx2.p1.4.m4.1.1.3a.cmml">th</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.4.m4.1b"><apply id="S2.SS4.SSSx2.p1.4.m4.1.1.cmml" xref="S2.SS4.SSSx2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.SSSx2.p1.4.m4.1.1.1.cmml" xref="S2.SS4.SSSx2.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS4.SSSx2.p1.4.m4.1.1.2.cmml" xref="S2.SS4.SSSx2.p1.4.m4.1.1.2">𝑢</ci><ci id="S2.SS4.SSSx2.p1.4.m4.1.1.3a.cmml" xref="S2.SS4.SSSx2.p1.4.m4.1.1.3"><mtext id="S2.SS4.SSSx2.p1.4.m4.1.1.3.cmml" mathsize="70%" xref="S2.SS4.SSSx2.p1.4.m4.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.4.m4.1c">u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.4.m4.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math>. A constant gradient <math alttext="g" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.5.m5.1"><semantics id="S2.SS4.SSSx2.p1.5.m5.1a"><mi id="S2.SS4.SSSx2.p1.5.m5.1.1" xref="S2.SS4.SSSx2.p1.5.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.5.m5.1b"><ci id="S2.SS4.SSSx2.p1.5.m5.1.1.cmml" xref="S2.SS4.SSSx2.p1.5.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.5.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.5.m5.1d">italic_g</annotation></semantics></math> creates a spike rate</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx15"> <tbody id="S2.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle r(g)=\frac{1}{-\tau\ln(1-\frac{u_{\text{th}}}{g+u_{\text{rest}}}% )}." class="ltx_Math" display="inline" 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linearly proportional to <math alttext="g" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.6.m1.1"><semantics id="S2.SS4.SSSx2.p1.6.m1.1a"><mi id="S2.SS4.SSSx2.p1.6.m1.1.1" xref="S2.SS4.SSSx2.p1.6.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx2.p1.6.m1.1b"><ci id="S2.SS4.SSSx2.p1.6.m1.1.1.cmml" xref="S2.SS4.SSSx2.p1.6.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.6.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.6.m1.1d">italic_g</annotation></semantics></math> within the time window <math alttext="[0,T_{c}]" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.7.m2.2"><semantics id="S2.SS4.SSSx2.p1.7.m2.2a"><mrow id="S2.SS4.SSSx2.p1.7.m2.2.2.1" xref="S2.SS4.SSSx2.p1.7.m2.2.2.2.cmml"><mo id="S2.SS4.SSSx2.p1.7.m2.2.2.1.2" stretchy="false" xref="S2.SS4.SSSx2.p1.7.m2.2.2.2.cmml">[</mo><mn id="S2.SS4.SSSx2.p1.7.m2.1.1" xref="S2.SS4.SSSx2.p1.7.m2.1.1.cmml">0</mn><mo 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xref="S2.E19.m1.2.2.1.1.3.2.3.2">𝑢</ci><ci id="S2.E19.m1.2.2.1.1.3.2.3.3a.cmml" xref="S2.E19.m1.2.2.1.1.3.2.3.3"><mtext id="S2.E19.m1.2.2.1.1.3.2.3.3.cmml" mathsize="70%" xref="S2.E19.m1.2.2.1.1.3.2.3.3">th</mtext></ci></apply></apply><apply id="S2.E19.m1.2.2.1.1.3.3.cmml" xref="S2.E19.m1.2.2.1.1.3.3"><divide id="S2.E19.m1.2.2.1.1.3.3.1.cmml" xref="S2.E19.m1.2.2.1.1.3.3"></divide><apply id="S2.E19.m1.2.2.1.1.3.3.2.cmml" xref="S2.E19.m1.2.2.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.E19.m1.2.2.1.1.3.3.2.1.cmml" xref="S2.E19.m1.2.2.1.1.3.3.2">subscript</csymbol><ci id="S2.E19.m1.2.2.1.1.3.3.2.2.cmml" xref="S2.E19.m1.2.2.1.1.3.3.2.2">𝑇</ci><ci id="S2.E19.m1.2.2.1.1.3.3.2.3.cmml" xref="S2.E19.m1.2.2.1.1.3.3.2.3">𝑐</ci></apply><ci id="S2.E19.m1.2.2.1.1.3.3.3.cmml" xref="S2.E19.m1.2.2.1.1.3.3.3">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E19.m1.2c">\displaystyle r(g)=\frac{g+u_{\text{rest}}}{u_{\text{th}}}\cdot\frac{T_{c}}{% \tau}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.E19.m1.2d">italic_r ( italic_g ) = divide start_ARG italic_g + italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT end_ARG start_ARG italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT end_ARG ⋅ divide start_ARG italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_τ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx2.p1.8">For consecutive chirps, the membrane is rest to <math alttext="u(0)=0" class="ltx_Math" display="inline" id="S2.SS4.SSSx2.p1.8.m1.1"><semantics id="S2.SS4.SSSx2.p1.8.m1.1a"><mrow id="S2.SS4.SSSx2.p1.8.m1.1.2" xref="S2.SS4.SSSx2.p1.8.m1.1.2.cmml"><mrow id="S2.SS4.SSSx2.p1.8.m1.1.2.2" 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id="S2.SS4.SSSx2.p1.8.m1.1.2.2.cmml" xref="S2.SS4.SSSx2.p1.8.m1.1.2.2"><times id="S2.SS4.SSSx2.p1.8.m1.1.2.2.1.cmml" xref="S2.SS4.SSSx2.p1.8.m1.1.2.2.1"></times><ci id="S2.SS4.SSSx2.p1.8.m1.1.2.2.2.cmml" xref="S2.SS4.SSSx2.p1.8.m1.1.2.2.2">𝑢</ci><cn id="S2.SS4.SSSx2.p1.8.m1.1.1.cmml" type="integer" xref="S2.SS4.SSSx2.p1.8.m1.1.1">0</cn></apply><cn id="S2.SS4.SSSx2.p1.8.m1.1.2.3.cmml" type="integer" xref="S2.SS4.SSSx2.p1.8.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx2.p1.8.m1.1c">u(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx2.p1.8.m1.1d">italic_u ( 0 ) = 0</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS4.SSSx3"> <h4 class="ltx_title ltx_title_subsubsection">Time-coded LIF Spiking Function</h4> <div class="ltx_para" id="S2.SS4.SSSx3.p1"> <p class="ltx_p" id="S2.SS4.SSSx3.p1.1">Instead of transmitting the information about g(t) via the spike rate, we can use a more efficient encoding. As in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib18" title="">18</a>]</cite>, a single spike encodes a constant value by charging a LIF model and using an adaptive refractory period. Similar to the rate-coded approach, we calculate the spike time depending on the constant input <math alttext="g" class="ltx_Math" display="inline" id="S2.SS4.SSSx3.p1.1.m1.1"><semantics id="S2.SS4.SSSx3.p1.1.m1.1a"><mi id="S2.SS4.SSSx3.p1.1.m1.1.1" xref="S2.SS4.SSSx3.p1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx3.p1.1.m1.1b"><ci id="S2.SS4.SSSx3.p1.1.m1.1.1.cmml" xref="S2.SS4.SSSx3.p1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx3.p1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx3.p1.1.m1.1d">italic_g</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx17"> <tbody id="S2.E20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 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- divide start_ARG italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT end_ARG start_ARG italic_g + italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx3.p1.2">Once the neuron spikes, we can read out the value <math alttext="g" class="ltx_Math" display="inline" id="S2.SS4.SSSx3.p1.2.m1.1"><semantics id="S2.SS4.SSSx3.p1.2.m1.1a"><mi id="S2.SS4.SSSx3.p1.2.m1.1.1" xref="S2.SS4.SSSx3.p1.2.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx3.p1.2.m1.1b"><ci id="S2.SS4.SSSx3.p1.2.m1.1.1.cmml" xref="S2.SS4.SSSx3.p1.2.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx3.p1.2.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx3.p1.2.m1.1d">italic_g</annotation></semantics></math> by using a linear decoding scheme,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx18"> <tbody id="S2.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g\simeq T_{c}-t_{s}(g)\,." class="ltx_Math" display="inline" id="S2.E21.m1.2"><semantics id="S2.E21.m1.2a"><mrow id="S2.E21.m1.2.2.1" xref="S2.E21.m1.2.2.1.1.cmml"><mrow id="S2.E21.m1.2.2.1.1" xref="S2.E21.m1.2.2.1.1.cmml"><mi id="S2.E21.m1.2.2.1.1.2" xref="S2.E21.m1.2.2.1.1.2.cmml">g</mi><mo id="S2.E21.m1.2.2.1.1.1" xref="S2.E21.m1.2.2.1.1.1.cmml">≃</mo><mrow id="S2.E21.m1.2.2.1.1.3" xref="S2.E21.m1.2.2.1.1.3.cmml"><msub id="S2.E21.m1.2.2.1.1.3.2" xref="S2.E21.m1.2.2.1.1.3.2.cmml"><mi id="S2.E21.m1.2.2.1.1.3.2.2" xref="S2.E21.m1.2.2.1.1.3.2.2.cmml">T</mi><mi id="S2.E21.m1.2.2.1.1.3.2.3" xref="S2.E21.m1.2.2.1.1.3.2.3.cmml">c</mi></msub><mo id="S2.E21.m1.2.2.1.1.3.1" xref="S2.E21.m1.2.2.1.1.3.1.cmml">−</mo><mrow id="S2.E21.m1.2.2.1.1.3.3" xref="S2.E21.m1.2.2.1.1.3.3.cmml"><msub id="S2.E21.m1.2.2.1.1.3.3.2" xref="S2.E21.m1.2.2.1.1.3.3.2.cmml"><mi id="S2.E21.m1.2.2.1.1.3.3.2.2" xref="S2.E21.m1.2.2.1.1.3.3.2.2.cmml">t</mi><mi id="S2.E21.m1.2.2.1.1.3.3.2.3" xref="S2.E21.m1.2.2.1.1.3.3.2.3.cmml">s</mi></msub><mo id="S2.E21.m1.2.2.1.1.3.3.1" xref="S2.E21.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mrow id="S2.E21.m1.2.2.1.1.3.3.3.2" xref="S2.E21.m1.2.2.1.1.3.3.cmml"><mo id="S2.E21.m1.2.2.1.1.3.3.3.2.1" stretchy="false" xref="S2.E21.m1.2.2.1.1.3.3.cmml">(</mo><mi id="S2.E21.m1.1.1" xref="S2.E21.m1.1.1.cmml">g</mi><mo id="S2.E21.m1.2.2.1.1.3.3.3.2.2" stretchy="false" xref="S2.E21.m1.2.2.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E21.m1.2.2.1.2" lspace="0.170em" xref="S2.E21.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E21.m1.2b"><apply id="S2.E21.m1.2.2.1.1.cmml" xref="S2.E21.m1.2.2.1"><csymbol cd="latexml" id="S2.E21.m1.2.2.1.1.1.cmml" xref="S2.E21.m1.2.2.1.1.1">similar-to-or-equals</csymbol><ci id="S2.E21.m1.2.2.1.1.2.cmml" xref="S2.E21.m1.2.2.1.1.2">𝑔</ci><apply id="S2.E21.m1.2.2.1.1.3.cmml" xref="S2.E21.m1.2.2.1.1.3"><minus id="S2.E21.m1.2.2.1.1.3.1.cmml" xref="S2.E21.m1.2.2.1.1.3.1"></minus><apply id="S2.E21.m1.2.2.1.1.3.2.cmml" xref="S2.E21.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S2.E21.m1.2.2.1.1.3.2.1.cmml" xref="S2.E21.m1.2.2.1.1.3.2">subscript</csymbol><ci id="S2.E21.m1.2.2.1.1.3.2.2.cmml" xref="S2.E21.m1.2.2.1.1.3.2.2">𝑇</ci><ci id="S2.E21.m1.2.2.1.1.3.2.3.cmml" xref="S2.E21.m1.2.2.1.1.3.2.3">𝑐</ci></apply><apply id="S2.E21.m1.2.2.1.1.3.3.cmml" xref="S2.E21.m1.2.2.1.1.3.3"><times id="S2.E21.m1.2.2.1.1.3.3.1.cmml" xref="S2.E21.m1.2.2.1.1.3.3.1"></times><apply id="S2.E21.m1.2.2.1.1.3.3.2.cmml" xref="S2.E21.m1.2.2.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.E21.m1.2.2.1.1.3.3.2.1.cmml" xref="S2.E21.m1.2.2.1.1.3.3.2">subscript</csymbol><ci id="S2.E21.m1.2.2.1.1.3.3.2.2.cmml" xref="S2.E21.m1.2.2.1.1.3.3.2.2">𝑡</ci><ci id="S2.E21.m1.2.2.1.1.3.3.2.3.cmml" xref="S2.E21.m1.2.2.1.1.3.3.2.3">𝑠</ci></apply><ci id="S2.E21.m1.1.1.cmml" xref="S2.E21.m1.1.1">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E21.m1.2c">\displaystyle g\simeq T_{c}-t_{s}(g)\,.</annotation><annotation encoding="application/x-llamapun" id="S2.E21.m1.2d">italic_g ≃ italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_g ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.SSSx3.p1.3">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.F3" title="Figure 3 ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">3</span></a> illustrates the charging of the neuron and the transmitted spike. For consecutive chirps, the membrane is also reset to <math alttext="u(0)=0" class="ltx_Math" display="inline" id="S2.SS4.SSSx3.p1.3.m1.1"><semantics id="S2.SS4.SSSx3.p1.3.m1.1a"><mrow id="S2.SS4.SSSx3.p1.3.m1.1.2" xref="S2.SS4.SSSx3.p1.3.m1.1.2.cmml"><mrow id="S2.SS4.SSSx3.p1.3.m1.1.2.2" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.cmml"><mi id="S2.SS4.SSSx3.p1.3.m1.1.2.2.2" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.2.cmml">u</mi><mo id="S2.SS4.SSSx3.p1.3.m1.1.2.2.1" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.SSSx3.p1.3.m1.1.2.2.3.2" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.cmml"><mo id="S2.SS4.SSSx3.p1.3.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.cmml">(</mo><mn id="S2.SS4.SSSx3.p1.3.m1.1.1" xref="S2.SS4.SSSx3.p1.3.m1.1.1.cmml">0</mn><mo id="S2.SS4.SSSx3.p1.3.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.SSSx3.p1.3.m1.1.2.1" xref="S2.SS4.SSSx3.p1.3.m1.1.2.1.cmml">=</mo><mn id="S2.SS4.SSSx3.p1.3.m1.1.2.3" xref="S2.SS4.SSSx3.p1.3.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.SSSx3.p1.3.m1.1b"><apply id="S2.SS4.SSSx3.p1.3.m1.1.2.cmml" xref="S2.SS4.SSSx3.p1.3.m1.1.2"><eq id="S2.SS4.SSSx3.p1.3.m1.1.2.1.cmml" xref="S2.SS4.SSSx3.p1.3.m1.1.2.1"></eq><apply id="S2.SS4.SSSx3.p1.3.m1.1.2.2.cmml" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2"><times id="S2.SS4.SSSx3.p1.3.m1.1.2.2.1.cmml" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.1"></times><ci id="S2.SS4.SSSx3.p1.3.m1.1.2.2.2.cmml" xref="S2.SS4.SSSx3.p1.3.m1.1.2.2.2">𝑢</ci><cn id="S2.SS4.SSSx3.p1.3.m1.1.1.cmml" type="integer" xref="S2.SS4.SSSx3.p1.3.m1.1.1">0</cn></apply><cn id="S2.SS4.SSSx3.p1.3.m1.1.2.3.cmml" type="integer" xref="S2.SS4.SSSx3.p1.3.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.SSSx3.p1.3.m1.1c">u(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.SSSx3.p1.3.m1.1d">italic_u ( 0 ) = 0</annotation></semantics></math>. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S2.T2" title="TABLE II ‣ Time-coded LIF Spiking Function ‣ II-D Spiking Functions ‣ II Neuron model and network architecture ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">II</span></a> summarizes the variables and parameters of the time-coded spiking function and compares them with the alternative spiking functions.</p> </div> <figure class="ltx_table" id="S2.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE I: </span>Neuron states and parameters for neuron model excluding spiking functions</figcaption> <table class="ltx_tabular ltx_align_middle" id="S2.T1.19"> <tr class="ltx_tr" id="S2.T1.19.20"> <td class="ltx_td ltx_align_center" id="S2.T1.19.20.1" style="padding-top:1pt;padding-bottom:1pt;">Variable</td> <td class="ltx_td ltx_align_center" id="S2.T1.19.20.2" style="padding-top:1pt;padding-bottom:1pt;">Dim.</td> <td class="ltx_td ltx_align_center" id="S2.T1.19.20.3" style="padding-top:1pt;padding-bottom:1pt;">Time</td> <td class="ltx_td ltx_align_center" id="S2.T1.19.20.4" style="padding-top:1pt;padding-bottom:1pt;">Description</td> </tr> <tr class="ltx_tr" id="S2.T1.3.3"> <td class="ltx_td ltx_align_right ltx_border_t" id="S2.T1.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\textbf{x}(t)" class="ltx_Math" display="inline" id="S2.T1.1.1.1.m1.1"><semantics id="S2.T1.1.1.1.m1.1a"><mrow id="S2.T1.1.1.1.m1.1.2" xref="S2.T1.1.1.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_bold" id="S2.T1.1.1.1.m1.1.2.2" xref="S2.T1.1.1.1.m1.1.2.2a.cmml">x</mtext><mo id="S2.T1.1.1.1.m1.1.2.1" xref="S2.T1.1.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T1.1.1.1.m1.1.2.3.2" xref="S2.T1.1.1.1.m1.1.2.cmml"><mo id="S2.T1.1.1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T1.1.1.1.m1.1.2.cmml">(</mo><mi id="S2.T1.1.1.1.m1.1.1" xref="S2.T1.1.1.1.m1.1.1.cmml">t</mi><mo id="S2.T1.1.1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T1.1.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.1.1.1.m1.1b"><apply id="S2.T1.1.1.1.m1.1.2.cmml" xref="S2.T1.1.1.1.m1.1.2"><times id="S2.T1.1.1.1.m1.1.2.1.cmml" xref="S2.T1.1.1.1.m1.1.2.1"></times><ci id="S2.T1.1.1.1.m1.1.2.2a.cmml" xref="S2.T1.1.1.1.m1.1.2.2"><mtext class="ltx_mathvariant_bold" id="S2.T1.1.1.1.m1.1.2.2.cmml" xref="S2.T1.1.1.1.m1.1.2.2">x</mtext></ci><ci id="S2.T1.1.1.1.m1.1.1.cmml" xref="S2.T1.1.1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.1.1.1.m1.1c">\textbf{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T1.1.1.1.m1.1d">x ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="r" class="ltx_Math" display="inline" id="S2.T1.2.2.2.m1.1"><semantics id="S2.T1.2.2.2.m1.1a"><mi id="S2.T1.2.2.2.m1.1.1" xref="S2.T1.2.2.2.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.T1.2.2.2.m1.1b"><ci id="S2.T1.2.2.2.m1.1.1.cmml" xref="S2.T1.2.2.2.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.2.2.2.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.T1.2.2.2.m1.1d">italic_r</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.3.3.4" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.3.3.3" style="padding-top:1pt;padding-bottom:1pt;">radar data from <math alttext="N_{\text{vx}}" class="ltx_Math" display="inline" id="S2.T1.3.3.3.m1.1"><semantics id="S2.T1.3.3.3.m1.1a"><msub id="S2.T1.3.3.3.m1.1.1" xref="S2.T1.3.3.3.m1.1.1.cmml"><mi id="S2.T1.3.3.3.m1.1.1.2" xref="S2.T1.3.3.3.m1.1.1.2.cmml">N</mi><mtext id="S2.T1.3.3.3.m1.1.1.3" xref="S2.T1.3.3.3.m1.1.1.3a.cmml">vx</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.T1.3.3.3.m1.1b"><apply id="S2.T1.3.3.3.m1.1.1.cmml" xref="S2.T1.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S2.T1.3.3.3.m1.1.1.1.cmml" xref="S2.T1.3.3.3.m1.1.1">subscript</csymbol><ci id="S2.T1.3.3.3.m1.1.1.2.cmml" xref="S2.T1.3.3.3.m1.1.1.2">𝑁</ci><ci id="S2.T1.3.3.3.m1.1.1.3a.cmml" xref="S2.T1.3.3.3.m1.1.1.3"><mtext id="S2.T1.3.3.3.m1.1.1.3.cmml" mathsize="70%" xref="S2.T1.3.3.3.m1.1.1.3">vx</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.3.3.3.m1.1c">N_{\text{vx}}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.3.3.3.m1.1d">italic_N start_POSTSUBSCRIPT vx end_POSTSUBSCRIPT</annotation></semantics></math> antennas</td> </tr> <tr class="ltx_tr" id="S2.T1.6.6"> <td class="ltx_td ltx_align_right" id="S2.T1.4.4.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="y(t)" class="ltx_Math" display="inline" id="S2.T1.4.4.1.m1.1"><semantics id="S2.T1.4.4.1.m1.1a"><mrow id="S2.T1.4.4.1.m1.1.2" xref="S2.T1.4.4.1.m1.1.2.cmml"><mi id="S2.T1.4.4.1.m1.1.2.2" xref="S2.T1.4.4.1.m1.1.2.2.cmml">y</mi><mo id="S2.T1.4.4.1.m1.1.2.1" xref="S2.T1.4.4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T1.4.4.1.m1.1.2.3.2" xref="S2.T1.4.4.1.m1.1.2.cmml"><mo id="S2.T1.4.4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T1.4.4.1.m1.1.2.cmml">(</mo><mi id="S2.T1.4.4.1.m1.1.1" xref="S2.T1.4.4.1.m1.1.1.cmml">t</mi><mo id="S2.T1.4.4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T1.4.4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.4.4.1.m1.1b"><apply id="S2.T1.4.4.1.m1.1.2.cmml" xref="S2.T1.4.4.1.m1.1.2"><times id="S2.T1.4.4.1.m1.1.2.1.cmml" xref="S2.T1.4.4.1.m1.1.2.1"></times><ci id="S2.T1.4.4.1.m1.1.2.2.cmml" xref="S2.T1.4.4.1.m1.1.2.2">𝑦</ci><ci id="S2.T1.4.4.1.m1.1.1.cmml" xref="S2.T1.4.4.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.4.4.1.m1.1c">y(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T1.4.4.1.m1.1d">italic_y ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.5.5.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T1.5.5.2.m1.1"><semantics id="S2.T1.5.5.2.m1.1a"><mn id="S2.T1.5.5.2.m1.1.1" xref="S2.T1.5.5.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T1.5.5.2.m1.1b"><cn id="S2.T1.5.5.2.m1.1.1.cmml" type="integer" xref="S2.T1.5.5.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.5.5.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.5.5.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.6.6.4" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T1.6.6.3" style="padding-top:1pt;padding-bottom:1pt;">phase-shifted radar data <math alttext="y(t)=\vec{w}\vec{x}(t)" class="ltx_Math" display="inline" 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stretchy="false" xref="S2.T1.6.6.3.m1.2.3.3.2.1.cmml">→</mo></mover><mo id="S2.T1.6.6.3.m1.2.3.3.1" xref="S2.T1.6.6.3.m1.2.3.3.1.cmml">⁢</mo><mover accent="true" id="S2.T1.6.6.3.m1.2.3.3.3" xref="S2.T1.6.6.3.m1.2.3.3.3.cmml"><mi id="S2.T1.6.6.3.m1.2.3.3.3.2" xref="S2.T1.6.6.3.m1.2.3.3.3.2.cmml">x</mi><mo id="S2.T1.6.6.3.m1.2.3.3.3.1" stretchy="false" xref="S2.T1.6.6.3.m1.2.3.3.3.1.cmml">→</mo></mover><mo id="S2.T1.6.6.3.m1.2.3.3.1a" xref="S2.T1.6.6.3.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.T1.6.6.3.m1.2.3.3.4.2" xref="S2.T1.6.6.3.m1.2.3.3.cmml"><mo id="S2.T1.6.6.3.m1.2.3.3.4.2.1" stretchy="false" xref="S2.T1.6.6.3.m1.2.3.3.cmml">(</mo><mi id="S2.T1.6.6.3.m1.2.2" xref="S2.T1.6.6.3.m1.2.2.cmml">t</mi><mo id="S2.T1.6.6.3.m1.2.3.3.4.2.2" stretchy="false" xref="S2.T1.6.6.3.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.6.6.3.m1.2b"><apply id="S2.T1.6.6.3.m1.2.3.cmml" xref="S2.T1.6.6.3.m1.2.3"><eq id="S2.T1.6.6.3.m1.2.3.1.cmml" xref="S2.T1.6.6.3.m1.2.3.1"></eq><apply id="S2.T1.6.6.3.m1.2.3.2.cmml" xref="S2.T1.6.6.3.m1.2.3.2"><times id="S2.T1.6.6.3.m1.2.3.2.1.cmml" xref="S2.T1.6.6.3.m1.2.3.2.1"></times><ci id="S2.T1.6.6.3.m1.2.3.2.2.cmml" xref="S2.T1.6.6.3.m1.2.3.2.2">𝑦</ci><ci id="S2.T1.6.6.3.m1.1.1.cmml" xref="S2.T1.6.6.3.m1.1.1">𝑡</ci></apply><apply id="S2.T1.6.6.3.m1.2.3.3.cmml" xref="S2.T1.6.6.3.m1.2.3.3"><times id="S2.T1.6.6.3.m1.2.3.3.1.cmml" xref="S2.T1.6.6.3.m1.2.3.3.1"></times><apply id="S2.T1.6.6.3.m1.2.3.3.2.cmml" xref="S2.T1.6.6.3.m1.2.3.3.2"><ci id="S2.T1.6.6.3.m1.2.3.3.2.1.cmml" xref="S2.T1.6.6.3.m1.2.3.3.2.1">→</ci><ci id="S2.T1.6.6.3.m1.2.3.3.2.2.cmml" xref="S2.T1.6.6.3.m1.2.3.3.2.2">𝑤</ci></apply><apply id="S2.T1.6.6.3.m1.2.3.3.3.cmml" xref="S2.T1.6.6.3.m1.2.3.3.3"><ci id="S2.T1.6.6.3.m1.2.3.3.3.1.cmml" xref="S2.T1.6.6.3.m1.2.3.3.3.1">→</ci><ci id="S2.T1.6.6.3.m1.2.3.3.3.2.cmml" xref="S2.T1.6.6.3.m1.2.3.3.3.2">𝑥</ci></apply><ci id="S2.T1.6.6.3.m1.2.2.cmml" xref="S2.T1.6.6.3.m1.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.6.6.3.m1.2c">y(t)=\vec{w}\vec{x}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T1.6.6.3.m1.2d">italic_y ( italic_t ) = over→ start_ARG italic_w end_ARG over→ start_ARG italic_x end_ARG ( italic_t )</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S2.T1.8.8"> <td class="ltx_td ltx_align_right" id="S2.T1.7.7.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\textbf{s}(t)" class="ltx_Math" display="inline" id="S2.T1.7.7.1.m1.1"><semantics id="S2.T1.7.7.1.m1.1a"><mrow id="S2.T1.7.7.1.m1.1.2" xref="S2.T1.7.7.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_bold" id="S2.T1.7.7.1.m1.1.2.2" xref="S2.T1.7.7.1.m1.1.2.2a.cmml">s</mtext><mo id="S2.T1.7.7.1.m1.1.2.1" xref="S2.T1.7.7.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T1.7.7.1.m1.1.2.3.2" xref="S2.T1.7.7.1.m1.1.2.cmml"><mo id="S2.T1.7.7.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T1.7.7.1.m1.1.2.cmml">(</mo><mi id="S2.T1.7.7.1.m1.1.1" xref="S2.T1.7.7.1.m1.1.1.cmml">t</mi><mo id="S2.T1.7.7.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T1.7.7.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.7.7.1.m1.1b"><apply id="S2.T1.7.7.1.m1.1.2.cmml" xref="S2.T1.7.7.1.m1.1.2"><times id="S2.T1.7.7.1.m1.1.2.1.cmml" xref="S2.T1.7.7.1.m1.1.2.1"></times><ci id="S2.T1.7.7.1.m1.1.2.2a.cmml" xref="S2.T1.7.7.1.m1.1.2.2"><mtext class="ltx_mathvariant_bold" id="S2.T1.7.7.1.m1.1.2.2.cmml" xref="S2.T1.7.7.1.m1.1.2.2">s</mtext></ci><ci id="S2.T1.7.7.1.m1.1.1.cmml" xref="S2.T1.7.7.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.7.7.1.m1.1c">\textbf{s}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T1.7.7.1.m1.1d">s ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.8.8.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="2" class="ltx_Math" display="inline" id="S2.T1.8.8.2.m1.1"><semantics id="S2.T1.8.8.2.m1.1a"><mn id="S2.T1.8.8.2.m1.1.1" xref="S2.T1.8.8.2.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S2.T1.8.8.2.m1.1b"><cn id="S2.T1.8.8.2.m1.1.1.cmml" type="integer" xref="S2.T1.8.8.2.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.8.8.2.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S2.T1.8.8.2.m1.1d">2</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.8.8.3" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T1.8.8.4" style="padding-top:1pt;padding-bottom:1pt;">neuron state of RF neuron</td> </tr> <tr class="ltx_tr" id="S2.T1.10.10"> <td class="ltx_td ltx_align_right" id="S2.T1.9.9.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\|s(t)\|" class="ltx_Math" display="inline" id="S2.T1.9.9.1.m1.2"><semantics id="S2.T1.9.9.1.m1.2a"><mrow id="S2.T1.9.9.1.m1.2.2.1" xref="S2.T1.9.9.1.m1.2.2.2.cmml"><mo id="S2.T1.9.9.1.m1.2.2.1.2" stretchy="false" xref="S2.T1.9.9.1.m1.2.2.2.1.cmml">‖</mo><mrow id="S2.T1.9.9.1.m1.2.2.1.1" xref="S2.T1.9.9.1.m1.2.2.1.1.cmml"><mi id="S2.T1.9.9.1.m1.2.2.1.1.2" xref="S2.T1.9.9.1.m1.2.2.1.1.2.cmml">s</mi><mo id="S2.T1.9.9.1.m1.2.2.1.1.1" xref="S2.T1.9.9.1.m1.2.2.1.1.1.cmml">⁢</mo><mrow id="S2.T1.9.9.1.m1.2.2.1.1.3.2" xref="S2.T1.9.9.1.m1.2.2.1.1.cmml"><mo id="S2.T1.9.9.1.m1.2.2.1.1.3.2.1" stretchy="false" xref="S2.T1.9.9.1.m1.2.2.1.1.cmml">(</mo><mi id="S2.T1.9.9.1.m1.1.1" xref="S2.T1.9.9.1.m1.1.1.cmml">t</mi><mo id="S2.T1.9.9.1.m1.2.2.1.1.3.2.2" stretchy="false" xref="S2.T1.9.9.1.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.T1.9.9.1.m1.2.2.1.3" stretchy="false" xref="S2.T1.9.9.1.m1.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.9.9.1.m1.2b"><apply id="S2.T1.9.9.1.m1.2.2.2.cmml" xref="S2.T1.9.9.1.m1.2.2.1"><csymbol cd="latexml" id="S2.T1.9.9.1.m1.2.2.2.1.cmml" xref="S2.T1.9.9.1.m1.2.2.1.2">norm</csymbol><apply id="S2.T1.9.9.1.m1.2.2.1.1.cmml" xref="S2.T1.9.9.1.m1.2.2.1.1"><times id="S2.T1.9.9.1.m1.2.2.1.1.1.cmml" xref="S2.T1.9.9.1.m1.2.2.1.1.1"></times><ci id="S2.T1.9.9.1.m1.2.2.1.1.2.cmml" xref="S2.T1.9.9.1.m1.2.2.1.1.2">𝑠</ci><ci id="S2.T1.9.9.1.m1.1.1.cmml" xref="S2.T1.9.9.1.m1.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.9.9.1.m1.2c">\|s(t)\|</annotation><annotation encoding="application/x-llamapun" id="S2.T1.9.9.1.m1.2d">∥ italic_s ( italic_t ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.10.10.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T1.10.10.2.m1.1"><semantics id="S2.T1.10.10.2.m1.1a"><mn id="S2.T1.10.10.2.m1.1.1" xref="S2.T1.10.10.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T1.10.10.2.m1.1b"><cn id="S2.T1.10.10.2.m1.1.1.cmml" type="integer" xref="S2.T1.10.10.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.10.10.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.10.10.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.10.10.3" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T1.10.10.4" style="padding-top:1pt;padding-bottom:1pt;">magnitude of neuron state</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12"> <td class="ltx_td ltx_align_right" id="S2.T1.11.11.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="s_{\text{max}}(t)" class="ltx_Math" display="inline" id="S2.T1.11.11.1.m1.1"><semantics id="S2.T1.11.11.1.m1.1a"><mrow id="S2.T1.11.11.1.m1.1.2" xref="S2.T1.11.11.1.m1.1.2.cmml"><msub id="S2.T1.11.11.1.m1.1.2.2" xref="S2.T1.11.11.1.m1.1.2.2.cmml"><mi id="S2.T1.11.11.1.m1.1.2.2.2" xref="S2.T1.11.11.1.m1.1.2.2.2.cmml">s</mi><mtext id="S2.T1.11.11.1.m1.1.2.2.3" 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xref="S2.T1.13.13.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T1.13.13.1.m1.1.2.3.2" xref="S2.T1.13.13.1.m1.1.2.cmml"><mo id="S2.T1.13.13.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T1.13.13.1.m1.1.2.cmml">(</mo><mi id="S2.T1.13.13.1.m1.1.1" xref="S2.T1.13.13.1.m1.1.1.cmml">t</mi><mo id="S2.T1.13.13.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T1.13.13.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.13.13.1.m1.1b"><apply id="S2.T1.13.13.1.m1.1.2.cmml" xref="S2.T1.13.13.1.m1.1.2"><times id="S2.T1.13.13.1.m1.1.2.1.cmml" xref="S2.T1.13.13.1.m1.1.2.1"></times><apply id="S2.T1.13.13.1.m1.1.2.2.cmml" xref="S2.T1.13.13.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.T1.13.13.1.m1.1.2.2.1.cmml" xref="S2.T1.13.13.1.m1.1.2.2">subscript</csymbol><ci id="S2.T1.13.13.1.m1.1.2.2.2.cmml" xref="S2.T1.13.13.1.m1.1.2.2.2">𝑤</ci><ci id="S2.T1.13.13.1.m1.1.2.2.3a.cmml" xref="S2.T1.13.13.1.m1.1.2.2.3"><mtext id="S2.T1.13.13.1.m1.1.2.2.3.cmml" mathsize="70%" xref="S2.T1.13.13.1.m1.1.2.2.3">max</mtext></ci></apply><ci id="S2.T1.13.13.1.m1.1.1.cmml" xref="S2.T1.13.13.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.13.13.1.m1.1c">w_{\text{max}}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T1.13.13.1.m1.1d">italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.14.14.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T1.14.14.2.m1.1"><semantics id="S2.T1.14.14.2.m1.1a"><mn id="S2.T1.14.14.2.m1.1.1" xref="S2.T1.14.14.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T1.14.14.2.m1.1b"><cn id="S2.T1.14.14.2.m1.1.1.cmml" type="integer" xref="S2.T1.14.14.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.14.14.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.14.14.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.14.14.3" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T1.14.14.4" style="padding-top:1pt;padding-bottom:1pt;"> <span class="ltx_text" id="S2.T1.14.14.4.1"></span><span class="ltx_text" id="S2.T1.14.14.4.2"> <span class="ltx_tabular ltx_align_top" id="S2.T1.14.14.4.2.1"> <span class="ltx_tr" id="S2.T1.14.14.4.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S2.T1.14.14.4.2.1.1.1" style="padding-top:1pt;padding-bottom:1pt;">estimated maximum of sum of width between</span></span> <span class="ltx_tr" id="S2.T1.14.14.4.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S2.T1.14.14.4.2.1.2.1" style="padding-top:1pt;padding-bottom:1pt;">magnitude and its maximum</span></span> </span></span><span class="ltx_text" id="S2.T1.14.14.4.3"></span></td> </tr> <tr class="ltx_tr" id="S2.T1.17.17"> <td class="ltx_td ltx_align_right ltx_border_t" id="S2.T1.15.15.1" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_markedasmath ltx_font_bold" id="S2.T1.15.15.1.1">w</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.16.16.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="r" class="ltx_Math" display="inline" id="S2.T1.16.16.2.m1.1"><semantics id="S2.T1.16.16.2.m1.1a"><mi id="S2.T1.16.16.2.m1.1.1" xref="S2.T1.16.16.2.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.T1.16.16.2.m1.1b"><ci id="S2.T1.16.16.2.m1.1.1.cmml" xref="S2.T1.16.16.2.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.16.16.2.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.T1.16.16.2.m1.1d">italic_r</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.17.17.4" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T1.17.17.3" style="padding-top:1pt;padding-bottom:1pt;">weight vector of complex weight matrix <math alttext="W" class="ltx_Math" display="inline" id="S2.T1.17.17.3.m1.1"><semantics id="S2.T1.17.17.3.m1.1a"><mi id="S2.T1.17.17.3.m1.1.1" xref="S2.T1.17.17.3.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.T1.17.17.3.m1.1b"><ci id="S2.T1.17.17.3.m1.1.1.cmml" xref="S2.T1.17.17.3.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.17.17.3.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.T1.17.17.3.m1.1d">italic_W</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S2.T1.19.19"> <td class="ltx_td ltx_align_right" id="S2.T1.18.18.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\omega" class="ltx_Math" display="inline" id="S2.T1.18.18.1.m1.1"><semantics id="S2.T1.18.18.1.m1.1a"><mi id="S2.T1.18.18.1.m1.1.1" xref="S2.T1.18.18.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S2.T1.18.18.1.m1.1b"><ci id="S2.T1.18.18.1.m1.1.1.cmml" xref="S2.T1.18.18.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.18.18.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S2.T1.18.18.1.m1.1d">italic_ω</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.19.19.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T1.19.19.2.m1.1"><semantics id="S2.T1.19.19.2.m1.1a"><mn id="S2.T1.19.19.2.m1.1.1" xref="S2.T1.19.19.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T1.19.19.2.m1.1b"><cn id="S2.T1.19.19.2.m1.1.1.cmml" type="integer" xref="S2.T1.19.19.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.19.19.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.19.19.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T1.19.19.3" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left" id="S2.T1.19.19.4" style="padding-top:1pt;padding-bottom:1pt;">eigenfrequency</td> </tr> </table> </figure> <figure class="ltx_table" id="S2.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE II: </span>Neuron states and parameters for spiking functions</figcaption> <table class="ltx_tabular ltx_align_middle" id="S2.T2.28"> <tr class="ltx_tr" id="S2.T2.28.29"> <td class="ltx_td ltx_align_center ltx_border_r" colspan="3" id="S2.T2.28.29.1" style="padding-top:1pt;padding-bottom:1pt;">Adaptive threshold</td> <td class="ltx_td ltx_align_center ltx_border_r" colspan="3" id="S2.T2.28.29.2" style="padding-top:1pt;padding-bottom:1pt;">Rate-coded LIF</td> <td class="ltx_td ltx_align_center" colspan="3" id="S2.T2.28.29.3" style="padding-top:1pt;padding-bottom:1pt;">Time-coded LIF</td> </tr> <tr class="ltx_tr" id="S2.T2.28.30"> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.1" style="padding-top:1pt;padding-bottom:1pt;">Var.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.2" style="padding-top:1pt;padding-bottom:1pt;">Dim.</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.28.30.3" style="padding-top:1pt;padding-bottom:1pt;">Time</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.4" style="padding-top:1pt;padding-bottom:1pt;">Var.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.5" style="padding-top:1pt;padding-bottom:1pt;">Dim.</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.28.30.6" style="padding-top:1pt;padding-bottom:1pt;">Time</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.7" style="padding-top:1pt;padding-bottom:1pt;">Var.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.8" style="padding-top:1pt;padding-bottom:1pt;">Dim.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.28.30.9" style="padding-top:1pt;padding-bottom:1pt;">Time</td> </tr> <tr class="ltx_tr" id="S2.T2.4.4"> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.4.4.5" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.4.4.6" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.4.4.7" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="g(t)" class="ltx_Math" display="inline" id="S2.T2.1.1.1.m1.1"><semantics id="S2.T2.1.1.1.m1.1a"><mrow id="S2.T2.1.1.1.m1.1.2" xref="S2.T2.1.1.1.m1.1.2.cmml"><mi id="S2.T2.1.1.1.m1.1.2.2" xref="S2.T2.1.1.1.m1.1.2.2.cmml">g</mi><mo id="S2.T2.1.1.1.m1.1.2.1" xref="S2.T2.1.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T2.1.1.1.m1.1.2.3.2" xref="S2.T2.1.1.1.m1.1.2.cmml"><mo id="S2.T2.1.1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T2.1.1.1.m1.1.2.cmml">(</mo><mi id="S2.T2.1.1.1.m1.1.1" xref="S2.T2.1.1.1.m1.1.1.cmml">t</mi><mo id="S2.T2.1.1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T2.1.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T2.1.1.1.m1.1b"><apply id="S2.T2.1.1.1.m1.1.2.cmml" xref="S2.T2.1.1.1.m1.1.2"><times id="S2.T2.1.1.1.m1.1.2.1.cmml" xref="S2.T2.1.1.1.m1.1.2.1"></times><ci id="S2.T2.1.1.1.m1.1.2.2.cmml" xref="S2.T2.1.1.1.m1.1.2.2">𝑔</ci><ci id="S2.T2.1.1.1.m1.1.1.cmml" xref="S2.T2.1.1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.1.1.1.m1.1c">g(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T2.1.1.1.m1.1d">italic_g ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.2.2.2.m1.1"><semantics id="S2.T2.2.2.2.m1.1a"><mn id="S2.T2.2.2.2.m1.1.1" xref="S2.T2.2.2.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.2.2.2.m1.1b"><cn id="S2.T2.2.2.2.m1.1.1.cmml" type="integer" xref="S2.T2.2.2.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.2.2.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.2.2.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.4.4.8" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.3.3.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="g(t)" class="ltx_Math" display="inline" id="S2.T2.3.3.3.m1.1"><semantics id="S2.T2.3.3.3.m1.1a"><mrow id="S2.T2.3.3.3.m1.1.2" xref="S2.T2.3.3.3.m1.1.2.cmml"><mi id="S2.T2.3.3.3.m1.1.2.2" xref="S2.T2.3.3.3.m1.1.2.2.cmml">g</mi><mo id="S2.T2.3.3.3.m1.1.2.1" xref="S2.T2.3.3.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T2.3.3.3.m1.1.2.3.2" xref="S2.T2.3.3.3.m1.1.2.cmml"><mo id="S2.T2.3.3.3.m1.1.2.3.2.1" stretchy="false" xref="S2.T2.3.3.3.m1.1.2.cmml">(</mo><mi id="S2.T2.3.3.3.m1.1.1" xref="S2.T2.3.3.3.m1.1.1.cmml">t</mi><mo id="S2.T2.3.3.3.m1.1.2.3.2.2" stretchy="false" xref="S2.T2.3.3.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T2.3.3.3.m1.1b"><apply id="S2.T2.3.3.3.m1.1.2.cmml" xref="S2.T2.3.3.3.m1.1.2"><times id="S2.T2.3.3.3.m1.1.2.1.cmml" xref="S2.T2.3.3.3.m1.1.2.1"></times><ci id="S2.T2.3.3.3.m1.1.2.2.cmml" xref="S2.T2.3.3.3.m1.1.2.2">𝑔</ci><ci id="S2.T2.3.3.3.m1.1.1.cmml" xref="S2.T2.3.3.3.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.3.3.3.m1.1c">g(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T2.3.3.3.m1.1d">italic_g ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.4.4.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.4.4.4.m1.1"><semantics id="S2.T2.4.4.4.m1.1a"><mn id="S2.T2.4.4.4.m1.1.1" xref="S2.T2.4.4.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.4.4.4.m1.1b"><cn id="S2.T2.4.4.4.m1.1.1.cmml" type="integer" xref="S2.T2.4.4.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.4.4.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.4.4.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.4.4.9" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> </tr> <tr class="ltx_tr" id="S2.T2.10.10"> <td class="ltx_td ltx_align_left" id="S2.T2.5.5.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\Gamma^{\pm}(t)" class="ltx_Math" display="inline" id="S2.T2.5.5.1.m1.1"><semantics id="S2.T2.5.5.1.m1.1a"><mrow id="S2.T2.5.5.1.m1.1.2" xref="S2.T2.5.5.1.m1.1.2.cmml"><msup id="S2.T2.5.5.1.m1.1.2.2" xref="S2.T2.5.5.1.m1.1.2.2.cmml"><mi id="S2.T2.5.5.1.m1.1.2.2.2" mathvariant="normal" xref="S2.T2.5.5.1.m1.1.2.2.2.cmml">Γ</mi><mo id="S2.T2.5.5.1.m1.1.2.2.3" xref="S2.T2.5.5.1.m1.1.2.2.3.cmml">±</mo></msup><mo id="S2.T2.5.5.1.m1.1.2.1" xref="S2.T2.5.5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T2.5.5.1.m1.1.2.3.2" xref="S2.T2.5.5.1.m1.1.2.cmml"><mo id="S2.T2.5.5.1.m1.1.2.3.2.1" stretchy="false" xref="S2.T2.5.5.1.m1.1.2.cmml">(</mo><mi id="S2.T2.5.5.1.m1.1.1" xref="S2.T2.5.5.1.m1.1.1.cmml">t</mi><mo id="S2.T2.5.5.1.m1.1.2.3.2.2" stretchy="false" xref="S2.T2.5.5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T2.5.5.1.m1.1b"><apply id="S2.T2.5.5.1.m1.1.2.cmml" xref="S2.T2.5.5.1.m1.1.2"><times id="S2.T2.5.5.1.m1.1.2.1.cmml" xref="S2.T2.5.5.1.m1.1.2.1"></times><apply id="S2.T2.5.5.1.m1.1.2.2.cmml" xref="S2.T2.5.5.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.T2.5.5.1.m1.1.2.2.1.cmml" xref="S2.T2.5.5.1.m1.1.2.2">superscript</csymbol><ci id="S2.T2.5.5.1.m1.1.2.2.2.cmml" xref="S2.T2.5.5.1.m1.1.2.2.2">Γ</ci><csymbol cd="latexml" id="S2.T2.5.5.1.m1.1.2.2.3.cmml" xref="S2.T2.5.5.1.m1.1.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.T2.5.5.1.m1.1.1.cmml" xref="S2.T2.5.5.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.5.5.1.m1.1c">\Gamma^{\pm}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T2.5.5.1.m1.1d">roman_Γ start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.6.6.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="2" class="ltx_Math" display="inline" id="S2.T2.6.6.2.m1.1"><semantics id="S2.T2.6.6.2.m1.1a"><mn id="S2.T2.6.6.2.m1.1.1" xref="S2.T2.6.6.2.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S2.T2.6.6.2.m1.1b"><cn id="S2.T2.6.6.2.m1.1.1.cmml" type="integer" xref="S2.T2.6.6.2.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.6.6.2.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S2.T2.6.6.2.m1.1d">2</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.10.10.7" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T2.7.7.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u(t)" class="ltx_Math" display="inline" id="S2.T2.7.7.3.m1.1"><semantics id="S2.T2.7.7.3.m1.1a"><mrow id="S2.T2.7.7.3.m1.1.2" xref="S2.T2.7.7.3.m1.1.2.cmml"><mi id="S2.T2.7.7.3.m1.1.2.2" xref="S2.T2.7.7.3.m1.1.2.2.cmml">u</mi><mo id="S2.T2.7.7.3.m1.1.2.1" xref="S2.T2.7.7.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T2.7.7.3.m1.1.2.3.2" xref="S2.T2.7.7.3.m1.1.2.cmml"><mo id="S2.T2.7.7.3.m1.1.2.3.2.1" stretchy="false" xref="S2.T2.7.7.3.m1.1.2.cmml">(</mo><mi id="S2.T2.7.7.3.m1.1.1" xref="S2.T2.7.7.3.m1.1.1.cmml">t</mi><mo id="S2.T2.7.7.3.m1.1.2.3.2.2" stretchy="false" xref="S2.T2.7.7.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T2.7.7.3.m1.1b"><apply id="S2.T2.7.7.3.m1.1.2.cmml" xref="S2.T2.7.7.3.m1.1.2"><times id="S2.T2.7.7.3.m1.1.2.1.cmml" xref="S2.T2.7.7.3.m1.1.2.1"></times><ci id="S2.T2.7.7.3.m1.1.2.2.cmml" xref="S2.T2.7.7.3.m1.1.2.2">𝑢</ci><ci id="S2.T2.7.7.3.m1.1.1.cmml" xref="S2.T2.7.7.3.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.7.7.3.m1.1c">u(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T2.7.7.3.m1.1d">italic_u ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.8.8.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.8.8.4.m1.1"><semantics id="S2.T2.8.8.4.m1.1a"><mn id="S2.T2.8.8.4.m1.1.1" xref="S2.T2.8.8.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.8.8.4.m1.1b"><cn id="S2.T2.8.8.4.m1.1.1.cmml" type="integer" xref="S2.T2.8.8.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.8.8.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.8.8.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.10.10.8" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> <td class="ltx_td ltx_align_left" id="S2.T2.9.9.5" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u(t)" class="ltx_Math" display="inline" id="S2.T2.9.9.5.m1.1"><semantics id="S2.T2.9.9.5.m1.1a"><mrow id="S2.T2.9.9.5.m1.1.2" xref="S2.T2.9.9.5.m1.1.2.cmml"><mi id="S2.T2.9.9.5.m1.1.2.2" xref="S2.T2.9.9.5.m1.1.2.2.cmml">u</mi><mo id="S2.T2.9.9.5.m1.1.2.1" xref="S2.T2.9.9.5.m1.1.2.1.cmml">⁢</mo><mrow id="S2.T2.9.9.5.m1.1.2.3.2" xref="S2.T2.9.9.5.m1.1.2.cmml"><mo id="S2.T2.9.9.5.m1.1.2.3.2.1" stretchy="false" xref="S2.T2.9.9.5.m1.1.2.cmml">(</mo><mi id="S2.T2.9.9.5.m1.1.1" xref="S2.T2.9.9.5.m1.1.1.cmml">t</mi><mo id="S2.T2.9.9.5.m1.1.2.3.2.2" stretchy="false" xref="S2.T2.9.9.5.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T2.9.9.5.m1.1b"><apply id="S2.T2.9.9.5.m1.1.2.cmml" xref="S2.T2.9.9.5.m1.1.2"><times id="S2.T2.9.9.5.m1.1.2.1.cmml" xref="S2.T2.9.9.5.m1.1.2.1"></times><ci id="S2.T2.9.9.5.m1.1.2.2.cmml" xref="S2.T2.9.9.5.m1.1.2.2">𝑢</ci><ci id="S2.T2.9.9.5.m1.1.1.cmml" xref="S2.T2.9.9.5.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.9.9.5.m1.1c">u(t)</annotation><annotation encoding="application/x-llamapun" id="S2.T2.9.9.5.m1.1d">italic_u ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.10.10.6" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.10.10.6.m1.1"><semantics id="S2.T2.10.10.6.m1.1a"><mn id="S2.T2.10.10.6.m1.1.1" xref="S2.T2.10.10.6.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.10.10.6.m1.1b"><cn id="S2.T2.10.10.6.m1.1.1.cmml" type="integer" xref="S2.T2.10.10.6.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.10.10.6.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.10.10.6.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.10.10.9" style="padding-top:1pt;padding-bottom:1pt;">dyn.</td> </tr> <tr class="ltx_tr" id="S2.T2.14.14"> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.14.14.5" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.14.14.6" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.14.14.7" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.11.11.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\alpha_{g}" class="ltx_Math" display="inline" id="S2.T2.11.11.1.m1.1"><semantics id="S2.T2.11.11.1.m1.1a"><msub id="S2.T2.11.11.1.m1.1.1" xref="S2.T2.11.11.1.m1.1.1.cmml"><mi id="S2.T2.11.11.1.m1.1.1.2" xref="S2.T2.11.11.1.m1.1.1.2.cmml">α</mi><mi id="S2.T2.11.11.1.m1.1.1.3" xref="S2.T2.11.11.1.m1.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S2.T2.11.11.1.m1.1b"><apply id="S2.T2.11.11.1.m1.1.1.cmml" xref="S2.T2.11.11.1.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.11.11.1.m1.1.1.1.cmml" xref="S2.T2.11.11.1.m1.1.1">subscript</csymbol><ci id="S2.T2.11.11.1.m1.1.1.2.cmml" xref="S2.T2.11.11.1.m1.1.1.2">𝛼</ci><ci id="S2.T2.11.11.1.m1.1.1.3.cmml" xref="S2.T2.11.11.1.m1.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.11.11.1.m1.1c">\alpha_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.11.11.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.12.12.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.12.12.2.m1.1"><semantics id="S2.T2.12.12.2.m1.1a"><mn id="S2.T2.12.12.2.m1.1.1" xref="S2.T2.12.12.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.12.12.2.m1.1b"><cn id="S2.T2.12.12.2.m1.1.1.cmml" type="integer" xref="S2.T2.12.12.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.12.12.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.12.12.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T2.14.14.8" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.13.13.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\alpha_{g}" class="ltx_Math" display="inline" id="S2.T2.13.13.3.m1.1"><semantics id="S2.T2.13.13.3.m1.1a"><msub id="S2.T2.13.13.3.m1.1.1" xref="S2.T2.13.13.3.m1.1.1.cmml"><mi id="S2.T2.13.13.3.m1.1.1.2" xref="S2.T2.13.13.3.m1.1.1.2.cmml">α</mi><mi id="S2.T2.13.13.3.m1.1.1.3" xref="S2.T2.13.13.3.m1.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S2.T2.13.13.3.m1.1b"><apply id="S2.T2.13.13.3.m1.1.1.cmml" xref="S2.T2.13.13.3.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.13.13.3.m1.1.1.1.cmml" xref="S2.T2.13.13.3.m1.1.1">subscript</csymbol><ci id="S2.T2.13.13.3.m1.1.1.2.cmml" xref="S2.T2.13.13.3.m1.1.1.2">𝛼</ci><ci id="S2.T2.13.13.3.m1.1.1.3.cmml" xref="S2.T2.13.13.3.m1.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.13.13.3.m1.1c">\alpha_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.13.13.3.m1.1d">italic_α start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.14.14.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.14.14.4.m1.1"><semantics id="S2.T2.14.14.4.m1.1a"><mn id="S2.T2.14.14.4.m1.1.1" xref="S2.T2.14.14.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.14.14.4.m1.1b"><cn id="S2.T2.14.14.4.m1.1.1.cmml" type="integer" xref="S2.T2.14.14.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.14.14.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.14.14.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S2.T2.14.14.9" style="padding-top:1pt;padding-bottom:1pt;">static</td> </tr> <tr class="ltx_tr" id="S2.T2.20.20"> <td class="ltx_td ltx_align_left" id="S2.T2.15.15.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\gamma" class="ltx_Math" display="inline" id="S2.T2.15.15.1.m1.1"><semantics id="S2.T2.15.15.1.m1.1a"><mi id="S2.T2.15.15.1.m1.1.1" xref="S2.T2.15.15.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.T2.15.15.1.m1.1b"><ci id="S2.T2.15.15.1.m1.1.1.cmml" xref="S2.T2.15.15.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.15.15.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.T2.15.15.1.m1.1d">italic_γ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.16.16.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.16.16.2.m1.1"><semantics id="S2.T2.16.16.2.m1.1a"><mn id="S2.T2.16.16.2.m1.1.1" xref="S2.T2.16.16.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.16.16.2.m1.1b"><cn id="S2.T2.16.16.2.m1.1.1.cmml" type="integer" xref="S2.T2.16.16.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.16.16.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.16.16.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.20.20.7" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left" id="S2.T2.17.17.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u_{\text{th}}" class="ltx_Math" display="inline" id="S2.T2.17.17.3.m1.1"><semantics id="S2.T2.17.17.3.m1.1a"><msub id="S2.T2.17.17.3.m1.1.1" xref="S2.T2.17.17.3.m1.1.1.cmml"><mi id="S2.T2.17.17.3.m1.1.1.2" xref="S2.T2.17.17.3.m1.1.1.2.cmml">u</mi><mtext id="S2.T2.17.17.3.m1.1.1.3" xref="S2.T2.17.17.3.m1.1.1.3a.cmml">th</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.T2.17.17.3.m1.1b"><apply id="S2.T2.17.17.3.m1.1.1.cmml" xref="S2.T2.17.17.3.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.17.17.3.m1.1.1.1.cmml" xref="S2.T2.17.17.3.m1.1.1">subscript</csymbol><ci id="S2.T2.17.17.3.m1.1.1.2.cmml" xref="S2.T2.17.17.3.m1.1.1.2">𝑢</ci><ci id="S2.T2.17.17.3.m1.1.1.3a.cmml" xref="S2.T2.17.17.3.m1.1.1.3"><mtext id="S2.T2.17.17.3.m1.1.1.3.cmml" mathsize="70%" xref="S2.T2.17.17.3.m1.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.17.17.3.m1.1c">u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.17.17.3.m1.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.18.18.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.18.18.4.m1.1"><semantics id="S2.T2.18.18.4.m1.1a"><mn id="S2.T2.18.18.4.m1.1.1" xref="S2.T2.18.18.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.18.18.4.m1.1b"><cn id="S2.T2.18.18.4.m1.1.1.cmml" type="integer" xref="S2.T2.18.18.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.18.18.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.18.18.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.20.20.8" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left" id="S2.T2.19.19.5" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u_{\text{th}}" class="ltx_Math" display="inline" id="S2.T2.19.19.5.m1.1"><semantics id="S2.T2.19.19.5.m1.1a"><msub id="S2.T2.19.19.5.m1.1.1" xref="S2.T2.19.19.5.m1.1.1.cmml"><mi id="S2.T2.19.19.5.m1.1.1.2" xref="S2.T2.19.19.5.m1.1.1.2.cmml">u</mi><mtext id="S2.T2.19.19.5.m1.1.1.3" xref="S2.T2.19.19.5.m1.1.1.3a.cmml">th</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.T2.19.19.5.m1.1b"><apply id="S2.T2.19.19.5.m1.1.1.cmml" xref="S2.T2.19.19.5.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.19.19.5.m1.1.1.1.cmml" xref="S2.T2.19.19.5.m1.1.1">subscript</csymbol><ci id="S2.T2.19.19.5.m1.1.1.2.cmml" xref="S2.T2.19.19.5.m1.1.1.2">𝑢</ci><ci id="S2.T2.19.19.5.m1.1.1.3a.cmml" xref="S2.T2.19.19.5.m1.1.1.3"><mtext id="S2.T2.19.19.5.m1.1.1.3.cmml" mathsize="70%" xref="S2.T2.19.19.5.m1.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.19.19.5.m1.1c">u_{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.19.19.5.m1.1d">italic_u start_POSTSUBSCRIPT th end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.20.20.6" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.20.20.6.m1.1"><semantics id="S2.T2.20.20.6.m1.1a"><mn id="S2.T2.20.20.6.m1.1.1" xref="S2.T2.20.20.6.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.20.20.6.m1.1b"><cn id="S2.T2.20.20.6.m1.1.1.cmml" type="integer" xref="S2.T2.20.20.6.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.20.20.6.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.20.20.6.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.20.20.9" style="padding-top:1pt;padding-bottom:1pt;">static</td> </tr> <tr class="ltx_tr" id="S2.T2.24.24"> <td class="ltx_td ltx_align_left" id="S2.T2.24.24.5" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left" id="S2.T2.24.24.6" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.24.24.7" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left" id="S2.T2.21.21.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u_{\text{rest}}" class="ltx_Math" display="inline" id="S2.T2.21.21.1.m1.1"><semantics id="S2.T2.21.21.1.m1.1a"><msub id="S2.T2.21.21.1.m1.1.1" xref="S2.T2.21.21.1.m1.1.1.cmml"><mi id="S2.T2.21.21.1.m1.1.1.2" xref="S2.T2.21.21.1.m1.1.1.2.cmml">u</mi><mtext id="S2.T2.21.21.1.m1.1.1.3" xref="S2.T2.21.21.1.m1.1.1.3a.cmml">rest</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.T2.21.21.1.m1.1b"><apply id="S2.T2.21.21.1.m1.1.1.cmml" xref="S2.T2.21.21.1.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.21.21.1.m1.1.1.1.cmml" xref="S2.T2.21.21.1.m1.1.1">subscript</csymbol><ci id="S2.T2.21.21.1.m1.1.1.2.cmml" xref="S2.T2.21.21.1.m1.1.1.2">𝑢</ci><ci id="S2.T2.21.21.1.m1.1.1.3a.cmml" xref="S2.T2.21.21.1.m1.1.1.3"><mtext id="S2.T2.21.21.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.T2.21.21.1.m1.1.1.3">rest</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.21.21.1.m1.1c">u_{\text{rest}}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.21.21.1.m1.1d">italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.22.22.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.22.22.2.m1.1"><semantics id="S2.T2.22.22.2.m1.1a"><mn id="S2.T2.22.22.2.m1.1.1" xref="S2.T2.22.22.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.22.22.2.m1.1b"><cn id="S2.T2.22.22.2.m1.1.1.cmml" type="integer" xref="S2.T2.22.22.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.22.22.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.22.22.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.24.24.8" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left" id="S2.T2.23.23.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="u_{\text{rest}}" class="ltx_Math" display="inline" id="S2.T2.23.23.3.m1.1"><semantics id="S2.T2.23.23.3.m1.1a"><msub id="S2.T2.23.23.3.m1.1.1" xref="S2.T2.23.23.3.m1.1.1.cmml"><mi id="S2.T2.23.23.3.m1.1.1.2" xref="S2.T2.23.23.3.m1.1.1.2.cmml">u</mi><mtext id="S2.T2.23.23.3.m1.1.1.3" xref="S2.T2.23.23.3.m1.1.1.3a.cmml">rest</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.T2.23.23.3.m1.1b"><apply id="S2.T2.23.23.3.m1.1.1.cmml" xref="S2.T2.23.23.3.m1.1.1"><csymbol cd="ambiguous" id="S2.T2.23.23.3.m1.1.1.1.cmml" xref="S2.T2.23.23.3.m1.1.1">subscript</csymbol><ci id="S2.T2.23.23.3.m1.1.1.2.cmml" xref="S2.T2.23.23.3.m1.1.1.2">𝑢</ci><ci id="S2.T2.23.23.3.m1.1.1.3a.cmml" xref="S2.T2.23.23.3.m1.1.1.3"><mtext id="S2.T2.23.23.3.m1.1.1.3.cmml" mathsize="70%" xref="S2.T2.23.23.3.m1.1.1.3">rest</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.23.23.3.m1.1c">u_{\text{rest}}</annotation><annotation encoding="application/x-llamapun" id="S2.T2.23.23.3.m1.1d">italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.24.24.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.24.24.4.m1.1"><semantics id="S2.T2.24.24.4.m1.1a"><mn id="S2.T2.24.24.4.m1.1.1" xref="S2.T2.24.24.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.24.24.4.m1.1b"><cn id="S2.T2.24.24.4.m1.1.1.cmml" type="integer" xref="S2.T2.24.24.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.24.24.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.24.24.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.24.24.9" style="padding-top:1pt;padding-bottom:1pt;">static</td> </tr> <tr class="ltx_tr" id="S2.T2.28.28"> <td class="ltx_td ltx_align_left" id="S2.T2.28.28.5" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left" id="S2.T2.28.28.6" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.28.28.7" style="padding-top:1pt;padding-bottom:1pt;">-</td> <td class="ltx_td ltx_align_left" id="S2.T2.25.25.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\tau" class="ltx_Math" display="inline" id="S2.T2.25.25.1.m1.1"><semantics id="S2.T2.25.25.1.m1.1a"><mi id="S2.T2.25.25.1.m1.1.1" xref="S2.T2.25.25.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.T2.25.25.1.m1.1b"><ci id="S2.T2.25.25.1.m1.1.1.cmml" xref="S2.T2.25.25.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.25.25.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.T2.25.25.1.m1.1d">italic_τ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.26.26.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.26.26.2.m1.1"><semantics id="S2.T2.26.26.2.m1.1a"><mn id="S2.T2.26.26.2.m1.1.1" xref="S2.T2.26.26.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.26.26.2.m1.1b"><cn id="S2.T2.26.26.2.m1.1.1.cmml" type="integer" xref="S2.T2.26.26.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.26.26.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.26.26.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T2.28.28.8" style="padding-top:1pt;padding-bottom:1pt;">static</td> <td class="ltx_td ltx_align_left" id="S2.T2.27.27.3" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\tau" class="ltx_Math" display="inline" id="S2.T2.27.27.3.m1.1"><semantics id="S2.T2.27.27.3.m1.1a"><mi id="S2.T2.27.27.3.m1.1.1" xref="S2.T2.27.27.3.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.T2.27.27.3.m1.1b"><ci id="S2.T2.27.27.3.m1.1.1.cmml" xref="S2.T2.27.27.3.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.27.27.3.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.T2.27.27.3.m1.1d">italic_τ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.28.28.4" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S2.T2.28.28.4.m1.1"><semantics id="S2.T2.28.28.4.m1.1a"><mn id="S2.T2.28.28.4.m1.1.1" xref="S2.T2.28.28.4.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.T2.28.28.4.m1.1b"><cn id="S2.T2.28.28.4.m1.1.1.cmml" type="integer" xref="S2.T2.28.28.4.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.T2.28.28.4.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.T2.28.28.4.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.T2.28.28.9" style="padding-top:1pt;padding-bottom:1pt;">static</td> </tr> </table> </figure> <figure class="ltx_figure" id="S2.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="1107" id="S2.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span> Comparison of neuron dynamics and spiking functions. The left column shows the neuron behavior when a target is present, whereas in the right column no target is present. The first row shows the the neuron’s magnitude <math alttext="\|s\|" class="ltx_Math" display="inline" id="S2.F3.11.m1.1"><semantics id="S2.F3.11.m1.1b"><mrow id="S2.F3.11.m1.1.2.2" xref="S2.F3.11.m1.1.2.1.cmml"><mo id="S2.F3.11.m1.1.2.2.1" stretchy="false" xref="S2.F3.11.m1.1.2.1.1.cmml">‖</mo><mi id="S2.F3.11.m1.1.1" xref="S2.F3.11.m1.1.1.cmml">s</mi><mo id="S2.F3.11.m1.1.2.2.2" stretchy="false" xref="S2.F3.11.m1.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.11.m1.1c"><apply id="S2.F3.11.m1.1.2.1.cmml" xref="S2.F3.11.m1.1.2.2"><csymbol cd="latexml" id="S2.F3.11.m1.1.2.1.1.cmml" xref="S2.F3.11.m1.1.2.2.1">norm</csymbol><ci id="S2.F3.11.m1.1.1.cmml" xref="S2.F3.11.m1.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.11.m1.1d">\|s\|</annotation><annotation encoding="application/x-llamapun" id="S2.F3.11.m1.1e">∥ italic_s ∥</annotation></semantics></math>, and its estimated maximum <math alttext="s_{\text{max}}" class="ltx_Math" display="inline" id="S2.F3.12.m2.1"><semantics id="S2.F3.12.m2.1b"><msub id="S2.F3.12.m2.1.1" xref="S2.F3.12.m2.1.1.cmml"><mi id="S2.F3.12.m2.1.1.2" xref="S2.F3.12.m2.1.1.2.cmml">s</mi><mtext id="S2.F3.12.m2.1.1.3" xref="S2.F3.12.m2.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.F3.12.m2.1c"><apply id="S2.F3.12.m2.1.1.cmml" xref="S2.F3.12.m2.1.1"><csymbol cd="ambiguous" id="S2.F3.12.m2.1.1.1.cmml" xref="S2.F3.12.m2.1.1">subscript</csymbol><ci id="S2.F3.12.m2.1.1.2.cmml" xref="S2.F3.12.m2.1.1.2">𝑠</ci><ci id="S2.F3.12.m2.1.1.3a.cmml" xref="S2.F3.12.m2.1.1.3"><mtext id="S2.F3.12.m2.1.1.3.cmml" mathsize="70%" xref="S2.F3.12.m2.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.12.m2.1d">s_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.F3.12.m2.1e">italic_s start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> and the estimated maximum width <math alttext="w_{\text{max}}" class="ltx_Math" display="inline" id="S2.F3.13.m3.1"><semantics id="S2.F3.13.m3.1b"><msub id="S2.F3.13.m3.1.1" xref="S2.F3.13.m3.1.1.cmml"><mi id="S2.F3.13.m3.1.1.2" xref="S2.F3.13.m3.1.1.2.cmml">w</mi><mtext id="S2.F3.13.m3.1.1.3" xref="S2.F3.13.m3.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.F3.13.m3.1c"><apply id="S2.F3.13.m3.1.1.cmml" xref="S2.F3.13.m3.1.1"><csymbol cd="ambiguous" id="S2.F3.13.m3.1.1.1.cmml" xref="S2.F3.13.m3.1.1">subscript</csymbol><ci id="S2.F3.13.m3.1.1.2.cmml" xref="S2.F3.13.m3.1.1.2">𝑤</ci><ci id="S2.F3.13.m3.1.1.3a.cmml" xref="S2.F3.13.m3.1.1.3"><mtext id="S2.F3.13.m3.1.1.3.cmml" mathsize="70%" xref="S2.F3.13.m3.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.13.m3.1d">w_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.F3.13.m3.1e">italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> between the magnitude and its maximum. The estimated envelope <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.F3.14.m4.1"><semantics id="S2.F3.14.m4.1b"><mi id="S2.F3.14.m4.1.1" mathvariant="normal" xref="S2.F3.14.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.F3.14.m4.1c"><ci id="S2.F3.14.m4.1.1.cmml" xref="S2.F3.14.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.14.m4.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.F3.14.m4.1e">roman_Λ</annotation></semantics></math> (grey) follows a lower boundary of the magnitude <math alttext="\|s\|" class="ltx_Math" display="inline" id="S2.F3.15.m5.1"><semantics id="S2.F3.15.m5.1b"><mrow id="S2.F3.15.m5.1.2.2" xref="S2.F3.15.m5.1.2.1.cmml"><mo id="S2.F3.15.m5.1.2.2.1" stretchy="false" xref="S2.F3.15.m5.1.2.1.1.cmml">‖</mo><mi id="S2.F3.15.m5.1.1" xref="S2.F3.15.m5.1.1.cmml">s</mi><mo id="S2.F3.15.m5.1.2.2.2" stretchy="false" xref="S2.F3.15.m5.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.15.m5.1c"><apply id="S2.F3.15.m5.1.2.1.cmml" xref="S2.F3.15.m5.1.2.2"><csymbol cd="latexml" id="S2.F3.15.m5.1.2.1.1.cmml" xref="S2.F3.15.m5.1.2.2.1">norm</csymbol><ci id="S2.F3.15.m5.1.1.cmml" xref="S2.F3.15.m5.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.15.m5.1d">\|s\|</annotation><annotation encoding="application/x-llamapun" id="S2.F3.15.m5.1e">∥ italic_s ∥</annotation></semantics></math>. The second row shows the behaviour of the adaptive threshold spiking function. In the top row, positive spikes are indicated black and negative spikes red. 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The negative width <math alttext="-w_{\text{max}}" class="ltx_Math" display="inline" id="S2.F3.20.m10.1"><semantics id="S2.F3.20.m10.1b"><mrow id="S2.F3.20.m10.1.1" xref="S2.F3.20.m10.1.1.cmml"><mo id="S2.F3.20.m10.1.1b" xref="S2.F3.20.m10.1.1.cmml">−</mo><msub id="S2.F3.20.m10.1.1.2" xref="S2.F3.20.m10.1.1.2.cmml"><mi id="S2.F3.20.m10.1.1.2.2" xref="S2.F3.20.m10.1.1.2.2.cmml">w</mi><mtext id="S2.F3.20.m10.1.1.2.3" xref="S2.F3.20.m10.1.1.2.3a.cmml">max</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.20.m10.1c"><apply id="S2.F3.20.m10.1.1.cmml" xref="S2.F3.20.m10.1.1"><minus id="S2.F3.20.m10.1.1.1.cmml" xref="S2.F3.20.m10.1.1"></minus><apply id="S2.F3.20.m10.1.1.2.cmml" xref="S2.F3.20.m10.1.1.2"><csymbol cd="ambiguous" id="S2.F3.20.m10.1.1.2.1.cmml" xref="S2.F3.20.m10.1.1.2">subscript</csymbol><ci id="S2.F3.20.m10.1.1.2.2.cmml" xref="S2.F3.20.m10.1.1.2.2">𝑤</ci><ci id="S2.F3.20.m10.1.1.2.3a.cmml" xref="S2.F3.20.m10.1.1.2.3"><mtext id="S2.F3.20.m10.1.1.2.3.cmml" mathsize="70%" xref="S2.F3.20.m10.1.1.2.3">max</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.20.m10.1d">-w_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S2.F3.20.m10.1e">- italic_w start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> is shown for better visibility. The third row shows the behaviour of the rate-coded LIF spiking function. A present target results in a spike rate, whereas no target does not produce any spike. The fourth row shows the behaviour of the time-coded LIF spiking function. A present target results in a single spike, whereas no target does not produce any spike. </figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Evaluation</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The evaluation of temporal neuron models is time- and energy-consuming, as the processor needs to calculate each time step of each neuron. Therefore, we implemented our neuron model on GPU to process all neurons of the range-angle map in parallel. Each neuron runs independently on a single CUDA core, removing waiting or communication time between different cores. This parallelization drastically improved the computation time by a factor <math alttext="\sim 1000" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml"></mi><mo id="S3.p1.1.m1.1.1.1" xref="S3.p1.1.m1.1.1.1.cmml">∼</mo><mn id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">1000</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">absent</csymbol><cn id="S3.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.p1.1.m1.1.1.3">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\sim 1000</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">∼ 1000</annotation></semantics></math> and allowed us to evaluate our network model on large amounts of data.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.5.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.6.2">Dataset Simulation</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">For the evaluation, we generated raw radar data from a radar simulator by Infineon Technologies AG. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T3" title="TABLE III ‣ III-A Dataset Simulation ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">III</span></a> summarizes the parameters of the simulated radar sensor. The simulation includes phase noise, thermal noise, and phase-locked loops. The simulator creates targets as single points with a given position, velocity, and radar cross-section (RCS). With these parameters, we created a labeled dataset consisting of the raw radar data and a list of point targets, including their attributes. Using two different seeds, we simulated five scenarios, summarized and described in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T4" title="TABLE IV ‣ III-A Dataset Simulation ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">IV</span></a>. The dataset with seed 0 will be used as a training dataset to optimize the parameters, whereas seed 1 is the evaluation dataset.</p> </div> <figure class="ltx_table" id="S3.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">TABLE III: </span>Parameters of the simulated radar sensor</figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S3.T3.18"> <tr class="ltx_tr" id="S3.T3.18.19"> <td class="ltx_td ltx_align_center" id="S3.T3.18.19.1" style="padding-top:1pt;padding-bottom:1pt;">Parameter name</td> <td class="ltx_td ltx_align_center" id="S3.T3.18.19.2" style="padding-top:1pt;padding-bottom:1pt;">Parameter value</td> </tr> <tr class="ltx_tr" id="S3.T3.2.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T3.1.1.1" style="padding-top:1pt;padding-bottom:1pt;">initial frequency <math alttext="f_{0}" class="ltx_Math" display="inline" id="S3.T3.1.1.1.m1.1"><semantics id="S3.T3.1.1.1.m1.1a"><msub id="S3.T3.1.1.1.m1.1.1" xref="S3.T3.1.1.1.m1.1.1.cmml"><mi id="S3.T3.1.1.1.m1.1.1.2" xref="S3.T3.1.1.1.m1.1.1.2.cmml">f</mi><mn id="S3.T3.1.1.1.m1.1.1.3" xref="S3.T3.1.1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T3.1.1.1.m1.1b"><apply id="S3.T3.1.1.1.m1.1.1.cmml" xref="S3.T3.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.1.1.1.m1.1.1.1.cmml" xref="S3.T3.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.T3.1.1.1.m1.1.1.2.cmml" xref="S3.T3.1.1.1.m1.1.1.2">𝑓</ci><cn id="S3.T3.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T3.1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.1.1.1.m1.1c">f_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.1.1.1.m1.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T3.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="76~{}\text{GHz}" class="ltx_Math" display="inline" id="S3.T3.2.2.2.m1.1"><semantics id="S3.T3.2.2.2.m1.1a"><mrow id="S3.T3.2.2.2.m1.1.1" xref="S3.T3.2.2.2.m1.1.1.cmml"><mn id="S3.T3.2.2.2.m1.1.1.2" xref="S3.T3.2.2.2.m1.1.1.2.cmml">76</mn><mo id="S3.T3.2.2.2.m1.1.1.1" lspace="0.330em" xref="S3.T3.2.2.2.m1.1.1.1.cmml">⁢</mo><mtext id="S3.T3.2.2.2.m1.1.1.3" xref="S3.T3.2.2.2.m1.1.1.3a.cmml">GHz</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T3.2.2.2.m1.1b"><apply id="S3.T3.2.2.2.m1.1.1.cmml" xref="S3.T3.2.2.2.m1.1.1"><times id="S3.T3.2.2.2.m1.1.1.1.cmml" xref="S3.T3.2.2.2.m1.1.1.1"></times><cn id="S3.T3.2.2.2.m1.1.1.2.cmml" type="integer" xref="S3.T3.2.2.2.m1.1.1.2">76</cn><ci id="S3.T3.2.2.2.m1.1.1.3a.cmml" xref="S3.T3.2.2.2.m1.1.1.3"><mtext id="S3.T3.2.2.2.m1.1.1.3.cmml" xref="S3.T3.2.2.2.m1.1.1.3">GHz</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.2.2.2.m1.1c">76~{}\text{GHz}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.2.2.2.m1.1d">76 GHz</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.4.4"> <td class="ltx_td ltx_align_left" id="S3.T3.3.3.1" style="padding-top:1pt;padding-bottom:1pt;">bandwidth <math alttext="B" class="ltx_Math" display="inline" id="S3.T3.3.3.1.m1.1"><semantics id="S3.T3.3.3.1.m1.1a"><mi id="S3.T3.3.3.1.m1.1.1" xref="S3.T3.3.3.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.T3.3.3.1.m1.1b"><ci id="S3.T3.3.3.1.m1.1.1.cmml" xref="S3.T3.3.3.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.3.3.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.T3.3.3.1.m1.1d">italic_B</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.4.4.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="507.6~{}\text{MHz}" class="ltx_Math" display="inline" id="S3.T3.4.4.2.m1.1"><semantics id="S3.T3.4.4.2.m1.1a"><mrow id="S3.T3.4.4.2.m1.1.1" xref="S3.T3.4.4.2.m1.1.1.cmml"><mn id="S3.T3.4.4.2.m1.1.1.2" xref="S3.T3.4.4.2.m1.1.1.2.cmml">507.6</mn><mo id="S3.T3.4.4.2.m1.1.1.1" lspace="0.330em" xref="S3.T3.4.4.2.m1.1.1.1.cmml">⁢</mo><mtext id="S3.T3.4.4.2.m1.1.1.3" xref="S3.T3.4.4.2.m1.1.1.3a.cmml">MHz</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T3.4.4.2.m1.1b"><apply id="S3.T3.4.4.2.m1.1.1.cmml" xref="S3.T3.4.4.2.m1.1.1"><times id="S3.T3.4.4.2.m1.1.1.1.cmml" xref="S3.T3.4.4.2.m1.1.1.1"></times><cn id="S3.T3.4.4.2.m1.1.1.2.cmml" type="float" xref="S3.T3.4.4.2.m1.1.1.2">507.6</cn><ci id="S3.T3.4.4.2.m1.1.1.3a.cmml" xref="S3.T3.4.4.2.m1.1.1.3"><mtext id="S3.T3.4.4.2.m1.1.1.3.cmml" xref="S3.T3.4.4.2.m1.1.1.3">MHz</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.4.4.2.m1.1c">507.6~{}\text{MHz}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.4.4.2.m1.1d">507.6 MHz</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.6.6"> <td class="ltx_td ltx_align_left" id="S3.T3.5.5.1" style="padding-top:1pt;padding-bottom:1pt;">number of samples <math alttext="N_{\text{samples}}" class="ltx_Math" display="inline" id="S3.T3.5.5.1.m1.1"><semantics id="S3.T3.5.5.1.m1.1a"><msub id="S3.T3.5.5.1.m1.1.1" xref="S3.T3.5.5.1.m1.1.1.cmml"><mi id="S3.T3.5.5.1.m1.1.1.2" xref="S3.T3.5.5.1.m1.1.1.2.cmml">N</mi><mtext id="S3.T3.5.5.1.m1.1.1.3" xref="S3.T3.5.5.1.m1.1.1.3a.cmml">samples</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.5.5.1.m1.1b"><apply id="S3.T3.5.5.1.m1.1.1.cmml" xref="S3.T3.5.5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.5.5.1.m1.1.1.1.cmml" xref="S3.T3.5.5.1.m1.1.1">subscript</csymbol><ci id="S3.T3.5.5.1.m1.1.1.2.cmml" xref="S3.T3.5.5.1.m1.1.1.2">𝑁</ci><ci id="S3.T3.5.5.1.m1.1.1.3a.cmml" xref="S3.T3.5.5.1.m1.1.1.3"><mtext id="S3.T3.5.5.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.5.5.1.m1.1.1.3">samples</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.5.5.1.m1.1c">N_{\text{samples}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.5.5.1.m1.1d">italic_N start_POSTSUBSCRIPT samples end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.6.6.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="512" class="ltx_Math" display="inline" id="S3.T3.6.6.2.m1.1"><semantics id="S3.T3.6.6.2.m1.1a"><mn id="S3.T3.6.6.2.m1.1.1" xref="S3.T3.6.6.2.m1.1.1.cmml">512</mn><annotation-xml encoding="MathML-Content" id="S3.T3.6.6.2.m1.1b"><cn id="S3.T3.6.6.2.m1.1.1.cmml" type="integer" xref="S3.T3.6.6.2.m1.1.1">512</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.6.6.2.m1.1c">512</annotation><annotation encoding="application/x-llamapun" id="S3.T3.6.6.2.m1.1d">512</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.8.8"> <td class="ltx_td ltx_align_left" id="S3.T3.7.7.1" style="padding-top:1pt;padding-bottom:1pt;">number of chirps <math alttext="N_{\text{chirps}}" class="ltx_Math" display="inline" id="S3.T3.7.7.1.m1.1"><semantics id="S3.T3.7.7.1.m1.1a"><msub id="S3.T3.7.7.1.m1.1.1" xref="S3.T3.7.7.1.m1.1.1.cmml"><mi id="S3.T3.7.7.1.m1.1.1.2" xref="S3.T3.7.7.1.m1.1.1.2.cmml">N</mi><mtext id="S3.T3.7.7.1.m1.1.1.3" xref="S3.T3.7.7.1.m1.1.1.3a.cmml">chirps</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.7.7.1.m1.1b"><apply id="S3.T3.7.7.1.m1.1.1.cmml" xref="S3.T3.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.7.7.1.m1.1.1.1.cmml" xref="S3.T3.7.7.1.m1.1.1">subscript</csymbol><ci id="S3.T3.7.7.1.m1.1.1.2.cmml" xref="S3.T3.7.7.1.m1.1.1.2">𝑁</ci><ci id="S3.T3.7.7.1.m1.1.1.3a.cmml" xref="S3.T3.7.7.1.m1.1.1.3"><mtext id="S3.T3.7.7.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.7.7.1.m1.1.1.3">chirps</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.7.7.1.m1.1c">N_{\text{chirps}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.7.7.1.m1.1d">italic_N start_POSTSUBSCRIPT chirps end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.8.8.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="32" class="ltx_Math" display="inline" id="S3.T3.8.8.2.m1.1"><semantics id="S3.T3.8.8.2.m1.1a"><mn id="S3.T3.8.8.2.m1.1.1" xref="S3.T3.8.8.2.m1.1.1.cmml">32</mn><annotation-xml encoding="MathML-Content" id="S3.T3.8.8.2.m1.1b"><cn id="S3.T3.8.8.2.m1.1.1.cmml" type="integer" xref="S3.T3.8.8.2.m1.1.1">32</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.8.8.2.m1.1c">32</annotation><annotation encoding="application/x-llamapun" id="S3.T3.8.8.2.m1.1d">32</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.10.10"> <td class="ltx_td ltx_align_left" id="S3.T3.9.9.1" style="padding-top:1pt;padding-bottom:1pt;">number of receiving antennas <math alttext="N_{\text{rx}}" class="ltx_Math" display="inline" id="S3.T3.9.9.1.m1.1"><semantics id="S3.T3.9.9.1.m1.1a"><msub id="S3.T3.9.9.1.m1.1.1" xref="S3.T3.9.9.1.m1.1.1.cmml"><mi id="S3.T3.9.9.1.m1.1.1.2" xref="S3.T3.9.9.1.m1.1.1.2.cmml">N</mi><mtext id="S3.T3.9.9.1.m1.1.1.3" xref="S3.T3.9.9.1.m1.1.1.3a.cmml">rx</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.9.9.1.m1.1b"><apply id="S3.T3.9.9.1.m1.1.1.cmml" xref="S3.T3.9.9.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.9.9.1.m1.1.1.1.cmml" xref="S3.T3.9.9.1.m1.1.1">subscript</csymbol><ci id="S3.T3.9.9.1.m1.1.1.2.cmml" xref="S3.T3.9.9.1.m1.1.1.2">𝑁</ci><ci id="S3.T3.9.9.1.m1.1.1.3a.cmml" xref="S3.T3.9.9.1.m1.1.1.3"><mtext id="S3.T3.9.9.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.9.9.1.m1.1.1.3">rx</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.9.9.1.m1.1c">N_{\text{rx}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.9.9.1.m1.1d">italic_N start_POSTSUBSCRIPT rx end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.10.10.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="32" class="ltx_Math" display="inline" id="S3.T3.10.10.2.m1.1"><semantics id="S3.T3.10.10.2.m1.1a"><mn id="S3.T3.10.10.2.m1.1.1" xref="S3.T3.10.10.2.m1.1.1.cmml">32</mn><annotation-xml encoding="MathML-Content" id="S3.T3.10.10.2.m1.1b"><cn id="S3.T3.10.10.2.m1.1.1.cmml" type="integer" xref="S3.T3.10.10.2.m1.1.1">32</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.10.10.2.m1.1c">32</annotation><annotation encoding="application/x-llamapun" id="S3.T3.10.10.2.m1.1d">32</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.12.12"> <td class="ltx_td ltx_align_left" id="S3.T3.11.11.1" style="padding-top:1pt;padding-bottom:1pt;">number of transmitting antennas <math alttext="N_{\text{tx}}" class="ltx_Math" display="inline" id="S3.T3.11.11.1.m1.1"><semantics id="S3.T3.11.11.1.m1.1a"><msub id="S3.T3.11.11.1.m1.1.1" xref="S3.T3.11.11.1.m1.1.1.cmml"><mi id="S3.T3.11.11.1.m1.1.1.2" xref="S3.T3.11.11.1.m1.1.1.2.cmml">N</mi><mtext id="S3.T3.11.11.1.m1.1.1.3" xref="S3.T3.11.11.1.m1.1.1.3a.cmml">tx</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.11.11.1.m1.1b"><apply id="S3.T3.11.11.1.m1.1.1.cmml" xref="S3.T3.11.11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.11.11.1.m1.1.1.1.cmml" xref="S3.T3.11.11.1.m1.1.1">subscript</csymbol><ci id="S3.T3.11.11.1.m1.1.1.2.cmml" xref="S3.T3.11.11.1.m1.1.1.2">𝑁</ci><ci id="S3.T3.11.11.1.m1.1.1.3a.cmml" xref="S3.T3.11.11.1.m1.1.1.3"><mtext id="S3.T3.11.11.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.11.11.1.m1.1.1.3">tx</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.11.11.1.m1.1c">N_{\text{tx}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.11.11.1.m1.1d">italic_N start_POSTSUBSCRIPT tx end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.12.12.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="1" class="ltx_Math" display="inline" id="S3.T3.12.12.2.m1.1"><semantics id="S3.T3.12.12.2.m1.1a"><mn id="S3.T3.12.12.2.m1.1.1" xref="S3.T3.12.12.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.T3.12.12.2.m1.1b"><cn id="S3.T3.12.12.2.m1.1.1.cmml" type="integer" xref="S3.T3.12.12.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.12.12.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.T3.12.12.2.m1.1d">1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.14.14"> <td class="ltx_td ltx_align_left" id="S3.T3.13.13.1" style="padding-top:1pt;padding-bottom:1pt;">duration of chirp <math alttext="t_{\text{chirp}}" class="ltx_Math" display="inline" id="S3.T3.13.13.1.m1.1"><semantics id="S3.T3.13.13.1.m1.1a"><msub id="S3.T3.13.13.1.m1.1.1" xref="S3.T3.13.13.1.m1.1.1.cmml"><mi id="S3.T3.13.13.1.m1.1.1.2" xref="S3.T3.13.13.1.m1.1.1.2.cmml">t</mi><mtext id="S3.T3.13.13.1.m1.1.1.3" xref="S3.T3.13.13.1.m1.1.1.3a.cmml">chirp</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.13.13.1.m1.1b"><apply id="S3.T3.13.13.1.m1.1.1.cmml" xref="S3.T3.13.13.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.13.13.1.m1.1.1.1.cmml" xref="S3.T3.13.13.1.m1.1.1">subscript</csymbol><ci id="S3.T3.13.13.1.m1.1.1.2.cmml" xref="S3.T3.13.13.1.m1.1.1.2">𝑡</ci><ci id="S3.T3.13.13.1.m1.1.1.3a.cmml" xref="S3.T3.13.13.1.m1.1.1.3"><mtext id="S3.T3.13.13.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.13.13.1.m1.1.1.3">chirp</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.13.13.1.m1.1c">t_{\text{chirp}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.13.13.1.m1.1d">italic_t start_POSTSUBSCRIPT chirp end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.14.14.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="20.52~{}\mu s" class="ltx_Math" display="inline" id="S3.T3.14.14.2.m1.1"><semantics id="S3.T3.14.14.2.m1.1a"><mrow id="S3.T3.14.14.2.m1.1.1" xref="S3.T3.14.14.2.m1.1.1.cmml"><mn id="S3.T3.14.14.2.m1.1.1.2" xref="S3.T3.14.14.2.m1.1.1.2.cmml">20.52</mn><mo id="S3.T3.14.14.2.m1.1.1.1" lspace="0.330em" xref="S3.T3.14.14.2.m1.1.1.1.cmml">⁢</mo><mi id="S3.T3.14.14.2.m1.1.1.3" xref="S3.T3.14.14.2.m1.1.1.3.cmml">μ</mi><mo id="S3.T3.14.14.2.m1.1.1.1a" xref="S3.T3.14.14.2.m1.1.1.1.cmml">⁢</mo><mi id="S3.T3.14.14.2.m1.1.1.4" xref="S3.T3.14.14.2.m1.1.1.4.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.T3.14.14.2.m1.1b"><apply id="S3.T3.14.14.2.m1.1.1.cmml" xref="S3.T3.14.14.2.m1.1.1"><times id="S3.T3.14.14.2.m1.1.1.1.cmml" xref="S3.T3.14.14.2.m1.1.1.1"></times><cn id="S3.T3.14.14.2.m1.1.1.2.cmml" type="float" xref="S3.T3.14.14.2.m1.1.1.2">20.52</cn><ci id="S3.T3.14.14.2.m1.1.1.3.cmml" xref="S3.T3.14.14.2.m1.1.1.3">𝜇</ci><ci id="S3.T3.14.14.2.m1.1.1.4.cmml" xref="S3.T3.14.14.2.m1.1.1.4">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.14.14.2.m1.1c">20.52~{}\mu s</annotation><annotation encoding="application/x-llamapun" id="S3.T3.14.14.2.m1.1d">20.52 italic_μ italic_s</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.16.16"> <td class="ltx_td ltx_align_left" id="S3.T3.15.15.1" style="padding-top:1pt;padding-bottom:1pt;">duration of wait time <math alttext="t_{\text{wait}}" class="ltx_Math" display="inline" id="S3.T3.15.15.1.m1.1"><semantics id="S3.T3.15.15.1.m1.1a"><msub id="S3.T3.15.15.1.m1.1.1" xref="S3.T3.15.15.1.m1.1.1.cmml"><mi id="S3.T3.15.15.1.m1.1.1.2" xref="S3.T3.15.15.1.m1.1.1.2.cmml">t</mi><mtext id="S3.T3.15.15.1.m1.1.1.3" xref="S3.T3.15.15.1.m1.1.1.3a.cmml">wait</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.15.15.1.m1.1b"><apply id="S3.T3.15.15.1.m1.1.1.cmml" xref="S3.T3.15.15.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.15.15.1.m1.1.1.1.cmml" xref="S3.T3.15.15.1.m1.1.1">subscript</csymbol><ci id="S3.T3.15.15.1.m1.1.1.2.cmml" xref="S3.T3.15.15.1.m1.1.1.2">𝑡</ci><ci id="S3.T3.15.15.1.m1.1.1.3a.cmml" xref="S3.T3.15.15.1.m1.1.1.3"><mtext id="S3.T3.15.15.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.15.15.1.m1.1.1.3">wait</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.15.15.1.m1.1c">t_{\text{wait}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.15.15.1.m1.1d">italic_t start_POSTSUBSCRIPT wait end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.16.16.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="5.96~{}\mu s" class="ltx_Math" display="inline" id="S3.T3.16.16.2.m1.1"><semantics id="S3.T3.16.16.2.m1.1a"><mrow id="S3.T3.16.16.2.m1.1.1" xref="S3.T3.16.16.2.m1.1.1.cmml"><mn id="S3.T3.16.16.2.m1.1.1.2" xref="S3.T3.16.16.2.m1.1.1.2.cmml">5.96</mn><mo id="S3.T3.16.16.2.m1.1.1.1" lspace="0.330em" xref="S3.T3.16.16.2.m1.1.1.1.cmml">⁢</mo><mi id="S3.T3.16.16.2.m1.1.1.3" xref="S3.T3.16.16.2.m1.1.1.3.cmml">μ</mi><mo id="S3.T3.16.16.2.m1.1.1.1a" xref="S3.T3.16.16.2.m1.1.1.1.cmml">⁢</mo><mi id="S3.T3.16.16.2.m1.1.1.4" xref="S3.T3.16.16.2.m1.1.1.4.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.T3.16.16.2.m1.1b"><apply id="S3.T3.16.16.2.m1.1.1.cmml" xref="S3.T3.16.16.2.m1.1.1"><times id="S3.T3.16.16.2.m1.1.1.1.cmml" xref="S3.T3.16.16.2.m1.1.1.1"></times><cn id="S3.T3.16.16.2.m1.1.1.2.cmml" type="float" xref="S3.T3.16.16.2.m1.1.1.2">5.96</cn><ci id="S3.T3.16.16.2.m1.1.1.3.cmml" xref="S3.T3.16.16.2.m1.1.1.3">𝜇</ci><ci id="S3.T3.16.16.2.m1.1.1.4.cmml" xref="S3.T3.16.16.2.m1.1.1.4">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.16.16.2.m1.1c">5.96~{}\mu s</annotation><annotation encoding="application/x-llamapun" id="S3.T3.16.16.2.m1.1d">5.96 italic_μ italic_s</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T3.18.18"> <td class="ltx_td ltx_align_left" id="S3.T3.17.17.1" style="padding-top:1pt;padding-bottom:1pt;">chirp frequency <math alttext="f_{\text{chirp}}" class="ltx_Math" display="inline" id="S3.T3.17.17.1.m1.1"><semantics id="S3.T3.17.17.1.m1.1a"><msub id="S3.T3.17.17.1.m1.1.1" xref="S3.T3.17.17.1.m1.1.1.cmml"><mi id="S3.T3.17.17.1.m1.1.1.2" xref="S3.T3.17.17.1.m1.1.1.2.cmml">f</mi><mtext id="S3.T3.17.17.1.m1.1.1.3" xref="S3.T3.17.17.1.m1.1.1.3a.cmml">chirp</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T3.17.17.1.m1.1b"><apply id="S3.T3.17.17.1.m1.1.1.cmml" xref="S3.T3.17.17.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T3.17.17.1.m1.1.1.1.cmml" xref="S3.T3.17.17.1.m1.1.1">subscript</csymbol><ci id="S3.T3.17.17.1.m1.1.1.2.cmml" xref="S3.T3.17.17.1.m1.1.1.2">𝑓</ci><ci id="S3.T3.17.17.1.m1.1.1.3a.cmml" xref="S3.T3.17.17.1.m1.1.1.3"><mtext id="S3.T3.17.17.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T3.17.17.1.m1.1.1.3">chirp</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.17.17.1.m1.1c">f_{\text{chirp}}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.17.17.1.m1.1d">italic_f start_POSTSUBSCRIPT chirp end_POSTSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_left" id="S3.T3.18.18.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="37.76~{}\text{kHz}" class="ltx_Math" display="inline" id="S3.T3.18.18.2.m1.1"><semantics id="S3.T3.18.18.2.m1.1a"><mrow id="S3.T3.18.18.2.m1.1.1" xref="S3.T3.18.18.2.m1.1.1.cmml"><mn id="S3.T3.18.18.2.m1.1.1.2" xref="S3.T3.18.18.2.m1.1.1.2.cmml">37.76</mn><mo id="S3.T3.18.18.2.m1.1.1.1" lspace="0.330em" xref="S3.T3.18.18.2.m1.1.1.1.cmml">⁢</mo><mtext id="S3.T3.18.18.2.m1.1.1.3" xref="S3.T3.18.18.2.m1.1.1.3a.cmml">kHz</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T3.18.18.2.m1.1b"><apply id="S3.T3.18.18.2.m1.1.1.cmml" xref="S3.T3.18.18.2.m1.1.1"><times id="S3.T3.18.18.2.m1.1.1.1.cmml" xref="S3.T3.18.18.2.m1.1.1.1"></times><cn id="S3.T3.18.18.2.m1.1.1.2.cmml" type="float" xref="S3.T3.18.18.2.m1.1.1.2">37.76</cn><ci id="S3.T3.18.18.2.m1.1.1.3a.cmml" xref="S3.T3.18.18.2.m1.1.1.3"><mtext id="S3.T3.18.18.2.m1.1.1.3.cmml" xref="S3.T3.18.18.2.m1.1.1.3">kHz</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T3.18.18.2.m1.1c">37.76~{}\text{kHz}</annotation><annotation encoding="application/x-llamapun" id="S3.T3.18.18.2.m1.1d">37.76 kHz</annotation></semantics></math></td> </tr> </table> </figure> <figure class="ltx_table" id="S3.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">TABLE IV: </span>List of simulated data sets</figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S3.T4.6"> <tr class="ltx_tr" id="S3.T4.6.7"> <td class="ltx_td ltx_align_center" id="S3.T4.6.7.1" style="padding-top:1pt;padding-bottom:1pt;">Dataset name</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.7.2" style="padding-top:1pt;padding-bottom:1pt;">Description</td> </tr> <tr class="ltx_tr" id="S3.T4.2.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T4.2.2.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_italic" id="S3.T4.2.2.3.1">close targets 2010</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T4.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"> <span class="ltx_text" id="S3.T4.2.2.2.3"></span><span class="ltx_text" id="S3.T4.2.2.2.2"> <span class="ltx_tabular ltx_align_top" id="S3.T4.2.2.2.2.2"> <span class="ltx_tr" id="S3.T4.2.2.2.2.2.3"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.2.2.2.2.2.3.1" style="padding-top:1pt;padding-bottom:1pt;">two targets close by with</span></span> <span class="ltx_tr" id="S3.T4.2.2.2.2.2.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.2.2.2.2.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sigma_{0}=10" class="ltx_Math" display="inline" id="S3.T4.1.1.1.1.1.1.1.m1.1"><semantics id="S3.T4.1.1.1.1.1.1.1.m1.1a"><mrow id="S3.T4.1.1.1.1.1.1.1.m1.1.1" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.cmml"><msub id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.cmml"><mi id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.2" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.2.cmml">σ</mi><mn id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.3" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.T4.1.1.1.1.1.1.1.m1.1.1.1" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.1.cmml">=</mo><mn id="S3.T4.1.1.1.1.1.1.1.m1.1.1.3" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T4.1.1.1.1.1.1.1.m1.1b"><apply id="S3.T4.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1"><eq id="S3.T4.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.1"></eq><apply id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.1.cmml" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.2.cmml" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.2">𝜎</ci><cn id="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.2.3">0</cn></apply><cn id="S3.T4.1.1.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T4.1.1.1.1.1.1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.1.1.1.1.1.1.1.m1.1c">\sigma_{0}=10</annotation><annotation encoding="application/x-llamapun" id="S3.T4.1.1.1.1.1.1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 10</annotation></semantics></math> and <math alttext="\sigma_{1}=20" class="ltx_Math" display="inline" id="S3.T4.2.2.2.2.2.2.2.m2.1"><semantics id="S3.T4.2.2.2.2.2.2.2.m2.1a"><mrow id="S3.T4.2.2.2.2.2.2.2.m2.1.1" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.cmml"><msub id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.cmml"><mi id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.2" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.2.cmml">σ</mi><mn id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.3" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.T4.2.2.2.2.2.2.2.m2.1.1.1" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.1.cmml">=</mo><mn id="S3.T4.2.2.2.2.2.2.2.m2.1.1.3" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T4.2.2.2.2.2.2.2.m2.1b"><apply id="S3.T4.2.2.2.2.2.2.2.m2.1.1.cmml" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1"><eq id="S3.T4.2.2.2.2.2.2.2.m2.1.1.1.cmml" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.1"></eq><apply id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.cmml" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.1.cmml" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2">subscript</csymbol><ci id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.2.cmml" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.2">𝜎</ci><cn id="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.3.cmml" type="integer" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.2.3">1</cn></apply><cn id="S3.T4.2.2.2.2.2.2.2.m2.1.1.3.cmml" type="integer" xref="S3.T4.2.2.2.2.2.2.2.m2.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.2.2.2.2.2.2.2.m2.1c">\sigma_{1}=20</annotation><annotation encoding="application/x-llamapun" id="S3.T4.2.2.2.2.2.2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 20</annotation></semantics></math></span></span> </span></span><span class="ltx_text" id="S3.T4.2.2.2.4"></span></td> </tr> <tr class="ltx_tr" id="S3.T4.4.4"> <td class="ltx_td ltx_align_left" id="S3.T4.4.4.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_italic" id="S3.T4.4.4.3.1">close targets 0010</span></td> <td class="ltx_td ltx_align_left" id="S3.T4.4.4.2" style="padding-top:1pt;padding-bottom:1pt;"> <span class="ltx_text" id="S3.T4.4.4.2.3"></span><span class="ltx_text" id="S3.T4.4.4.2.2"> <span class="ltx_tabular ltx_align_top" id="S3.T4.4.4.2.2.2"> <span class="ltx_tr" id="S3.T4.4.4.2.2.2.3"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.4.4.2.2.2.3.1" style="padding-top:1pt;padding-bottom:1pt;">two targets close by with</span></span> <span class="ltx_tr" id="S3.T4.4.4.2.2.2.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.4.4.2.2.2.2.2" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sigma_{0}=10" class="ltx_Math" display="inline" id="S3.T4.3.3.1.1.1.1.1.m1.1"><semantics id="S3.T4.3.3.1.1.1.1.1.m1.1a"><mrow id="S3.T4.3.3.1.1.1.1.1.m1.1.1" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.cmml"><msub id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.cmml"><mi id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.2" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.2.cmml">σ</mi><mn id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.3" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.T4.3.3.1.1.1.1.1.m1.1.1.1" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.1.cmml">=</mo><mn id="S3.T4.3.3.1.1.1.1.1.m1.1.1.3" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T4.3.3.1.1.1.1.1.m1.1b"><apply id="S3.T4.3.3.1.1.1.1.1.m1.1.1.cmml" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1"><eq id="S3.T4.3.3.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.1"></eq><apply id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.1.cmml" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.2.cmml" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.2">𝜎</ci><cn id="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.2.3">0</cn></apply><cn id="S3.T4.3.3.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T4.3.3.1.1.1.1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.3.3.1.1.1.1.1.m1.1c">\sigma_{0}=10</annotation><annotation encoding="application/x-llamapun" id="S3.T4.3.3.1.1.1.1.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 10</annotation></semantics></math> and <math alttext="\sigma_{1}=00" class="ltx_Math" display="inline" id="S3.T4.4.4.2.2.2.2.2.m2.1"><semantics id="S3.T4.4.4.2.2.2.2.2.m2.1a"><mrow id="S3.T4.4.4.2.2.2.2.2.m2.1.1" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.cmml"><msub id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.cmml"><mi id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.2" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.2.cmml">σ</mi><mn id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.3" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.T4.4.4.2.2.2.2.2.m2.1.1.1" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.1.cmml">=</mo><mn id="S3.T4.4.4.2.2.2.2.2.m2.1.1.3" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.3.cmml">00</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T4.4.4.2.2.2.2.2.m2.1b"><apply id="S3.T4.4.4.2.2.2.2.2.m2.1.1.cmml" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1"><eq id="S3.T4.4.4.2.2.2.2.2.m2.1.1.1.cmml" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.1"></eq><apply id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.cmml" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.1.cmml" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2">subscript</csymbol><ci id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.2.cmml" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.2">𝜎</ci><cn id="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.3.cmml" type="integer" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.2.3">1</cn></apply><cn id="S3.T4.4.4.2.2.2.2.2.m2.1.1.3.cmml" type="integer" xref="S3.T4.4.4.2.2.2.2.2.m2.1.1.3">00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.4.4.2.2.2.2.2.m2.1c">\sigma_{1}=00</annotation><annotation encoding="application/x-llamapun" id="S3.T4.4.4.2.2.2.2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 00</annotation></semantics></math></span></span> </span></span><span class="ltx_text" id="S3.T4.4.4.2.4"></span></td> </tr> <tr class="ltx_tr" id="S3.T4.5.5"> <td class="ltx_td ltx_align_left" id="S3.T4.5.5.2" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_italic" id="S3.T4.5.5.2.1">5 mixed objects</span></td> <td class="ltx_td ltx_align_left" id="S3.T4.5.5.1" style="padding-top:1pt;padding-bottom:1pt;"> <span class="ltx_text" id="S3.T4.5.5.1.2"></span><span class="ltx_text" id="S3.T4.5.5.1.1"> <span class="ltx_tabular ltx_align_top" id="S3.T4.5.5.1.1.1"> <span class="ltx_tr" id="S3.T4.5.5.1.1.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.5.5.1.1.1.2.1" style="padding-top:1pt;padding-bottom:1pt;">five targets randomly distributed with</span></span> <span class="ltx_tr" id="S3.T4.5.5.1.1.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.5.5.1.1.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\vec{\sigma}=[0,5,10,15,20]" class="ltx_Math" display="inline" id="S3.T4.5.5.1.1.1.1.1.m1.5"><semantics id="S3.T4.5.5.1.1.1.1.1.m1.5a"><mrow id="S3.T4.5.5.1.1.1.1.1.m1.5.6" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.cmml"><mover accent="true" id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.cmml"><mi id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.2" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.2.cmml">σ</mi><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.1" stretchy="false" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.1.cmml">→</mo></mover><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.1" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.1.cmml">=</mo><mrow id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml"><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.1" stretchy="false" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">[</mo><mn id="S3.T4.5.5.1.1.1.1.1.m1.1.1" xref="S3.T4.5.5.1.1.1.1.1.m1.1.1.cmml">0</mn><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.2" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">,</mo><mn id="S3.T4.5.5.1.1.1.1.1.m1.2.2" xref="S3.T4.5.5.1.1.1.1.1.m1.2.2.cmml">5</mn><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.3" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">,</mo><mn id="S3.T4.5.5.1.1.1.1.1.m1.3.3" xref="S3.T4.5.5.1.1.1.1.1.m1.3.3.cmml">10</mn><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.4" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">,</mo><mn id="S3.T4.5.5.1.1.1.1.1.m1.4.4" xref="S3.T4.5.5.1.1.1.1.1.m1.4.4.cmml">15</mn><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.5" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">,</mo><mn id="S3.T4.5.5.1.1.1.1.1.m1.5.5" xref="S3.T4.5.5.1.1.1.1.1.m1.5.5.cmml">20</mn><mo id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2.6" stretchy="false" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.T4.5.5.1.1.1.1.1.m1.5b"><apply id="S3.T4.5.5.1.1.1.1.1.m1.5.6.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6"><eq id="S3.T4.5.5.1.1.1.1.1.m1.5.6.1.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.1"></eq><apply id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2"><ci id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.1.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.1">→</ci><ci id="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.2.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.2.2">𝜎</ci></apply><list id="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.1.cmml" xref="S3.T4.5.5.1.1.1.1.1.m1.5.6.3.2"><cn id="S3.T4.5.5.1.1.1.1.1.m1.1.1.cmml" type="integer" xref="S3.T4.5.5.1.1.1.1.1.m1.1.1">0</cn><cn id="S3.T4.5.5.1.1.1.1.1.m1.2.2.cmml" type="integer" xref="S3.T4.5.5.1.1.1.1.1.m1.2.2">5</cn><cn id="S3.T4.5.5.1.1.1.1.1.m1.3.3.cmml" type="integer" xref="S3.T4.5.5.1.1.1.1.1.m1.3.3">10</cn><cn id="S3.T4.5.5.1.1.1.1.1.m1.4.4.cmml" type="integer" xref="S3.T4.5.5.1.1.1.1.1.m1.4.4">15</cn><cn id="S3.T4.5.5.1.1.1.1.1.m1.5.5.cmml" type="integer" xref="S3.T4.5.5.1.1.1.1.1.m1.5.5">20</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.5.5.1.1.1.1.1.m1.5c">\vec{\sigma}=[0,5,10,15,20]</annotation><annotation encoding="application/x-llamapun" id="S3.T4.5.5.1.1.1.1.1.m1.5d">over→ start_ARG italic_σ end_ARG = [ 0 , 5 , 10 , 15 , 20 ]</annotation></semantics></math></span></span> </span></span><span class="ltx_text" id="S3.T4.5.5.1.3"></span></td> </tr> <tr class="ltx_tr" id="S3.T4.6.6"> <td class="ltx_td ltx_align_left" id="S3.T4.6.6.2" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_italic" id="S3.T4.6.6.2.1">5 persons</span></td> <td class="ltx_td ltx_align_left" id="S3.T4.6.6.1" style="padding-top:1pt;padding-bottom:1pt;"> <span class="ltx_text" id="S3.T4.6.6.1.2"></span><span class="ltx_text" id="S3.T4.6.6.1.1"> <span class="ltx_tabular ltx_align_top" id="S3.T4.6.6.1.1.1"> <span class="ltx_tr" id="S3.T4.6.6.1.1.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.6.6.1.1.1.2.1" style="padding-top:1pt;padding-bottom:1pt;">five targets randomly distributed with</span></span> <span class="ltx_tr" id="S3.T4.6.6.1.1.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S3.T4.6.6.1.1.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\vec{\sigma}=[0,0,5,5,10]" class="ltx_Math" display="inline" id="S3.T4.6.6.1.1.1.1.1.m1.5"><semantics 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]</annotation></semantics></math></span></span> </span></span><span class="ltx_text" id="S3.T4.6.6.1.3"></span></td> </tr> <tr class="ltx_tr" id="S3.T4.6.8"> <td class="ltx_td ltx_align_left" id="S3.T4.6.8.1" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_italic" id="S3.T4.6.8.1.1">8 targets</span></td> <td class="ltx_td ltx_align_left" id="S3.T4.6.8.2" style="padding-top:1pt;padding-bottom:1pt;">up to 8 targets randomly distributed</td> </tr> </table> </figure> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.5.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.6.2">Metric</span> </h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">Since our model resembles the concept of frequency analysis, we compare our results with the FT. To evaluate the performance of our model, we measure the signal-to-noise ratio (SNR) and the accuracy of the target detection by applying the CA-CFAR algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib3" title="">3</a>]</cite> to the neurons’ output to create classification maps. These maps indicate whether a target is present at a given frequency bin for a distance or an angle. The two approaches, SNN and FT, are compared by their SNR, F-score, precision, and recall values.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">The sparsity of the SNN approaches leads to a majority of zeros in the range-angle map. 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xref="S3.E22.m1.1.1.1.1.3.3.2.2">𝑑</ci><apply id="S3.E22.m1.1.1.1.1.3.3.2.3.cmml" xref="S3.E22.m1.1.1.1.1.3.3.2.3"><times id="S3.E22.m1.1.1.1.1.3.3.2.3.1.cmml" xref="S3.E22.m1.1.1.1.1.3.3.2.3.1"></times><ci id="S3.E22.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S3.E22.m1.1.1.1.1.3.3.2.3.2">𝑖</ci><ci id="S3.E22.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S3.E22.m1.1.1.1.1.3.3.2.3.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E22.m1.1c">\displaystyle\text{SNR}_{\text{target}}=\frac{d_{\text{target}}}{\sum_{ij}d_{% ij}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E22.m1.1d">SNR start_POSTSUBSCRIPT target end_POSTSUBSCRIPT = divide start_ARG italic_d start_POSTSUBSCRIPT target end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p2.5">This adjustment leads to <math alttext="\text{SNR}=1" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m1.1"><semantics id="S3.SS2.p2.2.m1.1a"><mrow id="S3.SS2.p2.2.m1.1.1" xref="S3.SS2.p2.2.m1.1.1.cmml"><mtext id="S3.SS2.p2.2.m1.1.1.2" xref="S3.SS2.p2.2.m1.1.1.2a.cmml">SNR</mtext><mo id="S3.SS2.p2.2.m1.1.1.1" xref="S3.SS2.p2.2.m1.1.1.1.cmml">=</mo><mn id="S3.SS2.p2.2.m1.1.1.3" xref="S3.SS2.p2.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m1.1b"><apply id="S3.SS2.p2.2.m1.1.1.cmml" xref="S3.SS2.p2.2.m1.1.1"><eq id="S3.SS2.p2.2.m1.1.1.1.cmml" xref="S3.SS2.p2.2.m1.1.1.1"></eq><ci id="S3.SS2.p2.2.m1.1.1.2a.cmml" xref="S3.SS2.p2.2.m1.1.1.2"><mtext id="S3.SS2.p2.2.m1.1.1.2.cmml" xref="S3.SS2.p2.2.m1.1.1.2">SNR</mtext></ci><cn id="S3.SS2.p2.2.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p2.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m1.1c">\text{SNR}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m1.1d">SNR = 1</annotation></semantics></math> for no noise. For simplicity, we fix the number of neighboring cells and use no guard cells for all tests. Hence, we parameterize the CA-CFAR algorithm by only two parameters: an offset <math alttext="o" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m2.1"><semantics id="S3.SS2.p2.3.m2.1a"><mi id="S3.SS2.p2.3.m2.1.1" xref="S3.SS2.p2.3.m2.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m2.1b"><ci id="S3.SS2.p2.3.m2.1.1.cmml" xref="S3.SS2.p2.3.m2.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m2.1c">o</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m2.1d">italic_o</annotation></semantics></math> that adds to the threshold for the cell under test (CUT) derived from the window and a weight factor <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m3.1"><semantics id="S3.SS2.p2.4.m3.1a"><mi id="S3.SS2.p2.4.m3.1.1" xref="S3.SS2.p2.4.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m3.1b"><ci id="S3.SS2.p2.4.m3.1.1.cmml" xref="S3.SS2.p2.4.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m3.1d">italic_α</annotation></semantics></math> that scales the resulting threshold. The threshold for CUT <math alttext="c_{ij}" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m4.1"><semantics id="S3.SS2.p2.5.m4.1a"><msub id="S3.SS2.p2.5.m4.1.1" xref="S3.SS2.p2.5.m4.1.1.cmml"><mi id="S3.SS2.p2.5.m4.1.1.2" xref="S3.SS2.p2.5.m4.1.1.2.cmml">c</mi><mrow id="S3.SS2.p2.5.m4.1.1.3" xref="S3.SS2.p2.5.m4.1.1.3.cmml"><mi id="S3.SS2.p2.5.m4.1.1.3.2" xref="S3.SS2.p2.5.m4.1.1.3.2.cmml">i</mi><mo id="S3.SS2.p2.5.m4.1.1.3.1" xref="S3.SS2.p2.5.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.5.m4.1.1.3.3" xref="S3.SS2.p2.5.m4.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m4.1b"><apply id="S3.SS2.p2.5.m4.1.1.cmml" xref="S3.SS2.p2.5.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m4.1.1.1.cmml" xref="S3.SS2.p2.5.m4.1.1">subscript</csymbol><ci id="S3.SS2.p2.5.m4.1.1.2.cmml" xref="S3.SS2.p2.5.m4.1.1.2">𝑐</ci><apply id="S3.SS2.p2.5.m4.1.1.3.cmml" xref="S3.SS2.p2.5.m4.1.1.3"><times id="S3.SS2.p2.5.m4.1.1.3.1.cmml" xref="S3.SS2.p2.5.m4.1.1.3.1"></times><ci id="S3.SS2.p2.5.m4.1.1.3.2.cmml" xref="S3.SS2.p2.5.m4.1.1.3.2">𝑖</ci><ci id="S3.SS2.p2.5.m4.1.1.3.3.cmml" xref="S3.SS2.p2.5.m4.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m4.1c">c_{ij}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m4.1d">italic_c start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx20"> <tbody id="S3.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Theta_{\text{CUT}}(c_{ij})=\alpha\cdot\left(\frac{1}{K}\sum_{kl}% ^{K}c_{kl}+o\right)\,," class="ltx_Math" display="inline" id="S3.E23.m1.1"><semantics id="S3.E23.m1.1a"><mrow id="S3.E23.m1.1.1.1" xref="S3.E23.m1.1.1.1.1.cmml"><mrow 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italic_c start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) = italic_α ⋅ ( divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT + italic_o ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p2.8">with <math alttext="{K}" class="ltx_Math" display="inline" id="S3.SS2.p2.6.m1.1"><semantics id="S3.SS2.p2.6.m1.1a"><mi id="S3.SS2.p2.6.m1.1.1" xref="S3.SS2.p2.6.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.6.m1.1b"><ci id="S3.SS2.p2.6.m1.1.1.cmml" xref="S3.SS2.p2.6.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.6.m1.1c">{K}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m1.1d">italic_K</annotation></semantics></math> as a set of neighboring cells. We fixed <math alttext="K=3\times 5" class="ltx_Math" display="inline" id="S3.SS2.p2.7.m2.1"><semantics id="S3.SS2.p2.7.m2.1a"><mrow id="S3.SS2.p2.7.m2.1.1" xref="S3.SS2.p2.7.m2.1.1.cmml"><mi id="S3.SS2.p2.7.m2.1.1.2" xref="S3.SS2.p2.7.m2.1.1.2.cmml">K</mi><mo id="S3.SS2.p2.7.m2.1.1.1" xref="S3.SS2.p2.7.m2.1.1.1.cmml">=</mo><mrow id="S3.SS2.p2.7.m2.1.1.3" xref="S3.SS2.p2.7.m2.1.1.3.cmml"><mn id="S3.SS2.p2.7.m2.1.1.3.2" xref="S3.SS2.p2.7.m2.1.1.3.2.cmml">3</mn><mo id="S3.SS2.p2.7.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p2.7.m2.1.1.3.1.cmml">×</mo><mn id="S3.SS2.p2.7.m2.1.1.3.3" xref="S3.SS2.p2.7.m2.1.1.3.3.cmml">5</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.7.m2.1b"><apply id="S3.SS2.p2.7.m2.1.1.cmml" xref="S3.SS2.p2.7.m2.1.1"><eq id="S3.SS2.p2.7.m2.1.1.1.cmml" xref="S3.SS2.p2.7.m2.1.1.1"></eq><ci id="S3.SS2.p2.7.m2.1.1.2.cmml" xref="S3.SS2.p2.7.m2.1.1.2">𝐾</ci><apply id="S3.SS2.p2.7.m2.1.1.3.cmml" xref="S3.SS2.p2.7.m2.1.1.3"><times id="S3.SS2.p2.7.m2.1.1.3.1.cmml" xref="S3.SS2.p2.7.m2.1.1.3.1"></times><cn id="S3.SS2.p2.7.m2.1.1.3.2.cmml" type="integer" xref="S3.SS2.p2.7.m2.1.1.3.2">3</cn><cn id="S3.SS2.p2.7.m2.1.1.3.3.cmml" type="integer" xref="S3.SS2.p2.7.m2.1.1.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.7.m2.1c">K=3\times 5</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.7.m2.1d">italic_K = 3 × 5</annotation></semantics></math> for all experiments in the range and angle dimensions, respectively. Typically, the SNN output contains mainly zeros. Therefore, the offset <math alttext="o" class="ltx_Math" display="inline" id="S3.SS2.p2.8.m3.1"><semantics id="S3.SS2.p2.8.m3.1a"><mi id="S3.SS2.p2.8.m3.1.1" xref="S3.SS2.p2.8.m3.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.8.m3.1b"><ci id="S3.SS2.p2.8.m3.1.1.cmml" xref="S3.SS2.p2.8.m3.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.8.m3.1c">o</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.8.m3.1d">italic_o</annotation></semantics></math> is needed to generate useful results of the CA-CFAR algorithm.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS3.5.1.1">III-C</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS3.6.2">Parameter optimization</span> </h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">We distinguish two parts in our neuron model: the gradient estimation and the spiking function. To tune the parameters of the whole neuron model, we optimized the two parts consecutively. First, we optimized the parameters of the gradient estimation and CA-CFAR using a grid search on a training data set by maximizing the F-score. Second, we set the gradient estimation parameters and optimized the spiking functions’ parameters with the same approach.</p> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS4.5.1.1">III-D</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS4.6.2">Model evaluation for a single chirp</span> </h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.9">The three spiking functions were evaluated and compared to the FT results. In addition, we analyzed the results of the gradient estimation by directly using the value of the estimated gradient. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T5" title="TABLE V ‣ III-D Model evaluation for a single chirp ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">V</span></a> summarizes the optimized parameters of each model, and Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T6" title="TABLE VI ‣ III-D Model evaluation for a single chirp ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">VI</span></a> shows the F-score, precision, and recall for each spiking model and the comparable FT approach over all datasets. These results show that the rate- and time-coded spiking functions reach higher F-scores than the gradient or adaptive threshold model. Because the results of the gradient model and FT are similar, we can conclude that the advantage of the two spiking models mainly lies in the non-linear mapping of the spiking function itself. The adaptive threshold spiking function, which utilizes a linear mapping from values to spikes, shows only a slight loss in the F-score, affirming the previous assumption. An intrinsic cut-off can explain the increased SNR of all neuron models, as only estimated gradients larger than <math alttext="0" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.1"><semantics id="S3.SS4.p1.1.m1.1a"><mn id="S3.SS4.p1.1.m1.1.1" xref="S3.SS4.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.1.m1.1b"><cn id="S3.SS4.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS4.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math> are transmitted. Even though our model eliminates sensor data storage by continuous processing and transmits only spikes, it performs similarly in classification accuracy as the FT. The intrinsic cut-off and the usage of binary spikes drastically reduce the data bandwidth, as the number of generated spikes shows. The entire network consists of <math alttext="256\times 32=8192" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.1"><semantics id="S3.SS4.p1.2.m2.1a"><mrow id="S3.SS4.p1.2.m2.1.1" xref="S3.SS4.p1.2.m2.1.1.cmml"><mrow id="S3.SS4.p1.2.m2.1.1.2" xref="S3.SS4.p1.2.m2.1.1.2.cmml"><mn id="S3.SS4.p1.2.m2.1.1.2.2" xref="S3.SS4.p1.2.m2.1.1.2.2.cmml">256</mn><mo id="S3.SS4.p1.2.m2.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS4.p1.2.m2.1.1.2.1.cmml">×</mo><mn id="S3.SS4.p1.2.m2.1.1.2.3" xref="S3.SS4.p1.2.m2.1.1.2.3.cmml">32</mn></mrow><mo id="S3.SS4.p1.2.m2.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS4.p1.2.m2.1.1.3" xref="S3.SS4.p1.2.m2.1.1.3.cmml">8192</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.1b"><apply id="S3.SS4.p1.2.m2.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1"><eq id="S3.SS4.p1.2.m2.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1"></eq><apply id="S3.SS4.p1.2.m2.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.2"><times id="S3.SS4.p1.2.m2.1.1.2.1.cmml" xref="S3.SS4.p1.2.m2.1.1.2.1"></times><cn id="S3.SS4.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S3.SS4.p1.2.m2.1.1.2.2">256</cn><cn id="S3.SS4.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S3.SS4.p1.2.m2.1.1.2.3">32</cn></apply><cn id="S3.SS4.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS4.p1.2.m2.1.1.3">8192</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.1c">256\times 32=8192</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.1d">256 × 32 = 8192</annotation></semantics></math> neurons. A network utilizing the adaptive threshold spiking function transmits, on average, <math alttext="27128" class="ltx_Math" display="inline" id="S3.SS4.p1.3.m3.1"><semantics id="S3.SS4.p1.3.m3.1a"><mn id="S3.SS4.p1.3.m3.1.1" xref="S3.SS4.p1.3.m3.1.1.cmml">27128</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.3.m3.1b"><cn id="S3.SS4.p1.3.m3.1.1.cmml" type="integer" xref="S3.SS4.p1.3.m3.1.1">27128</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.3.m3.1c">27128</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.3.m3.1d">27128</annotation></semantics></math> spikes, which is more than the other spiking functions, as it sends positive and negative spikes. The rate-coded spiking function transmits <math alttext="910" class="ltx_Math" display="inline" id="S3.SS4.p1.4.m4.1"><semantics id="S3.SS4.p1.4.m4.1a"><mn id="S3.SS4.p1.4.m4.1.1" xref="S3.SS4.p1.4.m4.1.1.cmml">910</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.4.m4.1b"><cn id="S3.SS4.p1.4.m4.1.1.cmml" type="integer" xref="S3.SS4.p1.4.m4.1.1">910</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.4.m4.1c">910</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.4.m4.1d">910</annotation></semantics></math> spikes per network, leading already to less than one spike per neuron on average. By utilizing the time-coded spiking function, the entire network transmits, on average, only <math alttext="53" class="ltx_Math" display="inline" id="S3.SS4.p1.5.m5.1"><semantics id="S3.SS4.p1.5.m5.1a"><mn id="S3.SS4.p1.5.m5.1.1" xref="S3.SS4.p1.5.m5.1.1.cmml">53</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.5.m5.1b"><cn id="S3.SS4.p1.5.m5.1.1.cmml" type="integer" xref="S3.SS4.p1.5.m5.1.1">53</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.5.m5.1c">53</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.5.m5.1d">53</annotation></semantics></math> spikes. In contrast, the classic FT approach transmits <math alttext="256\times 32=8192" class="ltx_Math" display="inline" id="S3.SS4.p1.6.m6.1"><semantics id="S3.SS4.p1.6.m6.1a"><mrow id="S3.SS4.p1.6.m6.1.1" xref="S3.SS4.p1.6.m6.1.1.cmml"><mrow id="S3.SS4.p1.6.m6.1.1.2" xref="S3.SS4.p1.6.m6.1.1.2.cmml"><mn id="S3.SS4.p1.6.m6.1.1.2.2" xref="S3.SS4.p1.6.m6.1.1.2.2.cmml">256</mn><mo id="S3.SS4.p1.6.m6.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS4.p1.6.m6.1.1.2.1.cmml">×</mo><mn id="S3.SS4.p1.6.m6.1.1.2.3" xref="S3.SS4.p1.6.m6.1.1.2.3.cmml">32</mn></mrow><mo id="S3.SS4.p1.6.m6.1.1.1" xref="S3.SS4.p1.6.m6.1.1.1.cmml">=</mo><mn id="S3.SS4.p1.6.m6.1.1.3" xref="S3.SS4.p1.6.m6.1.1.3.cmml">8192</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.6.m6.1b"><apply id="S3.SS4.p1.6.m6.1.1.cmml" xref="S3.SS4.p1.6.m6.1.1"><eq id="S3.SS4.p1.6.m6.1.1.1.cmml" xref="S3.SS4.p1.6.m6.1.1.1"></eq><apply id="S3.SS4.p1.6.m6.1.1.2.cmml" xref="S3.SS4.p1.6.m6.1.1.2"><times id="S3.SS4.p1.6.m6.1.1.2.1.cmml" xref="S3.SS4.p1.6.m6.1.1.2.1"></times><cn id="S3.SS4.p1.6.m6.1.1.2.2.cmml" type="integer" xref="S3.SS4.p1.6.m6.1.1.2.2">256</cn><cn id="S3.SS4.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S3.SS4.p1.6.m6.1.1.2.3">32</cn></apply><cn id="S3.SS4.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS4.p1.6.m6.1.1.3">8192</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.6.m6.1c">256\times 32=8192</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.6.m6.1d">256 × 32 = 8192</annotation></semantics></math> floating values, i.e., <math alttext="262\,144" class="ltx_Math" display="inline" id="S3.SS4.p1.7.m7.1"><semantics id="S3.SS4.p1.7.m7.1a"><mn id="S3.SS4.p1.7.m7.1.1" xref="S3.SS4.p1.7.m7.1.1.cmml">262 144</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.7.m7.1b"><cn id="S3.SS4.p1.7.m7.1.1.cmml" type="integer" xref="S3.SS4.p1.7.m7.1.1">262144</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.7.m7.1c">262\,144</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.7.m7.1d">262 144</annotation></semantics></math> bits per chirp, assuming float-<math alttext="32" class="ltx_Math" display="inline" id="S3.SS4.p1.8.m8.1"><semantics id="S3.SS4.p1.8.m8.1a"><mn id="S3.SS4.p1.8.m8.1.1" xref="S3.SS4.p1.8.m8.1.1.cmml">32</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.8.m8.1b"><cn id="S3.SS4.p1.8.m8.1.1.cmml" type="integer" xref="S3.SS4.p1.8.m8.1.1">32</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.8.m8.1c">32</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.8.m8.1d">32</annotation></semantics></math>. The spiking neural resonator network transmits only 0.02 <math alttext="\%" class="ltx_Math" display="inline" id="S3.SS4.p1.9.m9.1"><semantics id="S3.SS4.p1.9.m9.1a"><mo id="S3.SS4.p1.9.m9.1.1" xref="S3.SS4.p1.9.m9.1.1.cmml">%</mo><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.9.m9.1b"><csymbol cd="latexml" id="S3.SS4.p1.9.m9.1.1.cmml" xref="S3.SS4.p1.9.m9.1.1">percent</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.9.m9.1c">\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.9.m9.1d">%</annotation></semantics></math> of the data transmitted by the classic float-32 FT approach.</p> </div> <figure class="ltx_table" id="S3.T5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE V: </span>Optimized parameters of the spiking and non-spiking gradient models for the single chirp use-case</figcaption> <table class="ltx_tabular ltx_align_middle" id="S3.T5.4"> <tr class="ltx_tr" id="S3.T5.4.5"> <td class="ltx_td ltx_align_center" id="S3.T5.4.5.1" style="padding-top:1pt;padding-bottom:1pt;">Model name</td> <td class="ltx_td ltx_align_center" id="S3.T5.4.5.2" style="padding-top:1pt;padding-bottom:1pt;">Optimized parameters</td> </tr> <tr class="ltx_tr" id="S3.T5.1.1"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T5.1.1.2" style="padding-top:1pt;padding-bottom:1pt;">gradient model</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T5.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\alpha_{x}=0.6,\alpha_{g}=0.001" class="ltx_Math" display="inline" id="S3.T5.1.1.1.m1.2"><semantics id="S3.T5.1.1.1.m1.2a"><mrow id="S3.T5.1.1.1.m1.2.2.2" xref="S3.T5.1.1.1.m1.2.2.3.cmml"><mrow id="S3.T5.1.1.1.m1.1.1.1.1" xref="S3.T5.1.1.1.m1.1.1.1.1.cmml"><msub id="S3.T5.1.1.1.m1.1.1.1.1.2" xref="S3.T5.1.1.1.m1.1.1.1.1.2.cmml"><mi id="S3.T5.1.1.1.m1.1.1.1.1.2.2" xref="S3.T5.1.1.1.m1.1.1.1.1.2.2.cmml">α</mi><mi id="S3.T5.1.1.1.m1.1.1.1.1.2.3" 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xref="S3.T5.4.4.1.m1.2.2.2.2.2.2.1"></eq><ci id="S3.T5.4.4.1.m1.2.2.2.2.2.2.2.cmml" xref="S3.T5.4.4.1.m1.2.2.2.2.2.2.2">𝜏</ci><cn id="S3.T5.4.4.1.m1.2.2.2.2.2.2.3.cmml" type="integer" xref="S3.T5.4.4.1.m1.2.2.2.2.2.2.3">200</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T5.4.4.1.m1.2c">u_{\text{threshold}}=231,u_{\text{rest}}=250,\tau=200</annotation><annotation encoding="application/x-llamapun" id="S3.T5.4.4.1.m1.2d">italic_u start_POSTSUBSCRIPT threshold end_POSTSUBSCRIPT = 231 , italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT = 250 , italic_τ = 200</annotation></semantics></math></td> </tr> </table> </figure> <figure class="ltx_table" id="S3.T6"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE VI: </span>Evaluation results on simulated data after a single chirp</figcaption> <table class="ltx_tabular ltx_align_middle" id="S3.T6.3"> <tr class="ltx_tr" id="S3.T6.3.4"> <td class="ltx_td ltx_align_center" id="S3.T6.3.4.1" style="padding-top:1pt;padding-bottom:1pt;">Model name</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.4.2" style="padding-top:1pt;padding-bottom:1pt;">F-score</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.4.3" style="padding-top:1pt;padding-bottom:1pt;">Prec.</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.4.4" style="padding-top:1pt;padding-bottom:1pt;">Recall</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.4.5" style="padding-top:1pt;padding-bottom:1pt;">SNR</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.4.6" style="padding-top:1pt;padding-bottom:1pt;">avg. # spikes</td> </tr> <tr class="ltx_tr" id="S3.T6.3.5"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T6.3.5.1" style="padding-top:1pt;padding-bottom:1pt;">gradient model</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T6.3.5.2" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.3.5.2.1">0.61</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T6.3.5.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.3.5.3.1">0.66</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T6.3.5.4" style="padding-top:1pt;padding-bottom:1pt;">0.58</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T6.3.5.5" style="padding-top:1pt;padding-bottom:1pt;">0.012</td> <td class="ltx_td ltx_border_t" id="S3.T6.3.5.6" style="padding-top:1pt;padding-bottom:1pt;"></td> </tr> <tr class="ltx_tr" id="S3.T6.1.1"> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.2" style="padding-top:1pt;padding-bottom:1pt;">adaptive threshold</td> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.3" style="padding-top:1pt;padding-bottom:1pt;">0.60</td> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.4" style="padding-top:1pt;padding-bottom:1pt;">0.65</td> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.5" style="padding-top:1pt;padding-bottom:1pt;">0.58</td> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.6" style="padding-top:1pt;padding-bottom:1pt;">0.010</td> <td class="ltx_td ltx_align_left" id="S3.T6.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 27128" class="ltx_Math" display="inline" id="S3.T6.1.1.1.m1.1"><semantics id="S3.T6.1.1.1.m1.1a"><mrow id="S3.T6.1.1.1.m1.1.1" xref="S3.T6.1.1.1.m1.1.1.cmml"><mi id="S3.T6.1.1.1.m1.1.1.2" xref="S3.T6.1.1.1.m1.1.1.2.cmml"></mi><mo id="S3.T6.1.1.1.m1.1.1.1" xref="S3.T6.1.1.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T6.1.1.1.m1.1.1.3" xref="S3.T6.1.1.1.m1.1.1.3.cmml">27128</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T6.1.1.1.m1.1b"><apply id="S3.T6.1.1.1.m1.1.1.cmml" xref="S3.T6.1.1.1.m1.1.1"><csymbol cd="latexml" id="S3.T6.1.1.1.m1.1.1.1.cmml" xref="S3.T6.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T6.1.1.1.m1.1.1.2.cmml" xref="S3.T6.1.1.1.m1.1.1.2">absent</csymbol><cn id="S3.T6.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T6.1.1.1.m1.1.1.3">27128</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T6.1.1.1.m1.1c">\sim 27128</annotation><annotation encoding="application/x-llamapun" id="S3.T6.1.1.1.m1.1d">∼ 27128</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T6.2.2"> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.2" style="padding-top:1pt;padding-bottom:1pt;">rate-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.2.2.3.1">0.61</span></td> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.4" style="padding-top:1pt;padding-bottom:1pt;">0.59</td> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.5" style="padding-top:1pt;padding-bottom:1pt;">0.64</td> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.6" style="padding-top:1pt;padding-bottom:1pt;">0.104</td> <td class="ltx_td ltx_align_left" id="S3.T6.2.2.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 910" class="ltx_Math" display="inline" id="S3.T6.2.2.1.m1.1"><semantics id="S3.T6.2.2.1.m1.1a"><mrow id="S3.T6.2.2.1.m1.1.1" xref="S3.T6.2.2.1.m1.1.1.cmml"><mi id="S3.T6.2.2.1.m1.1.1.2" xref="S3.T6.2.2.1.m1.1.1.2.cmml"></mi><mo id="S3.T6.2.2.1.m1.1.1.1" xref="S3.T6.2.2.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T6.2.2.1.m1.1.1.3" xref="S3.T6.2.2.1.m1.1.1.3.cmml">910</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T6.2.2.1.m1.1b"><apply id="S3.T6.2.2.1.m1.1.1.cmml" xref="S3.T6.2.2.1.m1.1.1"><csymbol cd="latexml" id="S3.T6.2.2.1.m1.1.1.1.cmml" xref="S3.T6.2.2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T6.2.2.1.m1.1.1.2.cmml" xref="S3.T6.2.2.1.m1.1.1.2">absent</csymbol><cn id="S3.T6.2.2.1.m1.1.1.3.cmml" type="integer" xref="S3.T6.2.2.1.m1.1.1.3">910</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T6.2.2.1.m1.1c">\sim 910</annotation><annotation encoding="application/x-llamapun" id="S3.T6.2.2.1.m1.1d">∼ 910</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T6.3.3"> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.2" style="padding-top:1pt;padding-bottom:1pt;">time-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.3.3.3.1">0.61</span></td> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.4" style="padding-top:1pt;padding-bottom:1pt;">0.57</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.5" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.3.3.5.1">0.66</span></td> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.6" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T6.3.3.6.1">0.105</span></td> <td class="ltx_td ltx_align_left" id="S3.T6.3.3.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim\textbf{53}" class="ltx_Math" display="inline" id="S3.T6.3.3.1.m1.1"><semantics id="S3.T6.3.3.1.m1.1a"><mrow id="S3.T6.3.3.1.m1.1.1" xref="S3.T6.3.3.1.m1.1.1.cmml"><mi id="S3.T6.3.3.1.m1.1.1.2" xref="S3.T6.3.3.1.m1.1.1.2.cmml"></mi><mo id="S3.T6.3.3.1.m1.1.1.1" xref="S3.T6.3.3.1.m1.1.1.1.cmml">∼</mo><mtext class="ltx_mathvariant_bold" id="S3.T6.3.3.1.m1.1.1.3" xref="S3.T6.3.3.1.m1.1.1.3a.cmml">53</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T6.3.3.1.m1.1b"><apply id="S3.T6.3.3.1.m1.1.1.cmml" xref="S3.T6.3.3.1.m1.1.1"><csymbol cd="latexml" id="S3.T6.3.3.1.m1.1.1.1.cmml" xref="S3.T6.3.3.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T6.3.3.1.m1.1.1.2.cmml" xref="S3.T6.3.3.1.m1.1.1.2">absent</csymbol><ci id="S3.T6.3.3.1.m1.1.1.3a.cmml" xref="S3.T6.3.3.1.m1.1.1.3"><mtext class="ltx_mathvariant_bold" id="S3.T6.3.3.1.m1.1.1.3.cmml" xref="S3.T6.3.3.1.m1.1.1.3">53</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T6.3.3.1.m1.1c">\sim\textbf{53}</annotation><annotation encoding="application/x-llamapun" id="S3.T6.3.3.1.m1.1d">∼ 53</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T6.3.6"> <td class="ltx_td ltx_align_left" id="S3.T6.3.6.1" style="padding-top:1pt;padding-bottom:1pt;">FT</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.6.2" style="padding-top:1pt;padding-bottom:1pt;">0.59</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.6.3" style="padding-top:1pt;padding-bottom:1pt;">0.64</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.6.4" style="padding-top:1pt;padding-bottom:1pt;">0.56</td> <td class="ltx_td ltx_align_left" id="S3.T6.3.6.5" style="padding-top:1pt;padding-bottom:1pt;">0.006</td> <td class="ltx_td" id="S3.T6.3.6.6" style="padding-top:1pt;padding-bottom:1pt;"></td> </tr> </table> </figure> </section> <section class="ltx_subsection" id="S3.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS5.5.1.1">III-E</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS5.6.2">Model evaluation of early detections for a single chirp</span> </h3> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="221" id="S3.F4.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span> Early detection evaluation for adaptive threshold (adapt.), time-coded (time), rate-coded (rate), and the gradient (grad) model. After every <math alttext="64" class="ltx_Math" display="inline" id="S3.F4.2.m1.1"><semantics id="S3.F4.2.m1.1b"><mn id="S3.F4.2.m1.1.1" xref="S3.F4.2.m1.1.1.cmml">64</mn><annotation-xml encoding="MathML-Content" id="S3.F4.2.m1.1c"><cn id="S3.F4.2.m1.1.1.cmml" type="integer" xref="S3.F4.2.m1.1.1">64</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.2.m1.1d">64</annotation><annotation encoding="application/x-llamapun" id="S3.F4.2.m1.1e">64</annotation></semantics></math> data sample, we determine the neural resonator network’s F-score, precision, and recall. The maximum value of each model normalizes the values. For comparison, a linear reference is shown in red. The F-Score of the adaptive threshold, rate-coded, and gradient model increases almost linearly, as the resolution of the Fourier transform also increases linearly with the number of samples. The time-coded model shows a different behavior, as most informative spikes happen at the end of a chirp. </figcaption> </figure> <div class="ltx_para" id="S3.SS5.p1"> <p class="ltx_p" id="S3.SS5.p1.3">Because our neuron model continuously processes sensor data and transmits information by sending spikes, objects are already detected during data sampling. In contrast, the FT relies on sampling and storing the data before performing any calculation. Therefore, we also studied the early detection accuracy of our neuron model by evaluating the spike output at every <math alttext="64" class="ltx_Math" display="inline" id="S3.SS5.p1.1.m1.1"><semantics id="S3.SS5.p1.1.m1.1a"><mn id="S3.SS5.p1.1.m1.1.1" xref="S3.SS5.p1.1.m1.1.1.cmml">64</mn><annotation-xml encoding="MathML-Content" id="S3.SS5.p1.1.m1.1b"><cn id="S3.SS5.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS5.p1.1.m1.1.1">64</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p1.1.m1.1c">64</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p1.1.m1.1d">64</annotation></semantics></math>th time step to analyze whether early predictions of close-by objects are possible. We performed the evaluation on the <span class="ltx_text ltx_font_italic" id="S3.SS5.p1.3.1">close targets 2010</span> dataset. The recall results depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.F4" title="Figure 4 ‣ III-E Model evaluation of early detections for a single chirp ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">4</span></a> show that for the adaptive threshold and rate-coded spiking functions already after half of the available samples <math alttext="N_{\text{samples}}/2" class="ltx_Math" display="inline" id="S3.SS5.p1.2.m2.1"><semantics id="S3.SS5.p1.2.m2.1a"><mrow id="S3.SS5.p1.2.m2.1.1" xref="S3.SS5.p1.2.m2.1.1.cmml"><msub id="S3.SS5.p1.2.m2.1.1.2" xref="S3.SS5.p1.2.m2.1.1.2.cmml"><mi id="S3.SS5.p1.2.m2.1.1.2.2" xref="S3.SS5.p1.2.m2.1.1.2.2.cmml">N</mi><mtext id="S3.SS5.p1.2.m2.1.1.2.3" xref="S3.SS5.p1.2.m2.1.1.2.3a.cmml">samples</mtext></msub><mo id="S3.SS5.p1.2.m2.1.1.1" xref="S3.SS5.p1.2.m2.1.1.1.cmml">/</mo><mn id="S3.SS5.p1.2.m2.1.1.3" xref="S3.SS5.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p1.2.m2.1b"><apply id="S3.SS5.p1.2.m2.1.1.cmml" xref="S3.SS5.p1.2.m2.1.1"><divide id="S3.SS5.p1.2.m2.1.1.1.cmml" xref="S3.SS5.p1.2.m2.1.1.1"></divide><apply id="S3.SS5.p1.2.m2.1.1.2.cmml" xref="S3.SS5.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS5.p1.2.m2.1.1.2.1.cmml" xref="S3.SS5.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS5.p1.2.m2.1.1.2.2.cmml" xref="S3.SS5.p1.2.m2.1.1.2.2">𝑁</ci><ci id="S3.SS5.p1.2.m2.1.1.2.3a.cmml" xref="S3.SS5.p1.2.m2.1.1.2.3"><mtext id="S3.SS5.p1.2.m2.1.1.2.3.cmml" mathsize="70%" xref="S3.SS5.p1.2.m2.1.1.2.3">samples</mtext></ci></apply><cn id="S3.SS5.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS5.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p1.2.m2.1c">N_{\text{samples}}/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p1.2.m2.1d">italic_N start_POSTSUBSCRIPT samples end_POSTSUBSCRIPT / 2</annotation></semantics></math> more than <math alttext="75" class="ltx_Math" display="inline" id="S3.SS5.p1.3.m3.1"><semantics id="S3.SS5.p1.3.m3.1a"><mn id="S3.SS5.p1.3.m3.1.1" xref="S3.SS5.p1.3.m3.1.1.cmml">75</mn><annotation-xml encoding="MathML-Content" id="S3.SS5.p1.3.m3.1b"><cn id="S3.SS5.p1.3.m3.1.1.cmml" type="integer" xref="S3.SS5.p1.3.m3.1.1">75</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p1.3.m3.1c">75</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p1.3.m3.1d">75</annotation></semantics></math> % of the detectable objects are recognized. The time-coded spiking function, however, needs due to its internal dynamics more time to produce a spike. Since most of the peaks in the frequency spectrum are close to zero, the spikes typically happen at the end of a chirp.</p> </div> </section> <section class="ltx_subsection" id="S3.SS6"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS6.5.1.1">III-F</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS6.6.2">Model evaluation for multiple consecutive chirps</span> </h3> <div class="ltx_para" id="S3.SS6.p1"> <p class="ltx_p" id="S3.SS6.p1.4">Due to a high chirp rate (<math alttext="\sim" class="ltx_Math" display="inline" id="S3.SS6.p1.1.m1.1"><semantics id="S3.SS6.p1.1.m1.1a"><mo id="S3.SS6.p1.1.m1.1.1" xref="S3.SS6.p1.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.SS6.p1.1.m1.1b"><csymbol cd="latexml" id="S3.SS6.p1.1.m1.1.1.cmml" xref="S3.SS6.p1.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.p1.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.p1.1.m1.1d">∼</annotation></semantics></math> kHz) of FMCW radar sensor <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib11" title="">11</a>]</cite> changes in the scenes between consecutive chirps are hardly detected in range-angle maps. Consequently, it is natural to exploit the continuous computation capabilities of the neurons to process consecutive chirps without resetting the entire network. Therefore, we also evaluate the classification accuracy after processing multiple chirps of the same scene. We follow the optimization strategy described before and summarize the new parameters for processing <math alttext="8" class="ltx_Math" display="inline" id="S3.SS6.p1.2.m2.1"><semantics id="S3.SS6.p1.2.m2.1a"><mn id="S3.SS6.p1.2.m2.1.1" xref="S3.SS6.p1.2.m2.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S3.SS6.p1.2.m2.1b"><cn id="S3.SS6.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS6.p1.2.m2.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.p1.2.m2.1c">8</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.p1.2.m2.1d">8</annotation></semantics></math> chirps in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T7" title="TABLE VII ‣ III-F Model evaluation for multiple consecutive chirps ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">VII</span></a>. It is important to emphasize that the optimal parameters depend not directly on the number of processed chirps but on the state of convergence to the gradient estimation. 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By optimizing the parameters with the assumption of a converged gradient estimation, results are optimal for all chirps <math alttext="n_{\text{chirps}}>\tilde{N}_{\text{chirps}}" class="ltx_Math" display="inline" id="S3.SS6.p1.4.m4.1"><semantics id="S3.SS6.p1.4.m4.1a"><mrow id="S3.SS6.p1.4.m4.1.1" xref="S3.SS6.p1.4.m4.1.1.cmml"><msub id="S3.SS6.p1.4.m4.1.1.2" xref="S3.SS6.p1.4.m4.1.1.2.cmml"><mi id="S3.SS6.p1.4.m4.1.1.2.2" xref="S3.SS6.p1.4.m4.1.1.2.2.cmml">n</mi><mtext id="S3.SS6.p1.4.m4.1.1.2.3" xref="S3.SS6.p1.4.m4.1.1.2.3a.cmml">chirps</mtext></msub><mo id="S3.SS6.p1.4.m4.1.1.1" xref="S3.SS6.p1.4.m4.1.1.1.cmml">></mo><msub id="S3.SS6.p1.4.m4.1.1.3" xref="S3.SS6.p1.4.m4.1.1.3.cmml"><mover accent="true" id="S3.SS6.p1.4.m4.1.1.3.2" xref="S3.SS6.p1.4.m4.1.1.3.2.cmml"><mi id="S3.SS6.p1.4.m4.1.1.3.2.2" xref="S3.SS6.p1.4.m4.1.1.3.2.2.cmml">N</mi><mo id="S3.SS6.p1.4.m4.1.1.3.2.1" xref="S3.SS6.p1.4.m4.1.1.3.2.1.cmml">~</mo></mover><mtext id="S3.SS6.p1.4.m4.1.1.3.3" xref="S3.SS6.p1.4.m4.1.1.3.3a.cmml">chirps</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS6.p1.4.m4.1b"><apply id="S3.SS6.p1.4.m4.1.1.cmml" xref="S3.SS6.p1.4.m4.1.1"><gt id="S3.SS6.p1.4.m4.1.1.1.cmml" xref="S3.SS6.p1.4.m4.1.1.1"></gt><apply id="S3.SS6.p1.4.m4.1.1.2.cmml" xref="S3.SS6.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS6.p1.4.m4.1.1.2.1.cmml" xref="S3.SS6.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS6.p1.4.m4.1.1.2.2.cmml" xref="S3.SS6.p1.4.m4.1.1.2.2">𝑛</ci><ci id="S3.SS6.p1.4.m4.1.1.2.3a.cmml" xref="S3.SS6.p1.4.m4.1.1.2.3"><mtext id="S3.SS6.p1.4.m4.1.1.2.3.cmml" mathsize="70%" xref="S3.SS6.p1.4.m4.1.1.2.3">chirps</mtext></ci></apply><apply id="S3.SS6.p1.4.m4.1.1.3.cmml" xref="S3.SS6.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS6.p1.4.m4.1.1.3.1.cmml" xref="S3.SS6.p1.4.m4.1.1.3">subscript</csymbol><apply id="S3.SS6.p1.4.m4.1.1.3.2.cmml" xref="S3.SS6.p1.4.m4.1.1.3.2"><ci id="S3.SS6.p1.4.m4.1.1.3.2.1.cmml" xref="S3.SS6.p1.4.m4.1.1.3.2.1">~</ci><ci id="S3.SS6.p1.4.m4.1.1.3.2.2.cmml" xref="S3.SS6.p1.4.m4.1.1.3.2.2">𝑁</ci></apply><ci id="S3.SS6.p1.4.m4.1.1.3.3a.cmml" xref="S3.SS6.p1.4.m4.1.1.3.3"><mtext id="S3.SS6.p1.4.m4.1.1.3.3.cmml" mathsize="70%" xref="S3.SS6.p1.4.m4.1.1.3.3">chirps</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.p1.4.m4.1c">n_{\text{chirps}}>\tilde{N}_{\text{chirps}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.p1.4.m4.1d">italic_n start_POSTSUBSCRIPT chirps end_POSTSUBSCRIPT > over~ start_ARG italic_N end_ARG start_POSTSUBSCRIPT chirps end_POSTSUBSCRIPT</annotation></semantics></math>. Earlier processed chirps will result in reduced accuracy.</p> </div> <div class="ltx_para" id="S3.SS6.p2"> <p class="ltx_p" id="S3.SS6.p2.1">The results summarized in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T8" title="TABLE VIII ‣ III-F Model evaluation for multiple consecutive chirps ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">VIII</span></a> show an increased performance compared to the single chirp evaluation. These results are also compared to an evaluation of averaging the output over the same <math alttext="8" class="ltx_Math" display="inline" id="S3.SS6.p2.1.m1.1"><semantics id="S3.SS6.p2.1.m1.1a"><mn id="S3.SS6.p2.1.m1.1.1" xref="S3.SS6.p2.1.m1.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S3.SS6.p2.1.m1.1b"><cn id="S3.SS6.p2.1.m1.1.1.cmml" type="integer" xref="S3.SS6.p2.1.m1.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.p2.1.m1.1c">8</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.p2.1.m1.1d">8</annotation></semantics></math> consecutive chirps. In this case, the same parameters of the single chirp evaluation are taken (Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T5" title="TABLE V ‣ III-D Model evaluation for a single chirp ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">V</span></a>). Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T9" title="TABLE IX ‣ III-F Model evaluation for multiple consecutive chirps ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">IX</span></a> summarizes the results for F-Scores, precision, recall, SNR, and spike numbers, including the results of averaging the FT calculations. Since every single chirp produces similar spike numbers, the average number is higher than the continuous processing of chirps in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.T8" title="TABLE VIII ‣ III-F Model evaluation for multiple consecutive chirps ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">VIII</span></a>. The detection accuracy of the neuron models is comparable and situated between the continuous processing and averaging approach, leading to the conclusion that storing outputs of multiple chirps is unnecessary. In general, the FT approach performs slightly worse than all neuron models.</p> </div> <figure class="ltx_table" id="S3.T7"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE VII: </span>Optimized parameters of the spiking and non-spiking gradient models for the consecutive chirps use-case</figcaption> <table class="ltx_tabular ltx_align_middle" id="S3.T7.4"> <tr class="ltx_tr" id="S3.T7.4.5"> <td class="ltx_td ltx_align_center" id="S3.T7.4.5.1" style="padding-top:1pt;padding-bottom:1pt;">Model name</td> <td class="ltx_td ltx_align_center" id="S3.T7.4.5.2" style="padding-top:1pt;padding-bottom:1pt;">Optimized parameters</td> </tr> <tr class="ltx_tr" id="S3.T7.1.1"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T7.1.1.2" style="padding-top:1pt;padding-bottom:1pt;">gradient model</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T7.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\alpha_{x}=0.6,\alpha_{g}=0.001" class="ltx_Math" 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0.001</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T7.2.2"> <td class="ltx_td ltx_align_left" id="S3.T7.2.2.2" style="padding-top:1pt;padding-bottom:1pt;">adaptive threshold</td> <td class="ltx_td ltx_align_left" id="S3.T7.2.2.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\gamma=0.1" class="ltx_Math" display="inline" id="S3.T7.2.2.1.m1.1"><semantics id="S3.T7.2.2.1.m1.1a"><mrow id="S3.T7.2.2.1.m1.1.1" xref="S3.T7.2.2.1.m1.1.1.cmml"><mi id="S3.T7.2.2.1.m1.1.1.2" xref="S3.T7.2.2.1.m1.1.1.2.cmml">γ</mi><mo id="S3.T7.2.2.1.m1.1.1.1" xref="S3.T7.2.2.1.m1.1.1.1.cmml">=</mo><mn id="S3.T7.2.2.1.m1.1.1.3" xref="S3.T7.2.2.1.m1.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T7.2.2.1.m1.1b"><apply id="S3.T7.2.2.1.m1.1.1.cmml" xref="S3.T7.2.2.1.m1.1.1"><eq id="S3.T7.2.2.1.m1.1.1.1.cmml" xref="S3.T7.2.2.1.m1.1.1.1"></eq><ci id="S3.T7.2.2.1.m1.1.1.2.cmml" xref="S3.T7.2.2.1.m1.1.1.2">𝛾</ci><cn id="S3.T7.2.2.1.m1.1.1.3.cmml" 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mathsize="70%" xref="S3.T7.4.4.1.m1.2.2.2.2.1.1.2.3">rest</mtext></ci></apply><cn id="S3.T7.4.4.1.m1.2.2.2.2.1.1.3.cmml" type="integer" xref="S3.T7.4.4.1.m1.2.2.2.2.1.1.3">250</cn></apply><apply id="S3.T7.4.4.1.m1.2.2.2.2.2.2.cmml" xref="S3.T7.4.4.1.m1.2.2.2.2.2.2"><eq id="S3.T7.4.4.1.m1.2.2.2.2.2.2.1.cmml" xref="S3.T7.4.4.1.m1.2.2.2.2.2.2.1"></eq><ci id="S3.T7.4.4.1.m1.2.2.2.2.2.2.2.cmml" xref="S3.T7.4.4.1.m1.2.2.2.2.2.2.2">𝜏</ci><cn id="S3.T7.4.4.1.m1.2.2.2.2.2.2.3.cmml" type="integer" xref="S3.T7.4.4.1.m1.2.2.2.2.2.2.3">200</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T7.4.4.1.m1.2c">u_{\text{threshold}}=232,u_{\text{rest}}=250,\tau=200</annotation><annotation encoding="application/x-llamapun" id="S3.T7.4.4.1.m1.2d">italic_u start_POSTSUBSCRIPT threshold end_POSTSUBSCRIPT = 232 , italic_u start_POSTSUBSCRIPT rest end_POSTSUBSCRIPT = 250 , italic_τ = 200</annotation></semantics></math></td> </tr> </table> </figure> <figure class="ltx_table" id="S3.T8"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE VIII: </span>Evaluation results on simulated data after eight consecutive chirps</figcaption> <table class="ltx_tabular ltx_align_middle" id="S3.T8.3"> <tr class="ltx_tr" id="S3.T8.3.4"> <td class="ltx_td ltx_align_center" id="S3.T8.3.4.1" style="padding-top:1pt;padding-bottom:1pt;">Model name</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.4.2" style="padding-top:1pt;padding-bottom:1pt;">F-score</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.4.3" style="padding-top:1pt;padding-bottom:1pt;">Prec.</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.4.4" style="padding-top:1pt;padding-bottom:1pt;">Recall</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.4.5" style="padding-top:1pt;padding-bottom:1pt;">SNR</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.4.6" style="padding-top:1pt;padding-bottom:1pt;">avg. # spikes</td> </tr> <tr class="ltx_tr" id="S3.T8.3.5"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T8.3.5.1" style="padding-top:1pt;padding-bottom:1pt;">gradient model</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T8.3.5.2" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T8.3.5.2.1">0.72</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T8.3.5.3" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T8.3.5.3.1">0.73</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T8.3.5.4" style="padding-top:1pt;padding-bottom:1pt;">0.71</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T8.3.5.5" style="padding-top:1pt;padding-bottom:1pt;">0.013</td> <td class="ltx_td ltx_border_t" id="S3.T8.3.5.6" style="padding-top:1pt;padding-bottom:1pt;"></td> </tr> <tr class="ltx_tr" id="S3.T8.1.1"> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.2" style="padding-top:1pt;padding-bottom:1pt;">adaptive threshold</td> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.3" style="padding-top:1pt;padding-bottom:1pt;">0.60</td> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.4" style="padding-top:1pt;padding-bottom:1pt;">0.65</td> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.5" style="padding-top:1pt;padding-bottom:1pt;">0.58</td> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.6" style="padding-top:1pt;padding-bottom:1pt;">0.010</td> <td class="ltx_td ltx_align_left" id="S3.T8.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 2579" class="ltx_Math" display="inline" id="S3.T8.1.1.1.m1.1"><semantics id="S3.T8.1.1.1.m1.1a"><mrow id="S3.T8.1.1.1.m1.1.1" xref="S3.T8.1.1.1.m1.1.1.cmml"><mi id="S3.T8.1.1.1.m1.1.1.2" xref="S3.T8.1.1.1.m1.1.1.2.cmml"></mi><mo id="S3.T8.1.1.1.m1.1.1.1" xref="S3.T8.1.1.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T8.1.1.1.m1.1.1.3" xref="S3.T8.1.1.1.m1.1.1.3.cmml">2579</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T8.1.1.1.m1.1b"><apply id="S3.T8.1.1.1.m1.1.1.cmml" xref="S3.T8.1.1.1.m1.1.1"><csymbol cd="latexml" id="S3.T8.1.1.1.m1.1.1.1.cmml" xref="S3.T8.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T8.1.1.1.m1.1.1.2.cmml" xref="S3.T8.1.1.1.m1.1.1.2">absent</csymbol><cn id="S3.T8.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T8.1.1.1.m1.1.1.3">2579</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T8.1.1.1.m1.1c">\sim 2579</annotation><annotation encoding="application/x-llamapun" id="S3.T8.1.1.1.m1.1d">∼ 2579</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T8.2.2"> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.2" style="padding-top:1pt;padding-bottom:1pt;">rate-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.3" style="padding-top:1pt;padding-bottom:1pt;">0.69</td> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.4" style="padding-top:1pt;padding-bottom:1pt;">0.67</td> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.5" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T8.2.2.5.1">0.72</span></td> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.6" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T8.2.2.6.1">0.117</span></td> <td class="ltx_td ltx_align_left" id="S3.T8.2.2.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 126" class="ltx_Math" display="inline" id="S3.T8.2.2.1.m1.1"><semantics id="S3.T8.2.2.1.m1.1a"><mrow id="S3.T8.2.2.1.m1.1.1" xref="S3.T8.2.2.1.m1.1.1.cmml"><mi id="S3.T8.2.2.1.m1.1.1.2" xref="S3.T8.2.2.1.m1.1.1.2.cmml"></mi><mo id="S3.T8.2.2.1.m1.1.1.1" xref="S3.T8.2.2.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T8.2.2.1.m1.1.1.3" xref="S3.T8.2.2.1.m1.1.1.3.cmml">126</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T8.2.2.1.m1.1b"><apply id="S3.T8.2.2.1.m1.1.1.cmml" xref="S3.T8.2.2.1.m1.1.1"><csymbol cd="latexml" id="S3.T8.2.2.1.m1.1.1.1.cmml" xref="S3.T8.2.2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T8.2.2.1.m1.1.1.2.cmml" xref="S3.T8.2.2.1.m1.1.1.2">absent</csymbol><cn id="S3.T8.2.2.1.m1.1.1.3.cmml" type="integer" xref="S3.T8.2.2.1.m1.1.1.3">126</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T8.2.2.1.m1.1c">\sim 126</annotation><annotation encoding="application/x-llamapun" id="S3.T8.2.2.1.m1.1d">∼ 126</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T8.3.3"> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.2" style="padding-top:1pt;padding-bottom:1pt;">time-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.3" style="padding-top:1pt;padding-bottom:1pt;">0.68</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.4" style="padding-top:1pt;padding-bottom:1pt;">0.66</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.5" style="padding-top:1pt;padding-bottom:1pt;">0.71</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.6" style="padding-top:1pt;padding-bottom:1pt;">0.103</td> <td class="ltx_td ltx_align_left" id="S3.T8.3.3.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim\textbf{5}" class="ltx_Math" display="inline" id="S3.T8.3.3.1.m1.1"><semantics id="S3.T8.3.3.1.m1.1a"><mrow id="S3.T8.3.3.1.m1.1.1" xref="S3.T8.3.3.1.m1.1.1.cmml"><mi id="S3.T8.3.3.1.m1.1.1.2" xref="S3.T8.3.3.1.m1.1.1.2.cmml"></mi><mo id="S3.T8.3.3.1.m1.1.1.1" xref="S3.T8.3.3.1.m1.1.1.1.cmml">∼</mo><mtext class="ltx_mathvariant_bold" id="S3.T8.3.3.1.m1.1.1.3" xref="S3.T8.3.3.1.m1.1.1.3a.cmml">5</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T8.3.3.1.m1.1b"><apply id="S3.T8.3.3.1.m1.1.1.cmml" xref="S3.T8.3.3.1.m1.1.1"><csymbol cd="latexml" id="S3.T8.3.3.1.m1.1.1.1.cmml" xref="S3.T8.3.3.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T8.3.3.1.m1.1.1.2.cmml" xref="S3.T8.3.3.1.m1.1.1.2">absent</csymbol><ci id="S3.T8.3.3.1.m1.1.1.3a.cmml" xref="S3.T8.3.3.1.m1.1.1.3"><mtext class="ltx_mathvariant_bold" id="S3.T8.3.3.1.m1.1.1.3.cmml" xref="S3.T8.3.3.1.m1.1.1.3">5</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T8.3.3.1.m1.1c">\sim\textbf{5}</annotation><annotation encoding="application/x-llamapun" id="S3.T8.3.3.1.m1.1d">∼ 5</annotation></semantics></math></td> </tr> </table> </figure> <figure class="ltx_table" id="S3.T9"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE IX: </span>Evaluation results on simulated data averaging eight consecutive chirps.</figcaption> <table class="ltx_tabular ltx_align_middle" id="S3.T9.3"> <tr class="ltx_tr" id="S3.T9.3.4"> <td class="ltx_td ltx_align_center" id="S3.T9.3.4.1" style="padding-top:1pt;padding-bottom:1pt;">Model name</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.4.2" style="padding-top:1pt;padding-bottom:1pt;">F-score</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.4.3" style="padding-top:1pt;padding-bottom:1pt;">Prec.</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.4.4" style="padding-top:1pt;padding-bottom:1pt;">Recall</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.4.5" style="padding-top:1pt;padding-bottom:1pt;">SNR</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.4.6" style="padding-top:1pt;padding-bottom:1pt;">avg. # spikes</td> </tr> <tr class="ltx_tr" id="S3.T9.3.5"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T9.3.5.1" style="padding-top:1pt;padding-bottom:1pt;">gradient model</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T9.3.5.2" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T9.3.5.2.1">0.72</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T9.3.5.3" style="padding-top:1pt;padding-bottom:1pt;">0.71</td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T9.3.5.4" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T9.3.5.4.1">0.72</span></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T9.3.5.5" style="padding-top:1pt;padding-bottom:1pt;">0.012</td> <td class="ltx_td ltx_border_t" id="S3.T9.3.5.6" style="padding-top:1pt;padding-bottom:1pt;"></td> </tr> <tr class="ltx_tr" id="S3.T9.1.1"> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.2" style="padding-top:1pt;padding-bottom:1pt;">adaptive threshold</td> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.3" style="padding-top:1pt;padding-bottom:1pt;">0.64</td> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.4" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T9.1.1.4.1">0.75</span></td> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.5" style="padding-top:1pt;padding-bottom:1pt;">0.56</td> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.6" style="padding-top:1pt;padding-bottom:1pt;">0.010</td> <td class="ltx_td ltx_align_left" id="S3.T9.1.1.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 20632" class="ltx_Math" display="inline" id="S3.T9.1.1.1.m1.1"><semantics id="S3.T9.1.1.1.m1.1a"><mrow id="S3.T9.1.1.1.m1.1.1" xref="S3.T9.1.1.1.m1.1.1.cmml"><mi id="S3.T9.1.1.1.m1.1.1.2" xref="S3.T9.1.1.1.m1.1.1.2.cmml"></mi><mo id="S3.T9.1.1.1.m1.1.1.1" xref="S3.T9.1.1.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T9.1.1.1.m1.1.1.3" xref="S3.T9.1.1.1.m1.1.1.3.cmml">20632</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T9.1.1.1.m1.1b"><apply id="S3.T9.1.1.1.m1.1.1.cmml" xref="S3.T9.1.1.1.m1.1.1"><csymbol cd="latexml" id="S3.T9.1.1.1.m1.1.1.1.cmml" xref="S3.T9.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T9.1.1.1.m1.1.1.2.cmml" xref="S3.T9.1.1.1.m1.1.1.2">absent</csymbol><cn id="S3.T9.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T9.1.1.1.m1.1.1.3">20632</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T9.1.1.1.m1.1c">\sim 20632</annotation><annotation encoding="application/x-llamapun" id="S3.T9.1.1.1.m1.1d">∼ 20632</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T9.2.2"> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.2" style="padding-top:1pt;padding-bottom:1pt;">rate-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.3" style="padding-top:1pt;padding-bottom:1pt;">0.63</td> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.4" style="padding-top:1pt;padding-bottom:1pt;">0.59</td> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.5" style="padding-top:1pt;padding-bottom:1pt;">0.67</td> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.6" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T9.2.2.6.1">0.105</span></td> <td class="ltx_td ltx_align_left" id="S3.T9.2.2.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim 372" class="ltx_Math" display="inline" id="S3.T9.2.2.1.m1.1"><semantics id="S3.T9.2.2.1.m1.1a"><mrow id="S3.T9.2.2.1.m1.1.1" xref="S3.T9.2.2.1.m1.1.1.cmml"><mi id="S3.T9.2.2.1.m1.1.1.2" xref="S3.T9.2.2.1.m1.1.1.2.cmml"></mi><mo id="S3.T9.2.2.1.m1.1.1.1" xref="S3.T9.2.2.1.m1.1.1.1.cmml">∼</mo><mn id="S3.T9.2.2.1.m1.1.1.3" xref="S3.T9.2.2.1.m1.1.1.3.cmml">372</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T9.2.2.1.m1.1b"><apply id="S3.T9.2.2.1.m1.1.1.cmml" xref="S3.T9.2.2.1.m1.1.1"><csymbol cd="latexml" id="S3.T9.2.2.1.m1.1.1.1.cmml" xref="S3.T9.2.2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T9.2.2.1.m1.1.1.2.cmml" xref="S3.T9.2.2.1.m1.1.1.2">absent</csymbol><cn id="S3.T9.2.2.1.m1.1.1.3.cmml" type="integer" xref="S3.T9.2.2.1.m1.1.1.3">372</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T9.2.2.1.m1.1c">\sim 372</annotation><annotation encoding="application/x-llamapun" id="S3.T9.2.2.1.m1.1d">∼ 372</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T9.3.3"> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.2" style="padding-top:1pt;padding-bottom:1pt;">time-coded LIF</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.3" style="padding-top:1pt;padding-bottom:1pt;">0.63</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.4" style="padding-top:1pt;padding-bottom:1pt;">0.59</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.5" style="padding-top:1pt;padding-bottom:1pt;">0.68</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.6" style="padding-top:1pt;padding-bottom:1pt;"><span class="ltx_text ltx_font_bold" id="S3.T9.3.3.6.1">0.105</span></td> <td class="ltx_td ltx_align_left" id="S3.T9.3.3.1" style="padding-top:1pt;padding-bottom:1pt;"><math alttext="\sim\textbf{43}" class="ltx_Math" display="inline" id="S3.T9.3.3.1.m1.1"><semantics id="S3.T9.3.3.1.m1.1a"><mrow id="S3.T9.3.3.1.m1.1.1" xref="S3.T9.3.3.1.m1.1.1.cmml"><mi id="S3.T9.3.3.1.m1.1.1.2" xref="S3.T9.3.3.1.m1.1.1.2.cmml"></mi><mo id="S3.T9.3.3.1.m1.1.1.1" xref="S3.T9.3.3.1.m1.1.1.1.cmml">∼</mo><mtext class="ltx_mathvariant_bold" id="S3.T9.3.3.1.m1.1.1.3" xref="S3.T9.3.3.1.m1.1.1.3a.cmml">43</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.T9.3.3.1.m1.1b"><apply id="S3.T9.3.3.1.m1.1.1.cmml" xref="S3.T9.3.3.1.m1.1.1"><csymbol cd="latexml" id="S3.T9.3.3.1.m1.1.1.1.cmml" xref="S3.T9.3.3.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.T9.3.3.1.m1.1.1.2.cmml" xref="S3.T9.3.3.1.m1.1.1.2">absent</csymbol><ci id="S3.T9.3.3.1.m1.1.1.3a.cmml" xref="S3.T9.3.3.1.m1.1.1.3"><mtext class="ltx_mathvariant_bold" id="S3.T9.3.3.1.m1.1.1.3.cmml" xref="S3.T9.3.3.1.m1.1.1.3">43</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T9.3.3.1.m1.1c">\sim\textbf{43}</annotation><annotation encoding="application/x-llamapun" id="S3.T9.3.3.1.m1.1d">∼ 43</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T9.3.6"> <td class="ltx_td ltx_align_left" id="S3.T9.3.6.1" style="padding-top:1pt;padding-bottom:1pt;">FT</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.6.2" style="padding-top:1pt;padding-bottom:1pt;">0.60</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.6.3" style="padding-top:1pt;padding-bottom:1pt;">0.63</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.6.4" style="padding-top:1pt;padding-bottom:1pt;">0.59</td> <td class="ltx_td ltx_align_left" id="S3.T9.3.6.5" style="padding-top:1pt;padding-bottom:1pt;">0.006</td> <td class="ltx_td" id="S3.T9.3.6.6" style="padding-top:1pt;padding-bottom:1pt;"></td> </tr> </table> </figure> </section> <section class="ltx_subsection" id="S3.SS7"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS7.5.1.1">III-G</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS7.6.2">Visual results on real data</span> </h3> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="277" id="S3.F5.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 5: </span> Comparison of range-angle maps from FMCW radar sensor data of the RADICAL dataset for one scene. (Left) RBG image of the RADICAL dataset of the corresponding scene. (Right, Top) Full-resolution range-angle maps showing the full spectrum. (Right, Bottom) Zoomed-in extract of the two people detected by the radar sensor. The noise level of the range-angle map of the Fourier Transform is higher than the non-spiking and spiking range-angle map of neural resonator networks. The gradient and adaptive threshold spiking function maps are similar, as are the rate-coded and time-coded maps. </figcaption> </figure> <div class="ltx_para" id="S3.SS7.p1"> <p class="ltx_p" id="S3.SS7.p1.1">To show the applicability of our model on real-world radar sensor data, we applied it to publicly available raw radar data. The availability of raw radar data including a multi-antenna layout is limited. The RADICAL dataset consists of a sample data file of indoor recordings of walking persons. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.F5" title="Figure 5 ‣ III-G Visual results on real data ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">5</span></a> shows a scene containing <math alttext="4" class="ltx_Math" display="inline" id="S3.SS7.p1.1.m1.1"><semantics id="S3.SS7.p1.1.m1.1a"><mn id="S3.SS7.p1.1.m1.1.1" xref="S3.SS7.p1.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S3.SS7.p1.1.m1.1b"><cn id="S3.SS7.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS7.p1.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS7.p1.1.m1.1c">4</annotation><annotation encoding="application/x-llamapun" id="S3.SS7.p1.1.m1.1d">4</annotation></semantics></math> persons, two walking close to the sensor. We compare the range-angle maps of the FT calculation, the gradient model, and the three different spiking functions. The comparison shows a similar frequency spectrum for all approaches (upper row in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.F5" title="Figure 5 ‣ III-G Visual results on real data ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">5</span></a>). However, our models show a significantly reduced noise level, which becomes evident in the zoomed-in extracts of the range-angle maps (lower row in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#S3.F5" title="Figure 5 ‣ III-G Visual results on real data ‣ III Evaluation ‣ Range and Angle Estimation with Spiking Neural Resonators for FMCW Radar"><span class="ltx_text ltx_ref_tag">5</span></a>).</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">This work extends the resonate-and-fire model to optimize it for radar processing applications. We focused on reducing latency and data bandwidth by allowing spikes during the sampling of radar data, eliminating the need to store any sensor data. Our continuous approach, therefore, stands out in contrast to the classic Fourier transformation, where all data points are sampled, stored, and processed afterward. In general, utilizing only a subset of all samples within a chirp reduces the resolution of the frequency analysis. However, the neuron model spikes during sampling time; we have shown that the object detection accuracy is comparable to or even better than the FT approach. This is achieved by first estimating the first-order part (linear in time) of the resonator dynamics, which filters out noise from non-resonating objects, and second, by emitting spikes proportional to the gradient of the linear part, which is comparable to a threshold function since only positive gradient above a specific value can produce spikes. The three different spiking functions compared in this work show similar target detection accuracy but have several trade-offs one must consider. The adaptive threshold spiking function is the most straightforward of these three but transmits the most spikes, including negative ones. The rate-coded spiking function has a similar complexity as the time-coded function but uses more spikes to convey information and is therefore suitable for early detection. The time-coded spiking function utilizes the least number of spikes but lacks the feature of early detection. To conclude, we have demonstrated that a network based on our spiking neural resonators can achieve the same performance as the FT while reducing the latency and data bandwidth. The data bandwidth can be reduced to 0.02 <math alttext="\%" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mo id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">%</mo><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><csymbol cd="latexml" id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">percent</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\%</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">%</annotation></semantics></math> of the data bandwidth for a float-32 FT.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Future work</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">The current state of the network does not implement any lateral connections; therefore, each neuron processes information independently, allowing complete parallel computations and optimizations. Introducing laterally inhibiting connections between neurons can theoretically improve the detection accuracy by suppressing neighboring neurons excited by the same target.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">So far, we evaluated a static data set, i.e. the targets were not moving, and we did not consider temporal changes. The continuous setup of the neuron model does not reset the internal variable of the gradient estimation and uses the information of previous chirps. Therefore, we plan to evaluate the performance and the reaction time of the network in a highly dynamic and realistic environment.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">The neuron model is continuous in time and can theoretically process analog data directly from the radar sensor. Analog processors that implement this model can be highly efficient in energy consumption and should be explored in the future. First implementations of analog resonate-and-fire models have been studied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00898v1#bib.bib14" title="">14</a>]</cite>. Directly processing analog signals and transmitting only sparse spikes for further post-processing paves the way for efficient neuromorphic radar sensors.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_font_smallcaps ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">This research has been funded by the Federal Ministry of Education and Research of Germany in the framework of the KI-ASIC project (16ES0995).</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Daniel Auge, Julian Hille, Etienne Mueller, and Alois Knoll. </span> <span class="ltx_bibblock">A survey of encoding techniques for signal processing in spiking neural networks. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Neural Processing Letters</span>, 53, 07 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> Daniel Auge and Etienne Mueller. </span> <span class="ltx_bibblock">Resonate-and-fire neurons as frequency selective input encoders for spiking neural networks. </span> <span class="ltx_bibblock">Technical report, Technical University Munich, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> Mourad Barkat and P.K. 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