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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning</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=" Channel knowledge map, environment-aware communication, deep-learning, cell-free networks. " lang="en" name="keywords"/> <base href="/html/2411.17716v1/"/></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/2411.17716v1#S1" title="In Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><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/2411.17716v1#S2" title="In Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps"> Problem Formulation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S3" title="In Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Model Training</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/2411.17716v1#S3.SS1" title="In III Model Training ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><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">Input Data Generation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S3.SS2" title="In III Model Training ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><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">UNet Design and Training</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S4" title="In Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Inference Results</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/2411.17716v1#S4.SS1" title="In IV Inference Results ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">Training Settings</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S4.SS2" title="In IV Inference Results ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Training Results</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S5" title="In Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Conclusion</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">Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Zhuoyin Dai, Di Wu, Xiaoli Xu, , and Yong Zeng, </span><span class="ltx_author_notes"> Z. Dai, D. Wu, X. Xu, and Y. Zeng are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China. Y. Zeng is also with the Purple Mountain Laboratories, Nanjing 211111, China (e-mail: {zhuoyin_dai, studywudi, xiaolixu, yong_zeng, }@seu.edu.cn). (<em class="ltx_emph ltx_font_italic" id="id1.1.id1">Corresponding author: Yong Zeng.</em>) </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.id1">Channel knowledge map (CKM) is a promising paradigm shift towards environment-aware communication and sensing by providing location-specific prior channel knowledge before real-time communication. Although CKM is particularly appealing for dense networks such as cell-free networks, it remains a challenge to efficiently generate CKMs in dense networks. For a dense network with CKMs of existing access points (APs), it will be useful to efficiently generate CKMs of potentially new APs with only AP location information. The generation of inferred CKMs across APs can help dense networks achieve convenient initial CKM generation, environment-aware AP deployment, and cost-effective CKM updates. Considering that different APs in the same region share the same physical environment, there exists a natural correlation between the channel knowledge of different APs. Therefore, by mining the implicit correlation between location-specific channel knowledge, cross-AP CKM inference can be realized using data from other APs. This paper proposes a cross-AP inference method to generate CKMs of potentially new APs with deep learning. The location of the target AP is fed into the UNet model in combination with the channel knowledge of other existing APs, and supervised learning is performed based on the channel knowledge of the target AP. Based on the trained UNet and the channel knowledge of the existing APs, the CKM inference of the potentially new AP can be generated across APs. The generation results of the inferred CKM validate the feasibility and effectiveness of cross-AP CKM inference with other APs’ channel knowledge. </p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Channel knowledge map, environment-aware communication, deep-learning, cell-free networks. </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 a promising paradigm shift from conventional environment-unaware to environment-aware communication and sensing, channel knowledge map (CKM) has been recently proposed to address the challenge of channel knowledge acquisition with the prior local environment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib2" title="">2</a>]</cite>. As a location-specific channel knowledge database, CKM can effectively reflect the channel knowledge related to the node location and dependent on the local environment, such as channel gain, time of arrival (ToA), angle of arrival (AoA), angle of departure (DoA), etc <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib3" title="">3</a>]</cite>. Different from the physical environment map <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib4" title="">4</a>]</cite>, CKM focuses on the intrinsic characteristics of wireless channels, which effectively avoids the complicated computation from physical environment to channel knowledge. Therefore, the prior local environment embedded in CKMs can greatly facilitate the performance optimization and resource allocation of future wireless communications.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Efficient construction of CKM is the key to realizing CKM-based environment-aware communication and sensing. Essentially, the construction of CKM is the process of combining limited data with prediction methods such as interpolation and inference and obtaining the location-specific channel knowledge in the region <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib2" title="">2</a>]</cite>. CKM construction based on existing channel knowledge can be categorized into same-AP construction and cross-AP construction. For the same AP, the CKM can be completed or the CKM resolution can be improved by using the physical environment map <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib5" title="">5</a>]</cite> or the channel knowledge of the nearest neighbor nodes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib6" title="">6</a>]</cite>. For the cross-AP construction, the mutual information proves the existence of channel state information (CSI) dependence across APs, and CSI features such as received power at a specific location are inferred from the source CSI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib7" title="">7</a>]</cite>. Learning-based channel mapping is also used for cross-antenna channel prediction at specific candidate locations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib8" title="">8</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Different from the CSI inference for a specific location across APs/antennas above, this letter focuses on the fundamental problem that constructing the complete CKMs of potentially new APs efficiently based on the CKMs of existing APs. Specifically, consider a dense network, such as a cell-free network, in which the existing APs are equipped with CKMs for the region. A potentially new AP is introduced into the network with only its location information known. A cross-AP CKM generation system needs to be designed whose inputs are the location information of the potentially new AP with the CKMs of other APs in the network, while the output is the complete CKM inference of the new AP. With the densification of network nodes, the generation of the cross-AP inferred CKMs of potentially new APs effectively expands the system potential. The overhead of constructing CKMs for all APs can be reduced during the initial CKM construction phase through a combination of measurement and cross-AP inference strategies. For newly introduced potential APs, traversing to generate CKMs in different locations can guide the environment-aware placement. CKM inference across APs also realizes cost-effective CKM updates during subsequent system maintenance.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">The cross-AP CKM inference is built on the location diversity of distributed APs and the same physical environment they share. Specifically, the wireless environment is an outward manifestation of the physical environment, which is also embedded in the spatial variations of the wireless environment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib2" title="">2</a>]</cite>. Therefore, there is a natural correlation and dependence between the CKMs of distributed APs at different locations. Although this implicit relationship is difficult to characterize concretely, it can be applied to cross-AP CKM inference with learning-based approaches. In this paper, neural network is used to learn the implicit correlation between CKMs of different APs related to the wireless environment, and ultimately to realize cross-AP CKM inference. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">1</span></a>, this learning-based cross-AP CKM inference avoids the complex computation from physical environment to channel knowledge. The UNet model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib9" title="">9</a>]</cite> is first trained with CKM datasets from different physical environments. After training, the locations of the potentially new APs are fed into the model along with the CKMs of other existing APs to output the CKMs of the potentially new APs. This cross-AP CKM inference learns the correlation between CKMs and is capable of generating complete CKMs of potentially new APs with high accuracy. The generated CKM results are compared with other benchmarks to validate the feasibility and effectiveness of CKM inference across APs.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="448" id="S1.F1.g1" src="x1.png" width="622"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Model of the Cross-AP CKM Inference.</figcaption> </figure> </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"> Problem Formulation</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.6">As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">1</span></a>, consider a cell-free network with <math alttext="N" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_N</annotation></semantics></math> APs.The coordinate of the <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_n</annotation></semantics></math>th AP is denoted as <math alttext="\mathbf{c}_{n}\in\mathbb{R}^{2\times 1}" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mrow id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml"><msub id="S2.p1.3.m3.1.1.2" xref="S2.p1.3.m3.1.1.2.cmml"><mi id="S2.p1.3.m3.1.1.2.2" xref="S2.p1.3.m3.1.1.2.2.cmml">𝐜</mi><mi id="S2.p1.3.m3.1.1.2.3" xref="S2.p1.3.m3.1.1.2.3.cmml">n</mi></msub><mo id="S2.p1.3.m3.1.1.1" xref="S2.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S2.p1.3.m3.1.1.3" xref="S2.p1.3.m3.1.1.3.cmml"><mi id="S2.p1.3.m3.1.1.3.2" xref="S2.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S2.p1.3.m3.1.1.3.3" xref="S2.p1.3.m3.1.1.3.3.cmml"><mn id="S2.p1.3.m3.1.1.3.3.2" xref="S2.p1.3.m3.1.1.3.3.2.cmml">2</mn><mo id="S2.p1.3.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p1.3.m3.1.1.3.3.1.cmml">×</mo><mn id="S2.p1.3.m3.1.1.3.3.3" xref="S2.p1.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><apply id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1"><in id="S2.p1.3.m3.1.1.1.cmml" xref="S2.p1.3.m3.1.1.1"></in><apply id="S2.p1.3.m3.1.1.2.cmml" xref="S2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.p1.3.m3.1.1.2.1.cmml" xref="S2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.p1.3.m3.1.1.2.2.cmml" xref="S2.p1.3.m3.1.1.2.2">𝐜</ci><ci id="S2.p1.3.m3.1.1.2.3.cmml" xref="S2.p1.3.m3.1.1.2.3">𝑛</ci></apply><apply id="S2.p1.3.m3.1.1.3.cmml" xref="S2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.p1.3.m3.1.1.3.1.cmml" xref="S2.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.p1.3.m3.1.1.3.2.cmml" xref="S2.p1.3.m3.1.1.3.2">ℝ</ci><apply id="S2.p1.3.m3.1.1.3.3.cmml" xref="S2.p1.3.m3.1.1.3.3"><times id="S2.p1.3.m3.1.1.3.3.1.cmml" xref="S2.p1.3.m3.1.1.3.3.1"></times><cn id="S2.p1.3.m3.1.1.3.3.2.cmml" type="integer" xref="S2.p1.3.m3.1.1.3.3.2">2</cn><cn id="S2.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S2.p1.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">\mathbf{c}_{n}\in\mathbb{R}^{2\times 1}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">bold_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 × 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. The problem to be solved is: how to construct a CKM for a potentially new AP at location <math alttext="\mathbf{c}_{0}" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><msub id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml">𝐜</mi><mn id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1">subscript</csymbol><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">𝐜</ci><cn id="S2.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">\mathbf{c}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">bold_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> based on the CKMs of existing APs in the cell-free network? Specifically, the problem is first analyzed with a typical kind of CKM, channel gain map (CGM), as an example. The entire network area is divided into <math alttext="W\times W" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><mrow id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml"><mi id="S2.p1.5.m5.1.1.2" xref="S2.p1.5.m5.1.1.2.cmml">W</mi><mo id="S2.p1.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.p1.5.m5.1.1.1.cmml">×</mo><mi id="S2.p1.5.m5.1.1.3" xref="S2.p1.5.m5.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><apply id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1"><times id="S2.p1.5.m5.1.1.1.cmml" xref="S2.p1.5.m5.1.1.1"></times><ci id="S2.p1.5.m5.1.1.2.cmml" xref="S2.p1.5.m5.1.1.2">𝑊</ci><ci id="S2.p1.5.m5.1.1.3.cmml" xref="S2.p1.5.m5.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">W\times W</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">italic_W × italic_W</annotation></semantics></math> grids, where each grid records a channel gain value. Therefore, for the <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_n</annotation></semantics></math>th AP, its CKM, which mainly stores the location of the AP itself and the channel gain corresponding to each grid, can be expressed as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{M}_{n}=\{\mathbf{c}_{n};\mathbf{G}_{n}\}," class="ltx_Math" display="block" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><msub id="S2.E1.m1.1.1.1.1.4" xref="S2.E1.m1.1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.1.1.1.4.2" xref="S2.E1.m1.1.1.1.1.4.2.cmml">ℳ</mi><mi id="S2.E1.m1.1.1.1.1.4.3" xref="S2.E1.m1.1.1.1.1.4.3.cmml">n</mi></msub><mo id="S2.E1.m1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.2.3.cmml"><mo id="S2.E1.m1.1.1.1.1.2.2.3" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.3.cmml">{</mo><msub id="S2.E1.m1.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.1.2.cmml">𝐜</mi><mi id="S2.E1.m1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.E1.m1.1.1.1.1.2.2.4" xref="S2.E1.m1.1.1.1.1.2.3.cmml">;</mo><msub id="S2.E1.m1.1.1.1.1.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2.2.2" xref="S2.E1.m1.1.1.1.1.2.2.2.2.cmml">𝐆</mi><mi id="S2.E1.m1.1.1.1.1.2.2.2.3" xref="S2.E1.m1.1.1.1.1.2.2.2.3.cmml">n</mi></msub><mo id="S2.E1.m1.1.1.1.1.2.2.5" stretchy="false" xref="S2.E1.m1.1.1.1.1.2.3.cmml">}</mo></mrow></mrow><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.1b"><apply id="S2.E1.m1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1"><eq id="S2.E1.m1.1.1.1.1.3.cmml" xref="S2.E1.m1.1.1.1.1.3"></eq><apply id="S2.E1.m1.1.1.1.1.4.cmml" xref="S2.E1.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.1.1.4.1.cmml" xref="S2.E1.m1.1.1.1.1.4">subscript</csymbol><ci id="S2.E1.m1.1.1.1.1.4.2.cmml" xref="S2.E1.m1.1.1.1.1.4.2">ℳ</ci><ci id="S2.E1.m1.1.1.1.1.4.3.cmml" xref="S2.E1.m1.1.1.1.1.4.3">𝑛</ci></apply><list id="S2.E1.m1.1.1.1.1.2.3.cmml" xref="S2.E1.m1.1.1.1.1.2.2"><apply id="S2.E1.m1.1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.E1.m1.1.1.1.1.1.1.1.2.cmml" xref="S2.E1.m1.1.1.1.1.1.1.1.2">𝐜</ci><ci id="S2.E1.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.E1.m1.1.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S2.E1.m1.1.1.1.1.2.2.2.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.1.1.2.2.2.1.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S2.E1.m1.1.1.1.1.2.2.2.2.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2.2">𝐆</ci><ci id="S2.E1.m1.1.1.1.1.2.2.2.3.cmml" xref="S2.E1.m1.1.1.1.1.2.2.2.3">𝑛</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.1c">\mathcal{M}_{n}=\{\mathbf{c}_{n};\mathbf{G}_{n}\},</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = { bold_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; bold_G start_POSTSUBSCRIPT 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.8">where <math alttext="\mathbf{G}_{n}\in\mathbb{R}^{W\times W}" class="ltx_Math" display="inline" id="S2.p1.7.m1.1"><semantics id="S2.p1.7.m1.1a"><mrow id="S2.p1.7.m1.1.1" xref="S2.p1.7.m1.1.1.cmml"><msub id="S2.p1.7.m1.1.1.2" xref="S2.p1.7.m1.1.1.2.cmml"><mi id="S2.p1.7.m1.1.1.2.2" xref="S2.p1.7.m1.1.1.2.2.cmml">𝐆</mi><mi id="S2.p1.7.m1.1.1.2.3" xref="S2.p1.7.m1.1.1.2.3.cmml">n</mi></msub><mo id="S2.p1.7.m1.1.1.1" xref="S2.p1.7.m1.1.1.1.cmml">∈</mo><msup id="S2.p1.7.m1.1.1.3" xref="S2.p1.7.m1.1.1.3.cmml"><mi id="S2.p1.7.m1.1.1.3.2" xref="S2.p1.7.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.p1.7.m1.1.1.3.3" xref="S2.p1.7.m1.1.1.3.3.cmml"><mi id="S2.p1.7.m1.1.1.3.3.2" xref="S2.p1.7.m1.1.1.3.3.2.cmml">W</mi><mo id="S2.p1.7.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p1.7.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.p1.7.m1.1.1.3.3.3" xref="S2.p1.7.m1.1.1.3.3.3.cmml">W</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m1.1b"><apply id="S2.p1.7.m1.1.1.cmml" xref="S2.p1.7.m1.1.1"><in id="S2.p1.7.m1.1.1.1.cmml" xref="S2.p1.7.m1.1.1.1"></in><apply id="S2.p1.7.m1.1.1.2.cmml" xref="S2.p1.7.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.7.m1.1.1.2.1.cmml" xref="S2.p1.7.m1.1.1.2">subscript</csymbol><ci id="S2.p1.7.m1.1.1.2.2.cmml" xref="S2.p1.7.m1.1.1.2.2">𝐆</ci><ci id="S2.p1.7.m1.1.1.2.3.cmml" xref="S2.p1.7.m1.1.1.2.3">𝑛</ci></apply><apply id="S2.p1.7.m1.1.1.3.cmml" xref="S2.p1.7.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.7.m1.1.1.3.1.cmml" xref="S2.p1.7.m1.1.1.3">superscript</csymbol><ci id="S2.p1.7.m1.1.1.3.2.cmml" xref="S2.p1.7.m1.1.1.3.2">ℝ</ci><apply id="S2.p1.7.m1.1.1.3.3.cmml" xref="S2.p1.7.m1.1.1.3.3"><times id="S2.p1.7.m1.1.1.3.3.1.cmml" xref="S2.p1.7.m1.1.1.3.3.1"></times><ci id="S2.p1.7.m1.1.1.3.3.2.cmml" xref="S2.p1.7.m1.1.1.3.3.2">𝑊</ci><ci id="S2.p1.7.m1.1.1.3.3.3.cmml" xref="S2.p1.7.m1.1.1.3.3.3">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m1.1c">\mathbf{G}_{n}\in\mathbb{R}^{W\times W}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m1.1d">bold_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_W × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> denotes the channel gain of the <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.8.m2.1"><semantics id="S2.p1.8.m2.1a"><mi id="S2.p1.8.m2.1.1" xref="S2.p1.8.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m2.1b"><ci id="S2.p1.8.m2.1.1.cmml" xref="S2.p1.8.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m2.1d">italic_n</annotation></semantics></math>th AP.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">Consider a potentially new AP 0 for which only its location information <math alttext="\mathbf{c}_{0}" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><msub id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mi id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml">𝐜</mi><mn id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1">subscript</csymbol><ci id="S2.p2.1.m1.1.1.2.cmml" xref="S2.p2.1.m1.1.1.2">𝐜</ci><cn id="S2.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\mathbf{c}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">bold_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is known. The focus of this paper is on how to reduce or even avoid actual measurements to generate the CKM for the AP 0. Based on the correlation on the physical environment, CKMs of other APs within the network area are naturally a source of information that can be mined. Therefore, the CKM construction problem for AP 0 can be formulated as a cross-AP inference problem from CKMs of other existing distributed APs to the CKM of AP 0 as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f:\big{\{}\mathcal{M}_{n}\big{\}}_{n=1}^{N};\mathbf{c}_{0}\rightarrow\mathbf{G% }_{0}." class="ltx_Math" display="block" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.1.4" xref="S2.E2.m1.1.1.1.1.4.cmml">f</mi><mo id="S2.E2.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.E2.m1.1.1.1.1.3.cmml">:</mo><mrow id="S2.E2.m1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.2.2.2" xref="S2.E2.m1.1.1.1.1.2.2.3.cmml"><msubsup 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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">(2)</span></td> </tr></tbody> </table> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="400" id="S2.F2.g1" src="x2.png" width="705"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Illustrating of Different Phases of the Cross-AP CKM Inference.</figcaption> </figure> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.2">The training and inference phases of cross-AP CKM inference are illustrated in Fig <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S2.F2" title="Figure 2 ‣ II Problem Formulation ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">2</span></a>. In the training phase, UNet learns the propagation characteristics in the wireless environment by performing supervised learning with the help of the channel knowledge database and optimizing the UNet parameters. In the inference phase, the coordinate <math alttext="\mathbf{c}_{0}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><msub id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml">𝐜</mi><mn id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1">subscript</csymbol><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">𝐜</ci><cn id="S2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\mathbf{c}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">bold_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of the introduced AP 0 and the CKMs of other APs in the area are combined and input into the trained UNet model, and the final inferred CKM <math alttext="\mathbf{G}_{0}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msub id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">𝐆</mi><mn id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝐆</ci><cn id="S2.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">\mathbf{G}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">bold_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is output.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">The proposed CKM inference method across APs is based on the fact that the physical environment shared by the APs determines the wireless channel conditions. We believe that the cross-AP CKM inference has great potential in cell-free networks. In the initial phase of the network deployment, the cross-AP CKM inference can effectively reduce the overhead of CKM construction. Specifically, CKMs are mapped for only part of the APs through actual measurements or ray tracing, while CKMs for the remaining APs are directly inferred to effectively reduce the overall construction overhead. Meanwhile, in a cell-free network already equipped with CKMs, cross-AP CKM inference can effectively guide the environment-aware deployment of potentially new APs. By inferring the corresponding CKMs, the coverage of the potentially new AP in different locations can be effectively modeled, so that the locations where the wireless environment best meets the requirements can be selected for AP deployment.</p> </div> </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">Model Training</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In this section, the construction and training of the UNet for cross-AP CKM inference is presented. Specifically, CKMs from distributed APs within the cell-free network are first transformed and combined to generate input data for UNet training. Then, the design of the UNet architecture for cross-AP CKM inference is presented.</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.SS1.4.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.5.2">Input Data Generation</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.2">The physical and wireless environments interact with each other. As an example, abrupt variations in channel strength in space often imply abrupt variations in the wireless environment, which can be further inferred from variations in the physical environment due to obstacles, buildings, etc. Therefore, effective cognition of the wireless environment can be obtained by synthesizing the CKMs of distributed APs in cell-free networks. Although this cognition of the wireless environment is difficult to express directly in a concrete mathematical form, it can be learned through neural networks. Specifically, the AP location stored in each CKM is first converted into a <math alttext="W\times W" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">W</mi><mo id="S3.SS1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.1.m1.1.1.1.cmml">×</mo><mi id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><times id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></times><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑊</ci><ci id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">W\times W</annotation><annotation encoding="application/x-llamapun" 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0,\quad\mathbf{c}\neq\mathbf{c}_{n},\end{aligned}\right." class="ltx_math_unparsed" display="block" id="S3.E3.m1.8"><semantics id="S3.E3.m1.8a"><mrow id="S3.E3.m1.8b"><msub id="S3.E3.m1.8.9"><mi id="S3.E3.m1.8.9.2">𝐌</mi><mrow id="S3.E3.m1.2.2.2.4"><mi id="S3.E3.m1.1.1.1.1">AP</mi><mo id="S3.E3.m1.2.2.2.4.1">,</mo><mi id="S3.E3.m1.2.2.2.2" mathvariant="normal">n</mi></mrow></msub><mrow id="S3.E3.m1.8.10"><mo id="S3.E3.m1.8.10.1" stretchy="false">(</mo><mi id="S3.E3.m1.8.8">𝐜</mi><mo id="S3.E3.m1.8.10.2" stretchy="false">)</mo></mrow><mo id="S3.E3.m1.8.11">=</mo><mrow id="S3.E3.m1.8.12"><mo id="S3.E3.m1.8.12.1">{</mo><mtable displaystyle="true" id="S3.E3.m1.7.7" rowspacing="0pt"><mtr id="S3.E3.m1.7.7a"><mtd class="ltx_align_right" columnalign="right" id="S3.E3.m1.7.7b"><mrow id="S3.E3.m1.4.4.2.2.2"><mrow id="S3.E3.m1.4.4.2.2.2.4.2"><mn id="S3.E3.m1.3.3.1.1.1.1">1</mn><mo id="S3.E3.m1.4.4.2.2.2.4.2.1" rspace="1.167em">,</mo><mi id="S3.E3.m1.4.4.2.2.2.2">𝐜</mi></mrow><mo id="S3.E3.m1.4.4.2.2.2.3">=</mo><msub id="S3.E3.m1.4.4.2.2.2.5"><mi id="S3.E3.m1.4.4.2.2.2.5.2">𝐜</mi><mi id="S3.E3.m1.4.4.2.2.2.5.3">n</mi></msub></mrow></mtd></mtr><mtr id="S3.E3.m1.7.7c"><mtd class="ltx_align_right" columnalign="right" id="S3.E3.m1.7.7d"><mrow id="S3.E3.m1.7.7.5.3.3.3"><mrow id="S3.E3.m1.7.7.5.3.3.3.1"><mrow id="S3.E3.m1.7.7.5.3.3.3.1.2.2"><mn id="S3.E3.m1.5.5.3.1.1.1">0</mn><mo id="S3.E3.m1.7.7.5.3.3.3.1.2.2.1" rspace="1.167em">,</mo><mi id="S3.E3.m1.6.6.4.2.2.2">𝐜</mi></mrow><mo id="S3.E3.m1.7.7.5.3.3.3.1.1">≠</mo><msub id="S3.E3.m1.7.7.5.3.3.3.1.3"><mi id="S3.E3.m1.7.7.5.3.3.3.1.3.2">𝐜</mi><mi id="S3.E3.m1.7.7.5.3.3.3.1.3.3">n</mi></msub></mrow><mo id="S3.E3.m1.7.7.5.3.3.3.2">,</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex" id="S3.E3.m1.8c">\mathbf{M}_{\rm{AP},n}(\mathbf{c})=\left\{\begin{aligned} 1,\quad\mathbf{c}=% \mathbf{c}_{n}\\ 0,\quad\mathbf{c}\neq\mathbf{c}_{n},\end{aligned}\right.</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.8d">bold_M start_POSTSUBSCRIPT roman_AP , roman_n end_POSTSUBSCRIPT ( bold_c ) = { start_ROW start_CELL 1 , bold_c = bold_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 , bold_c ≠ bold_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.3">where <math alttext="\mathbf{c}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m1.1"><semantics id="S3.SS1.p1.3.m1.1a"><mi id="S3.SS1.p1.3.m1.1.1" xref="S3.SS1.p1.3.m1.1.1.cmml">𝐜</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m1.1b"><ci id="S3.SS1.p1.3.m1.1.1.cmml" xref="S3.SS1.p1.3.m1.1.1">𝐜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m1.1c">\mathbf{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m1.1d">bold_c</annotation></semantics></math> is any poss coordinate in the CKM.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.4">Combining AP location maps with their corresponding CGMs can be a good way to help UNet learn wireless environment characteristics. In general, the grid of the AP tends to have the maximum value of channel gain in <math alttext="\mathbf{G}_{n}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><msub id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml"><mi id="S3.SS1.p2.1.m1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.2.cmml">𝐆</mi><mi id="S3.SS1.p2.1.m1.1.1.3" xref="S3.SS1.p2.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><apply id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2">𝐆</ci><ci id="S3.SS1.p2.1.m1.1.1.3.cmml" xref="S3.SS1.p2.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\mathbf{G}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">bold_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. To highlight the AP location feature and strengthen the influence of AP location on CKM, the AP location map <math alttext="\mathbf{M}_{\rm{AP},n}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.2"><semantics id="S3.SS1.p2.2.m2.2a"><msub id="S3.SS1.p2.2.m2.2.3" xref="S3.SS1.p2.2.m2.2.3.cmml"><mi id="S3.SS1.p2.2.m2.2.3.2" xref="S3.SS1.p2.2.m2.2.3.2.cmml">𝐌</mi><mrow id="S3.SS1.p2.2.m2.2.2.2.4" xref="S3.SS1.p2.2.m2.2.2.2.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.1.cmml">AP</mi><mo id="S3.SS1.p2.2.m2.2.2.2.4.1" xref="S3.SS1.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p2.2.m2.2.2.2.2" mathvariant="normal" xref="S3.SS1.p2.2.m2.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.2b"><apply id="S3.SS1.p2.2.m2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.2.3.1.cmml" xref="S3.SS1.p2.2.m2.2.3">subscript</csymbol><ci id="S3.SS1.p2.2.m2.2.3.2.cmml" xref="S3.SS1.p2.2.m2.2.3.2">𝐌</ci><list id="S3.SS1.p2.2.m2.2.2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.2.2.4"><ci id="S3.SS1.p2.2.m2.1.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1">AP</ci><ci id="S3.SS1.p2.2.m2.2.2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2">n</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.2c">\mathbf{M}_{\rm{AP},n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.2d">bold_M start_POSTSUBSCRIPT roman_AP , roman_n end_POSTSUBSCRIPT</annotation></semantics></math> is considered to be weighted with the corresponding channel gain matrix <math alttext="\mathbf{G}_{n}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><msub id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">𝐆</mi><mi id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝐆</ci><ci id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">\mathbf{G}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">bold_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as the feature map of the <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mi id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><ci id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_n</annotation></semantics></math>th AP</p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{M}_{n}=(1-\omega)\mathbf{G}_{n}+\omega\mathbf{M}_{\mathrm{AP},n}," class="ltx_Math" display="block" id="S3.E4.m1.3"><semantics id="S3.E4.m1.3a"><mrow id="S3.E4.m1.3.3.1" xref="S3.E4.m1.3.3.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1" xref="S3.E4.m1.3.3.1.1.cmml"><msub id="S3.E4.m1.3.3.1.1.3" xref="S3.E4.m1.3.3.1.1.3.cmml"><mi id="S3.E4.m1.3.3.1.1.3.2" xref="S3.E4.m1.3.3.1.1.3.2.cmml">𝐌</mi><mi id="S3.E4.m1.3.3.1.1.3.3" xref="S3.E4.m1.3.3.1.1.3.3.cmml">n</mi></msub><mo id="S3.E4.m1.3.3.1.1.2" xref="S3.E4.m1.3.3.1.1.2.cmml">=</mo><mrow id="S3.E4.m1.3.3.1.1.1" xref="S3.E4.m1.3.3.1.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S3.E4.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E4.m1.3.3.1.1.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.cmml"><mn id="S3.E4.m1.3.3.1.1.1.1.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.E4.m1.3.3.1.1.1.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.E4.m1.3.3.1.1.1.1.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.3.cmml">ω</mi></mrow><mo id="S3.E4.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S3.E4.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E4.m1.3.3.1.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msub id="S3.E4.m1.3.3.1.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.1.3.cmml"><mi id="S3.E4.m1.3.3.1.1.1.1.3.2" xref="S3.E4.m1.3.3.1.1.1.1.3.2.cmml">𝐆</mi><mi id="S3.E4.m1.3.3.1.1.1.1.3.3" xref="S3.E4.m1.3.3.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S3.E4.m1.3.3.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.2.cmml">+</mo><mrow id="S3.E4.m1.3.3.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.3.cmml"><mi id="S3.E4.m1.3.3.1.1.1.3.2" xref="S3.E4.m1.3.3.1.1.1.3.2.cmml">ω</mi><mo id="S3.E4.m1.3.3.1.1.1.3.1" xref="S3.E4.m1.3.3.1.1.1.3.1.cmml">⁢</mo><msub id="S3.E4.m1.3.3.1.1.1.3.3" xref="S3.E4.m1.3.3.1.1.1.3.3.cmml"><mi id="S3.E4.m1.3.3.1.1.1.3.3.2" xref="S3.E4.m1.3.3.1.1.1.3.3.2.cmml">𝐌</mi><mrow id="S3.E4.m1.2.2.2.4" xref="S3.E4.m1.2.2.2.3.cmml"><mi id="S3.E4.m1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.cmml">AP</mi><mo id="S3.E4.m1.2.2.2.4.1" xref="S3.E4.m1.2.2.2.3.cmml">,</mo><mi id="S3.E4.m1.2.2.2.2" xref="S3.E4.m1.2.2.2.2.cmml">n</mi></mrow></msub></mrow></mrow></mrow><mo id="S3.E4.m1.3.3.1.2" xref="S3.E4.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E4.m1.3b"><apply id="S3.E4.m1.3.3.1.1.cmml" xref="S3.E4.m1.3.3.1"><eq id="S3.E4.m1.3.3.1.1.2.cmml" xref="S3.E4.m1.3.3.1.1.2"></eq><apply id="S3.E4.m1.3.3.1.1.3.cmml" xref="S3.E4.m1.3.3.1.1.3"><csymbol 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encoding="application/x-tex" id="S3.E4.m1.3c">\mathbf{M}_{n}=(1-\omega)\mathbf{G}_{n}+\omega\mathbf{M}_{\mathrm{AP},n},</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.3d">bold_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ( 1 - italic_ω ) bold_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_ω bold_M start_POSTSUBSCRIPT roman_AP , 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">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p2.5">where increasing the weight <math alttext="\omega" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m1.1"><semantics id="S3.SS1.p2.5.m1.1a"><mi id="S3.SS1.p2.5.m1.1.1" xref="S3.SS1.p2.5.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m1.1b"><ci id="S3.SS1.p2.5.m1.1.1.cmml" xref="S3.SS1.p2.5.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m1.1d">italic_ω</annotation></semantics></math> strengthens the corresponding AP location feature but weakens the details of the CKM.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.9">Further, combining <math alttext="\mathbf{M}_{n}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><msub id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2.cmml">𝐌</mi><mi id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">𝐌</ci><ci id="S3.SS1.p3.1.m1.1.1.3.cmml" xref="S3.SS1.p3.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">\mathbf{M}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">bold_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> of all the <math alttext="N" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">italic_N</annotation></semantics></math> distributed APs in the cell-free network results in a feature map of <math alttext="N" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mi id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><ci id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_N</annotation></semantics></math>-dimensional channels, where the feature map of each AP corresponds to one dimension. Name the AP of which the CKM needs to be generated as the target AP. The next part is to combine the target AP location map with the <math alttext="N" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mi id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><ci id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_N</annotation></semantics></math>-dimensional feature maps of the other APs. Notice that the AP location information stored in map <math alttext="\mathbf{M}_{\rm{AP},{\rm target}}" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.2"><semantics id="S3.SS1.p3.5.m5.2a"><msub id="S3.SS1.p3.5.m5.2.3" xref="S3.SS1.p3.5.m5.2.3.cmml"><mi id="S3.SS1.p3.5.m5.2.3.2" xref="S3.SS1.p3.5.m5.2.3.2.cmml">𝐌</mi><mrow id="S3.SS1.p3.5.m5.2.2.2.4" xref="S3.SS1.p3.5.m5.2.2.2.3.cmml"><mi id="S3.SS1.p3.5.m5.1.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.1.cmml">AP</mi><mo id="S3.SS1.p3.5.m5.2.2.2.4.1" xref="S3.SS1.p3.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.5.m5.2.2.2.2" xref="S3.SS1.p3.5.m5.2.2.2.2.cmml">target</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.2b"><apply id="S3.SS1.p3.5.m5.2.3.cmml" xref="S3.SS1.p3.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.5.m5.2.3.1.cmml" xref="S3.SS1.p3.5.m5.2.3">subscript</csymbol><ci id="S3.SS1.p3.5.m5.2.3.2.cmml" xref="S3.SS1.p3.5.m5.2.3.2">𝐌</ci><list id="S3.SS1.p3.5.m5.2.2.2.3.cmml" xref="S3.SS1.p3.5.m5.2.2.2.4"><ci id="S3.SS1.p3.5.m5.1.1.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1.1.1">AP</ci><ci id="S3.SS1.p3.5.m5.2.2.2.2.cmml" xref="S3.SS1.p3.5.m5.2.2.2.2">target</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.2c">\mathbf{M}_{\rm{AP},{\rm target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.2d">bold_M start_POSTSUBSCRIPT roman_AP , roman_target end_POSTSUBSCRIPT</annotation></semantics></math> is sparse. To make the convolutional kernel fully capture the key AP location features and avoid feature omissions during the learning process, pre-convolution is first needed to reinforce the features of the <math alttext="\mathbf{M}_{\rm{AP},{\rm target}}" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.2"><semantics id="S3.SS1.p3.6.m6.2a"><msub id="S3.SS1.p3.6.m6.2.3" xref="S3.SS1.p3.6.m6.2.3.cmml"><mi id="S3.SS1.p3.6.m6.2.3.2" xref="S3.SS1.p3.6.m6.2.3.2.cmml">𝐌</mi><mrow id="S3.SS1.p3.6.m6.2.2.2.4" xref="S3.SS1.p3.6.m6.2.2.2.3.cmml"><mi id="S3.SS1.p3.6.m6.1.1.1.1" xref="S3.SS1.p3.6.m6.1.1.1.1.cmml">AP</mi><mo id="S3.SS1.p3.6.m6.2.2.2.4.1" xref="S3.SS1.p3.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.6.m6.2.2.2.2" xref="S3.SS1.p3.6.m6.2.2.2.2.cmml">target</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.2b"><apply id="S3.SS1.p3.6.m6.2.3.cmml" xref="S3.SS1.p3.6.m6.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.6.m6.2.3.1.cmml" xref="S3.SS1.p3.6.m6.2.3">subscript</csymbol><ci id="S3.SS1.p3.6.m6.2.3.2.cmml" xref="S3.SS1.p3.6.m6.2.3.2">𝐌</ci><list id="S3.SS1.p3.6.m6.2.2.2.3.cmml" xref="S3.SS1.p3.6.m6.2.2.2.4"><ci id="S3.SS1.p3.6.m6.1.1.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1.1.1">AP</ci><ci id="S3.SS1.p3.6.m6.2.2.2.2.cmml" xref="S3.SS1.p3.6.m6.2.2.2.2">target</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.2c">\mathbf{M}_{\rm{AP},{\rm target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.2d">bold_M start_POSTSUBSCRIPT roman_AP , roman_target end_POSTSUBSCRIPT</annotation></semantics></math>. By convolving with the kernel <math alttext="\mathbf{v}=\mathbf{1}\in\mathbb{R}^{3\times 3}" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><mrow id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml"><mi id="S3.SS1.p3.7.m7.1.1.2" xref="S3.SS1.p3.7.m7.1.1.2.cmml">𝐯</mi><mo id="S3.SS1.p3.7.m7.1.1.3" xref="S3.SS1.p3.7.m7.1.1.3.cmml">=</mo><mn id="S3.SS1.p3.7.m7.1.1.4" xref="S3.SS1.p3.7.m7.1.1.4.cmml">𝟏</mn><mo id="S3.SS1.p3.7.m7.1.1.5" xref="S3.SS1.p3.7.m7.1.1.5.cmml">∈</mo><msup id="S3.SS1.p3.7.m7.1.1.6" xref="S3.SS1.p3.7.m7.1.1.6.cmml"><mi id="S3.SS1.p3.7.m7.1.1.6.2" xref="S3.SS1.p3.7.m7.1.1.6.2.cmml">ℝ</mi><mrow id="S3.SS1.p3.7.m7.1.1.6.3" xref="S3.SS1.p3.7.m7.1.1.6.3.cmml"><mn id="S3.SS1.p3.7.m7.1.1.6.3.2" xref="S3.SS1.p3.7.m7.1.1.6.3.2.cmml">3</mn><mo id="S3.SS1.p3.7.m7.1.1.6.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p3.7.m7.1.1.6.3.1.cmml">×</mo><mn id="S3.SS1.p3.7.m7.1.1.6.3.3" xref="S3.SS1.p3.7.m7.1.1.6.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><apply id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1"><and id="S3.SS1.p3.7.m7.1.1a.cmml" xref="S3.SS1.p3.7.m7.1.1"></and><apply id="S3.SS1.p3.7.m7.1.1b.cmml" xref="S3.SS1.p3.7.m7.1.1"><eq id="S3.SS1.p3.7.m7.1.1.3.cmml" xref="S3.SS1.p3.7.m7.1.1.3"></eq><ci id="S3.SS1.p3.7.m7.1.1.2.cmml" xref="S3.SS1.p3.7.m7.1.1.2">𝐯</ci><cn id="S3.SS1.p3.7.m7.1.1.4.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.4">1</cn></apply><apply id="S3.SS1.p3.7.m7.1.1c.cmml" xref="S3.SS1.p3.7.m7.1.1"><in id="S3.SS1.p3.7.m7.1.1.5.cmml" xref="S3.SS1.p3.7.m7.1.1.5"></in><share href="https://arxiv.org/html/2411.17716v1#S3.SS1.p3.7.m7.1.1.4.cmml" id="S3.SS1.p3.7.m7.1.1d.cmml" xref="S3.SS1.p3.7.m7.1.1"></share><apply id="S3.SS1.p3.7.m7.1.1.6.cmml" xref="S3.SS1.p3.7.m7.1.1.6"><csymbol cd="ambiguous" id="S3.SS1.p3.7.m7.1.1.6.1.cmml" xref="S3.SS1.p3.7.m7.1.1.6">superscript</csymbol><ci id="S3.SS1.p3.7.m7.1.1.6.2.cmml" xref="S3.SS1.p3.7.m7.1.1.6.2">ℝ</ci><apply id="S3.SS1.p3.7.m7.1.1.6.3.cmml" xref="S3.SS1.p3.7.m7.1.1.6.3"><times id="S3.SS1.p3.7.m7.1.1.6.3.1.cmml" xref="S3.SS1.p3.7.m7.1.1.6.3.1"></times><cn id="S3.SS1.p3.7.m7.1.1.6.3.2.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.6.3.2">3</cn><cn id="S3.SS1.p3.7.m7.1.1.6.3.3.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.6.3.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">\mathbf{v}=\mathbf{1}\in\mathbb{R}^{3\times 3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">bold_v = bold_1 ∈ blackboard_R start_POSTSUPERSCRIPT 3 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, all elements in the surrounding <math alttext="3\times 3" class="ltx_Math" display="inline" id="S3.SS1.p3.8.m8.1"><semantics id="S3.SS1.p3.8.m8.1a"><mrow id="S3.SS1.p3.8.m8.1.1" xref="S3.SS1.p3.8.m8.1.1.cmml"><mn id="S3.SS1.p3.8.m8.1.1.2" xref="S3.SS1.p3.8.m8.1.1.2.cmml">3</mn><mo id="S3.SS1.p3.8.m8.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p3.8.m8.1.1.1.cmml">×</mo><mn id="S3.SS1.p3.8.m8.1.1.3" xref="S3.SS1.p3.8.m8.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.8.m8.1b"><apply id="S3.SS1.p3.8.m8.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1"><times id="S3.SS1.p3.8.m8.1.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1"></times><cn id="S3.SS1.p3.8.m8.1.1.2.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.2">3</cn><cn id="S3.SS1.p3.8.m8.1.1.3.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.8.m8.1c">3\times 3</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.8.m8.1d">3 × 3</annotation></semantics></math> grids of the AP location <math alttext="\mathbf{c}_{\rm{target}}" class="ltx_Math" display="inline" id="S3.SS1.p3.9.m9.1"><semantics id="S3.SS1.p3.9.m9.1a"><msub id="S3.SS1.p3.9.m9.1.1" xref="S3.SS1.p3.9.m9.1.1.cmml"><mi id="S3.SS1.p3.9.m9.1.1.2" xref="S3.SS1.p3.9.m9.1.1.2.cmml">𝐜</mi><mi id="S3.SS1.p3.9.m9.1.1.3" xref="S3.SS1.p3.9.m9.1.1.3.cmml">target</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.9.m9.1b"><apply id="S3.SS1.p3.9.m9.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.1.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.p3.9.m9.1.1.2.cmml" xref="S3.SS1.p3.9.m9.1.1.2">𝐜</ci><ci id="S3.SS1.p3.9.m9.1.1.3.cmml" xref="S3.SS1.p3.9.m9.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.9.m9.1c">\mathbf{c}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.9.m9.1d">bold_c start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math> are set to 1, effectively expanding the coverage of location information. The pre-convolution kernel can improve the extraction of AP location features, and make its performance more stable in the face of sparse location data. The target AP location map after pre-convolution is </p> <table class="ltx_equation ltx_eqn_table" id="S3.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{M}^{*}_{\rm{AP},{\rm target}}=\mathbf{M}_{\rm{AP},target}\ast\mathbf{v}." class="ltx_Math" display="block" id="S3.E5.m1.5"><semantics id="S3.E5.m1.5a"><mrow id="S3.E5.m1.5.5.1" xref="S3.E5.m1.5.5.1.1.cmml"><mrow id="S3.E5.m1.5.5.1.1" xref="S3.E5.m1.5.5.1.1.cmml"><msubsup id="S3.E5.m1.5.5.1.1.2" xref="S3.E5.m1.5.5.1.1.2.cmml"><mi id="S3.E5.m1.5.5.1.1.2.2.2" xref="S3.E5.m1.5.5.1.1.2.2.2.cmml">𝐌</mi><mrow id="S3.E5.m1.2.2.2.4" xref="S3.E5.m1.2.2.2.3.cmml"><mi id="S3.E5.m1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.cmml">AP</mi><mo id="S3.E5.m1.2.2.2.4.1" xref="S3.E5.m1.2.2.2.3.cmml">,</mo><mi id="S3.E5.m1.2.2.2.2" xref="S3.E5.m1.2.2.2.2.cmml">target</mi></mrow><mo id="S3.E5.m1.5.5.1.1.2.2.3" 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rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.3">Finally, the target AP location map <math alttext="\mathbf{M}^{*}_{\mathrm{AP},{\rm target}}" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.2"><semantics id="S3.SS1.p4.1.m1.2a"><msubsup id="S3.SS1.p4.1.m1.2.3" xref="S3.SS1.p4.1.m1.2.3.cmml"><mi id="S3.SS1.p4.1.m1.2.3.2.2" xref="S3.SS1.p4.1.m1.2.3.2.2.cmml">𝐌</mi><mrow id="S3.SS1.p4.1.m1.2.2.2.4" xref="S3.SS1.p4.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.p4.1.m1.1.1.1.1" xref="S3.SS1.p4.1.m1.1.1.1.1.cmml">AP</mi><mo id="S3.SS1.p4.1.m1.2.2.2.4.1" xref="S3.SS1.p4.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p4.1.m1.2.2.2.2" xref="S3.SS1.p4.1.m1.2.2.2.2.cmml">target</mi></mrow><mo id="S3.SS1.p4.1.m1.2.3.2.3" xref="S3.SS1.p4.1.m1.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.2b"><apply id="S3.SS1.p4.1.m1.2.3.cmml" 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The generation process and structure of <math alttext="\mathbf{M}_{\mathrm{input}}" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><msub id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2" xref="S3.SS1.p4.3.m3.1.1.2.cmml">𝐌</mi><mi id="S3.SS1.p4.3.m3.1.1.3" xref="S3.SS1.p4.3.m3.1.1.3.cmml">input</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><apply id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2">𝐌</ci><ci id="S3.SS1.p4.3.m3.1.1.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3">input</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">\mathbf{M}_{\mathrm{input}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">bold_M start_POSTSUBSCRIPT roman_input end_POSTSUBSCRIPT</annotation></semantics></math> is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S3.F3" title="Figure 3 ‣ III-A Input Data Generation ‣ III Model Training ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="733" id="S3.F3.g1" src="x3.png" width="622"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span> Input Structure Design.</figcaption> </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.SS2.4.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.5.2">UNet Design and Training</span> </h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">The structure of the UNet network for cross-AP inference is designed based on the CGMs and their corresponding AP location maps in RadioMapSeer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib10" title="">10</a>]</cite>. For other datasets like CKMImageNet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib11" title="">11</a>]</cite>, parameters such as the number of UNet input channels need to be adjusted according to the characteristics of the dataset.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.2">The details of the UNet structure are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S3.F4" title="Figure 4 ‣ III-B UNet Design and Training ‣ III Model Training ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">4</span></a>. Note that for each physical environment map in RadioMapSeer, there are 80 AP location maps and corresponding simulated CGMs. Therefore, the input data of the UNet for cross-AP inference is a high-dimensional matrix with 80 channels. According to the process of input data generation, the first channel dimension consists of the target AP location map <math alttext="\mathbf{M}^{*}_{\rm{AP},target}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.2"><semantics id="S3.SS2.p2.1.m1.2a"><msubsup id="S3.SS2.p2.1.m1.2.3" xref="S3.SS2.p2.1.m1.2.3.cmml"><mi id="S3.SS2.p2.1.m1.2.3.2.2" xref="S3.SS2.p2.1.m1.2.3.2.2.cmml">𝐌</mi><mrow id="S3.SS2.p2.1.m1.2.2.2.4" xref="S3.SS2.p2.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.cmml">AP</mi><mo id="S3.SS2.p2.1.m1.2.2.2.4.1" xref="S3.SS2.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p2.1.m1.2.2.2.2" xref="S3.SS2.p2.1.m1.2.2.2.2.cmml">target</mi></mrow><mo id="S3.SS2.p2.1.m1.2.3.2.3" xref="S3.SS2.p2.1.m1.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.2b"><apply id="S3.SS2.p2.1.m1.2.3.cmml" xref="S3.SS2.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.2.3.1.cmml" xref="S3.SS2.p2.1.m1.2.3">subscript</csymbol><apply id="S3.SS2.p2.1.m1.2.3.2.cmml" xref="S3.SS2.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.2.3.2.1.cmml" xref="S3.SS2.p2.1.m1.2.3">superscript</csymbol><ci id="S3.SS2.p2.1.m1.2.3.2.2.cmml" xref="S3.SS2.p2.1.m1.2.3.2.2">𝐌</ci><times id="S3.SS2.p2.1.m1.2.3.2.3.cmml" xref="S3.SS2.p2.1.m1.2.3.2.3"></times></apply><list id="S3.SS2.p2.1.m1.2.2.2.3.cmml" xref="S3.SS2.p2.1.m1.2.2.2.4"><ci id="S3.SS2.p2.1.m1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1">AP</ci><ci id="S3.SS2.p2.1.m1.2.2.2.2.cmml" xref="S3.SS2.p2.1.m1.2.2.2.2">target</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.2c">\mathbf{M}^{*}_{\rm{AP},target}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.2d">bold_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_AP , roman_target end_POSTSUBSCRIPT</annotation></semantics></math> after mask operation. All the other channels are feature maps composed of the weighted sums of CGMs and AP location maps of other APs in the same physical environment map. 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To enhance the learning of wireless environment features at the edges of the building, the <math alttext="5\times 5" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mn id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml">5</mn><mo id="S3.SS2.p3.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p3.1.m1.1.1.1.cmml">×</mo><mn id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><times id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1"></times><cn id="S3.SS2.p3.1.m1.1.1.2.cmml" type="integer" xref="S3.SS2.p3.1.m1.1.1.2">5</cn><cn id="S3.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p3.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">5\times 5</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">5 × 5</annotation></semantics></math> convolution kernel is used extensively, which helps to capture the local spatial information. For cross-AP CKM inference, the high-channel input data can lead to parameter redundancy. Dimensionality-reduction convolutions effectively reduce the parameter redundancy, significantly lowering computational overhead and improving model generalization. Further, in the process of down-sampling and up-sampling, the network extracts and recovers increasing high-level features layer by layer, while retaining low-level features with hop concatenation. Meanwhile, additional convolutions are added at the resolutions of <math alttext="64\times 64" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mn id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml">64</mn><mo id="S3.SS2.p3.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p3.2.m2.1.1.1.cmml">×</mo><mn id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">64</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><times id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1"></times><cn id="S3.SS2.p3.2.m2.1.1.2.cmml" type="integer" xref="S3.SS2.p3.2.m2.1.1.2">64</cn><cn id="S3.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p3.2.m2.1.1.3">64</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">64\times 64</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">64 × 64</annotation></semantics></math> and <math alttext="32\times 32" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mrow id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><mn id="S3.SS2.p3.3.m3.1.1.2" xref="S3.SS2.p3.3.m3.1.1.2.cmml">32</mn><mo id="S3.SS2.p3.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p3.3.m3.1.1.1.cmml">×</mo><mn id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><apply id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1"><times id="S3.SS2.p3.3.m3.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1"></times><cn id="S3.SS2.p3.3.m3.1.1.2.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.2">32</cn><cn id="S3.SS2.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.3">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">32\times 32</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">32 × 32</annotation></semantics></math>, which helps to enrich and strengthen the local feature representation and the CKM property characterization.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.5">During the training process, each AP randomly takes turns to play the role of target AP, while the remaining APs act as the other existing APs with CKMs. The input data <math alttext="\mathbf{M}_{\mathrm{input}}" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><msub id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml"><mi id="S3.SS2.p4.1.m1.1.1.2" xref="S3.SS2.p4.1.m1.1.1.2.cmml">𝐌</mi><mi id="S3.SS2.p4.1.m1.1.1.3" xref="S3.SS2.p4.1.m1.1.1.3.cmml">input</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><apply id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p4.1.m1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2">𝐌</ci><ci id="S3.SS2.p4.1.m1.1.1.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3">input</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">\mathbf{M}_{\mathrm{input}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">bold_M start_POSTSUBSCRIPT roman_input end_POSTSUBSCRIPT</annotation></semantics></math> is generated and fed into the UNet and outputs the corresponding inferred CKM <math alttext="\mathbf{G}^{\rm{infer}}_{\rm{target}}" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><msubsup id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml"><mi id="S3.SS2.p4.2.m2.1.1.2.2" xref="S3.SS2.p4.2.m2.1.1.2.2.cmml">𝐆</mi><mi id="S3.SS2.p4.2.m2.1.1.3" xref="S3.SS2.p4.2.m2.1.1.3.cmml">target</mi><mi id="S3.SS2.p4.2.m2.1.1.2.3" xref="S3.SS2.p4.2.m2.1.1.2.3.cmml">infer</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.1b"><apply id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.2.m2.1.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">subscript</csymbol><apply id="S3.SS2.p4.2.m2.1.1.2.cmml" xref="S3.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.2.m2.1.1.2.1.cmml" xref="S3.SS2.p4.2.m2.1.1">superscript</csymbol><ci id="S3.SS2.p4.2.m2.1.1.2.2.cmml" xref="S3.SS2.p4.2.m2.1.1.2.2">𝐆</ci><ci id="S3.SS2.p4.2.m2.1.1.2.3.cmml" xref="S3.SS2.p4.2.m2.1.1.2.3">infer</ci></apply><ci id="S3.SS2.p4.2.m2.1.1.3.cmml" xref="S3.SS2.p4.2.m2.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">\mathbf{G}^{\rm{infer}}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">bold_G start_POSTSUPERSCRIPT roman_infer end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math>. The corresponding CGM <math alttext="\mathbf{G}_{\rm{target}}" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><msub id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml"><mi id="S3.SS2.p4.3.m3.1.1.2" xref="S3.SS2.p4.3.m3.1.1.2.cmml">𝐆</mi><mi id="S3.SS2.p4.3.m3.1.1.3" xref="S3.SS2.p4.3.m3.1.1.3.cmml">target</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><apply id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.3.m3.1.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p4.3.m3.1.1.2.cmml" xref="S3.SS2.p4.3.m3.1.1.2">𝐆</ci><ci id="S3.SS2.p4.3.m3.1.1.3.cmml" xref="S3.SS2.p4.3.m3.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">\mathbf{G}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">bold_G start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math> of the target AP is the ground-truth. The mean square error (MSE) between <math alttext="\mathbf{G}_{\rm{target}}" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><msub id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml"><mi id="S3.SS2.p4.4.m4.1.1.2" xref="S3.SS2.p4.4.m4.1.1.2.cmml">𝐆</mi><mi id="S3.SS2.p4.4.m4.1.1.3" xref="S3.SS2.p4.4.m4.1.1.3.cmml">target</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><apply id="S3.SS2.p4.4.m4.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.4.m4.1.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.p4.4.m4.1.1.2.cmml" xref="S3.SS2.p4.4.m4.1.1.2">𝐆</ci><ci id="S3.SS2.p4.4.m4.1.1.3.cmml" xref="S3.SS2.p4.4.m4.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m4.1c">\mathbf{G}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m4.1d">bold_G start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{G}^{\rm{infer}}_{\rm{target}}" class="ltx_Math" display="inline" id="S3.SS2.p4.5.m5.1"><semantics id="S3.SS2.p4.5.m5.1a"><msubsup id="S3.SS2.p4.5.m5.1.1" xref="S3.SS2.p4.5.m5.1.1.cmml"><mi id="S3.SS2.p4.5.m5.1.1.2.2" xref="S3.SS2.p4.5.m5.1.1.2.2.cmml">𝐆</mi><mi id="S3.SS2.p4.5.m5.1.1.3" xref="S3.SS2.p4.5.m5.1.1.3.cmml">target</mi><mi id="S3.SS2.p4.5.m5.1.1.2.3" xref="S3.SS2.p4.5.m5.1.1.2.3.cmml">infer</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.5.m5.1b"><apply id="S3.SS2.p4.5.m5.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m5.1.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1">subscript</csymbol><apply id="S3.SS2.p4.5.m5.1.1.2.cmml" xref="S3.SS2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m5.1.1.2.1.cmml" xref="S3.SS2.p4.5.m5.1.1">superscript</csymbol><ci id="S3.SS2.p4.5.m5.1.1.2.2.cmml" xref="S3.SS2.p4.5.m5.1.1.2.2">𝐆</ci><ci id="S3.SS2.p4.5.m5.1.1.2.3.cmml" xref="S3.SS2.p4.5.m5.1.1.2.3">infer</ci></apply><ci id="S3.SS2.p4.5.m5.1.1.3.cmml" xref="S3.SS2.p4.5.m5.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m5.1c">\mathbf{G}^{\rm{infer}}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m5.1d">bold_G start_POSTSUPERSCRIPT roman_infer end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math> is used as the loss function as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="e=\frac{1}{W^{2}}\sum_{i=1}^{W^{2}}(\mathbf{G}_{\rm{target}}(i)-\mathbf{G}^{% \rm{infer}}_{\rm{target}}(i))^{2}." class="ltx_Math" display="block" id="S3.E6.m1.3"><semantics id="S3.E6.m1.3a"><mrow id="S3.E6.m1.3.3.1" xref="S3.E6.m1.3.3.1.1.cmml"><mrow 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start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_W start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( bold_G start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT ( italic_i ) - bold_G start_POSTSUPERSCRIPT roman_infer end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT ( italic_i ) ) 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p4.6">By updating the parameters according to the loss, the trained UNet model can be finally obtained. The detailed process of the training phase is shown in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#algorithm1" title="In III-B UNet Design and Training ‣ III Model Training ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm1"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm1.13"> <div class="ltx_listingline" id="algorithm1.1.1"> <span class="ltx_text" id="algorithm1.1.1.1"><span class="ltx_text ltx_font_bold" id="algorithm1.1.1.1.1">Data:</span> </span>the training CKM set <math alttext="\{\mathcal{M}_{n}\big{\}}_{n=1}^{N}" class="ltx_Math" display="inline" id="algorithm1.1.1.m1.1"><semantics id="algorithm1.1.1.m1.1a"><msubsup id="algorithm1.1.1.m1.1.1" xref="algorithm1.1.1.m1.1.1.cmml"><mrow id="algorithm1.1.1.m1.1.1.1.1.1" xref="algorithm1.1.1.m1.1.1.1.1.2.cmml"><mo id="algorithm1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="algorithm1.1.1.m1.1.1.1.1.2.cmml">{</mo><msub id="algorithm1.1.1.m1.1.1.1.1.1.1" xref="algorithm1.1.1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.1.1.m1.1.1.1.1.1.1.2" xref="algorithm1.1.1.m1.1.1.1.1.1.1.2.cmml">ℳ</mi><mi id="algorithm1.1.1.m1.1.1.1.1.1.1.3" xref="algorithm1.1.1.m1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="algorithm1.1.1.m1.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="algorithm1.1.1.m1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="algorithm1.1.1.m1.1.1.1.3" xref="algorithm1.1.1.m1.1.1.1.3.cmml"><mi id="algorithm1.1.1.m1.1.1.1.3.2" xref="algorithm1.1.1.m1.1.1.1.3.2.cmml">n</mi><mo id="algorithm1.1.1.m1.1.1.1.3.1" xref="algorithm1.1.1.m1.1.1.1.3.1.cmml">=</mo><mn id="algorithm1.1.1.m1.1.1.1.3.3" xref="algorithm1.1.1.m1.1.1.1.3.3.cmml">1</mn></mrow><mi id="algorithm1.1.1.m1.1.1.3" xref="algorithm1.1.1.m1.1.1.3.cmml">N</mi></msubsup><annotation-xml encoding="MathML-Content" id="algorithm1.1.1.m1.1b"><apply id="algorithm1.1.1.m1.1.1.cmml" xref="algorithm1.1.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.1.1.m1.1.1.2.cmml" xref="algorithm1.1.1.m1.1.1">superscript</csymbol><apply id="algorithm1.1.1.m1.1.1.1.cmml" xref="algorithm1.1.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.1.1.m1.1.1.1.2.cmml" xref="algorithm1.1.1.m1.1.1">subscript</csymbol><set id="algorithm1.1.1.m1.1.1.1.1.2.cmml" xref="algorithm1.1.1.m1.1.1.1.1.1"><apply id="algorithm1.1.1.m1.1.1.1.1.1.1.cmml" xref="algorithm1.1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="algorithm1.1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="algorithm1.1.1.m1.1.1.1.1.1.1.2">ℳ</ci><ci id="algorithm1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="algorithm1.1.1.m1.1.1.1.1.1.1.3">𝑛</ci></apply></set><apply id="algorithm1.1.1.m1.1.1.1.3.cmml" xref="algorithm1.1.1.m1.1.1.1.3"><eq id="algorithm1.1.1.m1.1.1.1.3.1.cmml" xref="algorithm1.1.1.m1.1.1.1.3.1"></eq><ci id="algorithm1.1.1.m1.1.1.1.3.2.cmml" xref="algorithm1.1.1.m1.1.1.1.3.2">𝑛</ci><cn id="algorithm1.1.1.m1.1.1.1.3.3.cmml" type="integer" xref="algorithm1.1.1.m1.1.1.1.3.3">1</cn></apply></apply><ci id="algorithm1.1.1.m1.1.1.3.cmml" xref="algorithm1.1.1.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.1.1.m1.1c">\{\mathcal{M}_{n}\big{\}}_{n=1}^{N}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.1.1.m1.1d">{ caligraphic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.2.2"> <span class="ltx_text" id="algorithm1.2.2.1"><span class="ltx_text ltx_font_bold" id="algorithm1.2.2.1.1">Result:</span> </span>the optimal parameters <math alttext="\{\bm{\theta^{\star}}\}" class="ltx_Math" display="inline" id="algorithm1.2.2.m1.1"><semantics id="algorithm1.2.2.m1.1a"><mrow id="algorithm1.2.2.m1.1.1.1" xref="algorithm1.2.2.m1.1.1.2.cmml"><mo id="algorithm1.2.2.m1.1.1.1.2" stretchy="false" xref="algorithm1.2.2.m1.1.1.2.cmml">{</mo><msup id="algorithm1.2.2.m1.1.1.1.1" xref="algorithm1.2.2.m1.1.1.1.1.cmml"><mi id="algorithm1.2.2.m1.1.1.1.1.2" xref="algorithm1.2.2.m1.1.1.1.1.2.cmml">𝜽</mi><mo class="ltx_mathvariant_bold" id="algorithm1.2.2.m1.1.1.1.1.3" mathvariant="bold" xref="algorithm1.2.2.m1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="algorithm1.2.2.m1.1.1.1.3" stretchy="false" xref="algorithm1.2.2.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.2.2.m1.1b"><set id="algorithm1.2.2.m1.1.1.2.cmml" xref="algorithm1.2.2.m1.1.1.1"><apply id="algorithm1.2.2.m1.1.1.1.1.cmml" xref="algorithm1.2.2.m1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.2.2.m1.1.1.1.1.1.cmml" xref="algorithm1.2.2.m1.1.1.1.1">superscript</csymbol><ci id="algorithm1.2.2.m1.1.1.1.1.2.cmml" xref="algorithm1.2.2.m1.1.1.1.1.2">𝜽</ci><ci id="algorithm1.2.2.m1.1.1.1.1.3.cmml" xref="algorithm1.2.2.m1.1.1.1.1.3">bold-⋆</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.2.2.m1.1c">\{\bm{\theta^{\star}}\}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.2.2.m1.1d">{ bold_italic_θ start_POSTSUPERSCRIPT bold_⋆ end_POSTSUPERSCRIPT }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.4.4"> <span class="ltx_tag ltx_tag_listingline">1</span> Get the AP location map <math alttext="\mathbf{M}_{\rm{AP},n}" class="ltx_Math" display="inline" id="algorithm1.3.3.m1.2"><semantics id="algorithm1.3.3.m1.2a"><msub id="algorithm1.3.3.m1.2.3" xref="algorithm1.3.3.m1.2.3.cmml"><mi id="algorithm1.3.3.m1.2.3.2" xref="algorithm1.3.3.m1.2.3.2.cmml">𝐌</mi><mrow id="algorithm1.3.3.m1.2.2.2.4" xref="algorithm1.3.3.m1.2.2.2.3.cmml"><mi id="algorithm1.3.3.m1.1.1.1.1" xref="algorithm1.3.3.m1.1.1.1.1.cmml">AP</mi><mo id="algorithm1.3.3.m1.2.2.2.4.1" xref="algorithm1.3.3.m1.2.2.2.3.cmml">,</mo><mi id="algorithm1.3.3.m1.2.2.2.2" mathvariant="normal" xref="algorithm1.3.3.m1.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="algorithm1.3.3.m1.2b"><apply id="algorithm1.3.3.m1.2.3.cmml" xref="algorithm1.3.3.m1.2.3"><csymbol cd="ambiguous" id="algorithm1.3.3.m1.2.3.1.cmml" xref="algorithm1.3.3.m1.2.3">subscript</csymbol><ci id="algorithm1.3.3.m1.2.3.2.cmml" xref="algorithm1.3.3.m1.2.3.2">𝐌</ci><list id="algorithm1.3.3.m1.2.2.2.3.cmml" xref="algorithm1.3.3.m1.2.2.2.4"><ci id="algorithm1.3.3.m1.1.1.1.1.cmml" xref="algorithm1.3.3.m1.1.1.1.1">AP</ci><ci id="algorithm1.3.3.m1.2.2.2.2.cmml" xref="algorithm1.3.3.m1.2.2.2.2">n</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.3.3.m1.2c">\mathbf{M}_{\rm{AP},n}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.3.3.m1.2d">bold_M start_POSTSUBSCRIPT roman_AP , roman_n end_POSTSUBSCRIPT</annotation></semantics></math> for any <math alttext="n" class="ltx_Math" display="inline" id="algorithm1.4.4.m2.1"><semantics id="algorithm1.4.4.m2.1a"><mi id="algorithm1.4.4.m2.1.1" xref="algorithm1.4.4.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="algorithm1.4.4.m2.1b"><ci id="algorithm1.4.4.m2.1.1.cmml" xref="algorithm1.4.4.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.4.4.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="algorithm1.4.4.m2.1d">italic_n</annotation></semantics></math> in dataset; </div> <div class="ltx_listingline" id="algorithm1.13.14"> <span class="ltx_tag ltx_tag_listingline">2</span> <span class="ltx_text ltx_font_bold" id="algorithm1.13.14.1">for</span> <em class="ltx_emph" id="algorithm1.13.14.2">each epoch</em> <span class="ltx_text ltx_font_bold" id="algorithm1.13.14.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.13.15"> <span class="ltx_tag ltx_tag_listingline">3</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.13.15.1">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.13.15.2"> <span class="ltx_text ltx_font_upright" id="algorithm1.13.15.2.1">each target AP in the training dataset </span></em> <span class="ltx_text ltx_font_bold" id="algorithm1.13.15.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.5.5"> <span class="ltx_tag ltx_tag_listingline">4</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Get <math alttext="\mathbf{M}^{*}_{\rm{AP},{\rm target}}" class="ltx_Math" display="inline" id="algorithm1.5.5.m1.2"><semantics id="algorithm1.5.5.m1.2a"><msubsup id="algorithm1.5.5.m1.2.3" xref="algorithm1.5.5.m1.2.3.cmml"><mi id="algorithm1.5.5.m1.2.3.2.2" xref="algorithm1.5.5.m1.2.3.2.2.cmml">𝐌</mi><mrow id="algorithm1.5.5.m1.2.2.2.4" xref="algorithm1.5.5.m1.2.2.2.3.cmml"><mi id="algorithm1.5.5.m1.1.1.1.1" xref="algorithm1.5.5.m1.1.1.1.1.cmml">AP</mi><mo id="algorithm1.5.5.m1.2.2.2.4.1" xref="algorithm1.5.5.m1.2.2.2.3.cmml">,</mo><mi id="algorithm1.5.5.m1.2.2.2.2" xref="algorithm1.5.5.m1.2.2.2.2.cmml">target</mi></mrow><mo id="algorithm1.5.5.m1.2.3.2.3" xref="algorithm1.5.5.m1.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="algorithm1.5.5.m1.2b"><apply id="algorithm1.5.5.m1.2.3.cmml" xref="algorithm1.5.5.m1.2.3"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.2.3.1.cmml" xref="algorithm1.5.5.m1.2.3">subscript</csymbol><apply id="algorithm1.5.5.m1.2.3.2.cmml" xref="algorithm1.5.5.m1.2.3"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.2.3.2.1.cmml" xref="algorithm1.5.5.m1.2.3">superscript</csymbol><ci id="algorithm1.5.5.m1.2.3.2.2.cmml" xref="algorithm1.5.5.m1.2.3.2.2">𝐌</ci><times id="algorithm1.5.5.m1.2.3.2.3.cmml" xref="algorithm1.5.5.m1.2.3.2.3"></times></apply><list id="algorithm1.5.5.m1.2.2.2.3.cmml" xref="algorithm1.5.5.m1.2.2.2.4"><ci id="algorithm1.5.5.m1.1.1.1.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1">AP</ci><ci id="algorithm1.5.5.m1.2.2.2.2.cmml" xref="algorithm1.5.5.m1.2.2.2.2">target</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.5.5.m1.2c">\mathbf{M}^{*}_{\rm{AP},{\rm target}}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.5.5.m1.2d">bold_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_AP , roman_target end_POSTSUBSCRIPT</annotation></semantics></math> through pre-convolution; </div> <div class="ltx_listingline" id="algorithm1.6.6"> <span class="ltx_tag ltx_tag_listingline">5</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Get the feature map <math alttext="\mathbf{M}_{n}" class="ltx_Math" display="inline" id="algorithm1.6.6.m1.1"><semantics id="algorithm1.6.6.m1.1a"><msub id="algorithm1.6.6.m1.1.1" xref="algorithm1.6.6.m1.1.1.cmml"><mi id="algorithm1.6.6.m1.1.1.2" xref="algorithm1.6.6.m1.1.1.2.cmml">𝐌</mi><mi id="algorithm1.6.6.m1.1.1.3" xref="algorithm1.6.6.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.6.6.m1.1b"><apply id="algorithm1.6.6.m1.1.1.cmml" xref="algorithm1.6.6.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.6.6.m1.1.1.1.cmml" xref="algorithm1.6.6.m1.1.1">subscript</csymbol><ci id="algorithm1.6.6.m1.1.1.2.cmml" xref="algorithm1.6.6.m1.1.1.2">𝐌</ci><ci id="algorithm1.6.6.m1.1.1.3.cmml" xref="algorithm1.6.6.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.6.6.m1.1c">\mathbf{M}_{n}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.6.6.m1.1d">bold_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> of all the other APs; </div> <div class="ltx_listingline" id="algorithm1.7.7"> <span class="ltx_tag ltx_tag_listingline">6</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Generate the input data <math alttext="\mathbf{M}_{\mathrm{input}}" class="ltx_Math" display="inline" id="algorithm1.7.7.m1.1"><semantics id="algorithm1.7.7.m1.1a"><msub id="algorithm1.7.7.m1.1.1" xref="algorithm1.7.7.m1.1.1.cmml"><mi id="algorithm1.7.7.m1.1.1.2" xref="algorithm1.7.7.m1.1.1.2.cmml">𝐌</mi><mi id="algorithm1.7.7.m1.1.1.3" xref="algorithm1.7.7.m1.1.1.3.cmml">input</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.7.7.m1.1b"><apply id="algorithm1.7.7.m1.1.1.cmml" xref="algorithm1.7.7.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.7.7.m1.1.1.1.cmml" xref="algorithm1.7.7.m1.1.1">subscript</csymbol><ci id="algorithm1.7.7.m1.1.1.2.cmml" xref="algorithm1.7.7.m1.1.1.2">𝐌</ci><ci id="algorithm1.7.7.m1.1.1.3.cmml" xref="algorithm1.7.7.m1.1.1.3">input</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.7.7.m1.1c">\mathbf{M}_{\mathrm{input}}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.7.7.m1.1d">bold_M start_POSTSUBSCRIPT roman_input end_POSTSUBSCRIPT</annotation></semantics></math>; </div> <div class="ltx_listingline" id="algorithm1.9.9"> <span class="ltx_tag ltx_tag_listingline">7</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Input <math alttext="\mathbf{M}_{\mathrm{input}}" class="ltx_Math" display="inline" id="algorithm1.8.8.m1.1"><semantics id="algorithm1.8.8.m1.1a"><msub id="algorithm1.8.8.m1.1.1" xref="algorithm1.8.8.m1.1.1.cmml"><mi id="algorithm1.8.8.m1.1.1.2" xref="algorithm1.8.8.m1.1.1.2.cmml">𝐌</mi><mi id="algorithm1.8.8.m1.1.1.3" xref="algorithm1.8.8.m1.1.1.3.cmml">input</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.8.8.m1.1b"><apply id="algorithm1.8.8.m1.1.1.cmml" xref="algorithm1.8.8.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.8.8.m1.1.1.1.cmml" xref="algorithm1.8.8.m1.1.1">subscript</csymbol><ci id="algorithm1.8.8.m1.1.1.2.cmml" xref="algorithm1.8.8.m1.1.1.2">𝐌</ci><ci id="algorithm1.8.8.m1.1.1.3.cmml" xref="algorithm1.8.8.m1.1.1.3">input</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.8.8.m1.1c">\mathbf{M}_{\mathrm{input}}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.8.8.m1.1d">bold_M start_POSTSUBSCRIPT roman_input end_POSTSUBSCRIPT</annotation></semantics></math> into the UNet and get <math alttext="\mathbf{G}^{\rm{infer}}_{\rm{target}}" class="ltx_Math" display="inline" id="algorithm1.9.9.m2.1"><semantics id="algorithm1.9.9.m2.1a"><msubsup id="algorithm1.9.9.m2.1.1" xref="algorithm1.9.9.m2.1.1.cmml"><mi id="algorithm1.9.9.m2.1.1.2.2" xref="algorithm1.9.9.m2.1.1.2.2.cmml">𝐆</mi><mi id="algorithm1.9.9.m2.1.1.3" xref="algorithm1.9.9.m2.1.1.3.cmml">target</mi><mi id="algorithm1.9.9.m2.1.1.2.3" xref="algorithm1.9.9.m2.1.1.2.3.cmml">infer</mi></msubsup><annotation-xml encoding="MathML-Content" id="algorithm1.9.9.m2.1b"><apply id="algorithm1.9.9.m2.1.1.cmml" xref="algorithm1.9.9.m2.1.1"><csymbol cd="ambiguous" id="algorithm1.9.9.m2.1.1.1.cmml" xref="algorithm1.9.9.m2.1.1">subscript</csymbol><apply id="algorithm1.9.9.m2.1.1.2.cmml" xref="algorithm1.9.9.m2.1.1"><csymbol cd="ambiguous" id="algorithm1.9.9.m2.1.1.2.1.cmml" xref="algorithm1.9.9.m2.1.1">superscript</csymbol><ci id="algorithm1.9.9.m2.1.1.2.2.cmml" xref="algorithm1.9.9.m2.1.1.2.2">𝐆</ci><ci id="algorithm1.9.9.m2.1.1.2.3.cmml" xref="algorithm1.9.9.m2.1.1.2.3">infer</ci></apply><ci id="algorithm1.9.9.m2.1.1.3.cmml" xref="algorithm1.9.9.m2.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.9.9.m2.1c">\mathbf{G}^{\rm{infer}}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.9.9.m2.1d">bold_G start_POSTSUPERSCRIPT roman_infer end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math>; </div> <div class="ltx_listingline" id="algorithm1.10.10"> <span class="ltx_tag ltx_tag_listingline">8</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Calculate the loss and gradients with <math alttext="\mathbf{G}_{\rm{target}}" class="ltx_Math" display="inline" id="algorithm1.10.10.m1.1"><semantics id="algorithm1.10.10.m1.1a"><msub id="algorithm1.10.10.m1.1.1" xref="algorithm1.10.10.m1.1.1.cmml"><mi id="algorithm1.10.10.m1.1.1.2" xref="algorithm1.10.10.m1.1.1.2.cmml">𝐆</mi><mi id="algorithm1.10.10.m1.1.1.3" xref="algorithm1.10.10.m1.1.1.3.cmml">target</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.10.10.m1.1b"><apply id="algorithm1.10.10.m1.1.1.cmml" xref="algorithm1.10.10.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.10.10.m1.1.1.1.cmml" xref="algorithm1.10.10.m1.1.1">subscript</csymbol><ci id="algorithm1.10.10.m1.1.1.2.cmml" xref="algorithm1.10.10.m1.1.1.2">𝐆</ci><ci id="algorithm1.10.10.m1.1.1.3.cmml" xref="algorithm1.10.10.m1.1.1.3">target</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.10.10.m1.1c">\mathbf{G}_{\rm{target}}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.10.10.m1.1d">bold_G start_POSTSUBSCRIPT roman_target end_POSTSUBSCRIPT</annotation></semantics></math>; </div> <div class="ltx_listingline" id="algorithm1.11.11"> <span class="ltx_tag ltx_tag_listingline">9</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Update the UNet parameters <math alttext="\{\bm{\theta}\}" class="ltx_Math" display="inline" id="algorithm1.11.11.m1.1"><semantics id="algorithm1.11.11.m1.1a"><mrow id="algorithm1.11.11.m1.1.2.2" xref="algorithm1.11.11.m1.1.2.1.cmml"><mo id="algorithm1.11.11.m1.1.2.2.1" stretchy="false" xref="algorithm1.11.11.m1.1.2.1.cmml">{</mo><mi id="algorithm1.11.11.m1.1.1" xref="algorithm1.11.11.m1.1.1.cmml">𝜽</mi><mo id="algorithm1.11.11.m1.1.2.2.2" stretchy="false" xref="algorithm1.11.11.m1.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.11.11.m1.1b"><set id="algorithm1.11.11.m1.1.2.1.cmml" xref="algorithm1.11.11.m1.1.2.2"><ci id="algorithm1.11.11.m1.1.1.cmml" xref="algorithm1.11.11.m1.1.1">𝜽</ci></set></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.11.11.m1.1c">\{\bm{\theta}\}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.11.11.m1.1d">{ bold_italic_θ }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.13.16"> <span class="ltx_tag ltx_tag_listingline">10</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> end for </div> <div class="ltx_listingline" id="algorithm1.13.17"> <span class="ltx_tag ltx_tag_listingline">11</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm1.12.12"> <span class="ltx_tag ltx_tag_listingline">12</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Save the parameters <math alttext="\{\bm{\theta^{\star}}\}" class="ltx_Math" display="inline" id="algorithm1.12.12.m1.1"><semantics id="algorithm1.12.12.m1.1a"><mrow id="algorithm1.12.12.m1.1.1.1" xref="algorithm1.12.12.m1.1.1.2.cmml"><mo id="algorithm1.12.12.m1.1.1.1.2" stretchy="false" xref="algorithm1.12.12.m1.1.1.2.cmml">{</mo><msup id="algorithm1.12.12.m1.1.1.1.1" xref="algorithm1.12.12.m1.1.1.1.1.cmml"><mi id="algorithm1.12.12.m1.1.1.1.1.2" xref="algorithm1.12.12.m1.1.1.1.1.2.cmml">𝜽</mi><mo class="ltx_mathvariant_bold" id="algorithm1.12.12.m1.1.1.1.1.3" mathvariant="bold" xref="algorithm1.12.12.m1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="algorithm1.12.12.m1.1.1.1.3" stretchy="false" xref="algorithm1.12.12.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.12.12.m1.1b"><set id="algorithm1.12.12.m1.1.1.2.cmml" xref="algorithm1.12.12.m1.1.1.1"><apply id="algorithm1.12.12.m1.1.1.1.1.cmml" xref="algorithm1.12.12.m1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.12.12.m1.1.1.1.1.1.cmml" xref="algorithm1.12.12.m1.1.1.1.1">superscript</csymbol><ci id="algorithm1.12.12.m1.1.1.1.1.2.cmml" xref="algorithm1.12.12.m1.1.1.1.1.2">𝜽</ci><ci id="algorithm1.12.12.m1.1.1.1.1.3.cmml" xref="algorithm1.12.12.m1.1.1.1.1.3">bold-⋆</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.12.12.m1.1c">\{\bm{\theta^{\star}}\}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.12.12.m1.1d">{ bold_italic_θ start_POSTSUPERSCRIPT bold_⋆ end_POSTSUPERSCRIPT }</annotation></semantics></math> with the best loss; </div> <div class="ltx_listingline" id="algorithm1.13.18"> <span class="ltx_tag ltx_tag_listingline">13</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm1.13.19"> <span class="ltx_tag ltx_tag_listingline">14</span> end for </div> <div class="ltx_listingline" id="algorithm1.13.13"> <span class="ltx_text ltx_font_bold" id="algorithm1.13.13.2">return</span> <em class="ltx_emph" id="algorithm1.13.13.1">the optimal parameters <math alttext="\{\bm{\theta^{\star}}\}" class="ltx_Math" display="inline" id="algorithm1.13.13.1.1.m1.1"><semantics id="algorithm1.13.13.1.1.m1.1a"><mrow id="algorithm1.13.13.1.1.m1.1.1.1" xref="algorithm1.13.13.1.1.m1.1.1.2.cmml"><mo id="algorithm1.13.13.1.1.m1.1.1.1.2" stretchy="false" xref="algorithm1.13.13.1.1.m1.1.1.2.cmml">{</mo><msup id="algorithm1.13.13.1.1.m1.1.1.1.1" xref="algorithm1.13.13.1.1.m1.1.1.1.1.cmml"><mi id="algorithm1.13.13.1.1.m1.1.1.1.1.2" xref="algorithm1.13.13.1.1.m1.1.1.1.1.2.cmml">𝜽</mi><mo class="ltx_mathvariant_bold" id="algorithm1.13.13.1.1.m1.1.1.1.1.3" mathvariant="bold" xref="algorithm1.13.13.1.1.m1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="algorithm1.13.13.1.1.m1.1.1.1.3" stretchy="false" xref="algorithm1.13.13.1.1.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.13.13.1.1.m1.1b"><set id="algorithm1.13.13.1.1.m1.1.1.2.cmml" xref="algorithm1.13.13.1.1.m1.1.1.1"><apply id="algorithm1.13.13.1.1.m1.1.1.1.1.cmml" xref="algorithm1.13.13.1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.13.13.1.1.m1.1.1.1.1.1.cmml" xref="algorithm1.13.13.1.1.m1.1.1.1.1">superscript</csymbol><ci id="algorithm1.13.13.1.1.m1.1.1.1.1.2.cmml" xref="algorithm1.13.13.1.1.m1.1.1.1.1.2">𝜽</ci><ci id="algorithm1.13.13.1.1.m1.1.1.1.1.3.cmml" xref="algorithm1.13.13.1.1.m1.1.1.1.1.3">bold-⋆</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.13.13.1.1.m1.1c">\{\bm{\theta^{\star}}\}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.13.13.1.1.m1.1d">{ bold_italic_θ start_POSTSUPERSCRIPT bold_⋆ end_POSTSUPERSCRIPT }</annotation></semantics></math></em> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm1.15.1.1">Algorithm 1</span> </span>Training phase of the cross-AP inference</figcaption> </figure> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="287" id="S3.F4.g1" src="x4.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Structure of the UNet for Cross-AP CKM Inference. </figcaption> </figure> </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">Inference Results</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, the UNet network is trained based on RadioMapSeer Dataset. The trained UNet network will perform cross-AP CKM inference for validation. The inference results for CKM will be compared with the benchmark schemes to demonstrate the feasibility of generating inferred CKM across APs in cell-free networks without physical environment.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2">Training Settings</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">To ensure the independence between the training dataset and the validation dataset, 500 of the 700 different physical environment maps in RadioMapSeer are arbitrarily selected for model training. The training dataset consists of 80 AP location maps and their CGMs corresponding to each physical environment map and does not contain the physical environment maps themselves. During the training process, each AP of each physical map acts as the target AP in turn and uses the corresponding target CGM as the ground-truth, constituting a total of 40000 sets of input-output training data. The training is performed based on Adam, where the initial learning rate is <math alttext="10^{-3}" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><msup id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mn id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">10</mn><mrow id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml"><mo id="S4.SS1.p1.1.m1.1.1.3a" xref="S4.SS1.p1.1.m1.1.1.3.cmml">−</mo><mn id="S4.SS1.p1.1.m1.1.1.3.2" xref="S4.SS1.p1.1.m1.1.1.3.2.cmml">3</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">superscript</csymbol><cn id="S4.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.2">10</cn><apply id="S4.SS1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.p1.1.m1.1.1.3"><minus id="S4.SS1.p1.1.m1.1.1.3.1.cmml" xref="S4.SS1.p1.1.m1.1.1.3"></minus><cn id="S4.SS1.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.3.2">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math>. The UNet training is carried out for 15 epochs, where the batch size is 15. To mitigate overfitting, the model that minimized the MSE loss in the validation set over the 15 epochs is saved.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2">Training Results</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">After training, CKM inference across APs was performed on the validation dataset with the UNet model. The 100 physical environment maps in RadioMapSeer Dataset that are disjoint from the training dataset are selected as the validation dataset. Similarly, the validation dataset is composed of 80 AP location maps with CGMs for each physical environment map. Two basic CKM construction methods are chosen as benchmarks.</p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1">Weighted cross-AP CKM inference scheme. 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id="S4.E7.m1.5.5.1.1.6.2.2.2.cmml" xref="S4.E7.m1.5.5.1.1.6.2.2.2">𝐆</ci><ci id="S4.E7.m1.5.5.1.1.6.2.2.3.cmml" xref="S4.E7.m1.5.5.1.1.6.2.2.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.5c">\mathbf{G}^{\rm{infer}}_{0}=\sum_{n=1}^{N}w_{n}\mathbf{G}_{n}=\sum_{n=1}^{N}% \frac{e^{-\beta d_{t,n}}}{\sum_{i=1}^{N}e^{-\beta d_{t,i}}}\mathbf{G}_{n},</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.5d">bold_G start_POSTSUPERSCRIPT roman_infer end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT bold_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_β italic_d start_POSTSUBSCRIPT italic_t , italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_β italic_d start_POSTSUBSCRIPT italic_t , italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG bold_G start_POSTSUBSCRIPT 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I1.i1.p2.4">where <math alttext="w_{n}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.1.m1.1"><semantics id="S4.I1.i1.p2.1.m1.1a"><msub id="S4.I1.i1.p2.1.m1.1.1" xref="S4.I1.i1.p2.1.m1.1.1.cmml"><mi id="S4.I1.i1.p2.1.m1.1.1.2" xref="S4.I1.i1.p2.1.m1.1.1.2.cmml">w</mi><mi id="S4.I1.i1.p2.1.m1.1.1.3" xref="S4.I1.i1.p2.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.1.m1.1b"><apply id="S4.I1.i1.p2.1.m1.1.1.cmml" xref="S4.I1.i1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.1.m1.1.1.1.cmml" xref="S4.I1.i1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.1.m1.1.1.2.cmml" xref="S4.I1.i1.p2.1.m1.1.1.2">𝑤</ci><ci id="S4.I1.i1.p2.1.m1.1.1.3.cmml" xref="S4.I1.i1.p2.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.1.m1.1c">w_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> denotes the weight based on the Euclidean distance, <math alttext="\beta=0.1" class="ltx_Math" display="inline" id="S4.I1.i1.p2.2.m2.1"><semantics id="S4.I1.i1.p2.2.m2.1a"><mrow id="S4.I1.i1.p2.2.m2.1.1" xref="S4.I1.i1.p2.2.m2.1.1.cmml"><mi id="S4.I1.i1.p2.2.m2.1.1.2" xref="S4.I1.i1.p2.2.m2.1.1.2.cmml">β</mi><mo id="S4.I1.i1.p2.2.m2.1.1.1" xref="S4.I1.i1.p2.2.m2.1.1.1.cmml">=</mo><mn id="S4.I1.i1.p2.2.m2.1.1.3" xref="S4.I1.i1.p2.2.m2.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.2.m2.1b"><apply id="S4.I1.i1.p2.2.m2.1.1.cmml" xref="S4.I1.i1.p2.2.m2.1.1"><eq id="S4.I1.i1.p2.2.m2.1.1.1.cmml" xref="S4.I1.i1.p2.2.m2.1.1.1"></eq><ci id="S4.I1.i1.p2.2.m2.1.1.2.cmml" xref="S4.I1.i1.p2.2.m2.1.1.2">𝛽</ci><cn id="S4.I1.i1.p2.2.m2.1.1.3.cmml" type="float" xref="S4.I1.i1.p2.2.m2.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.2.m2.1c">\beta=0.1</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.2.m2.1d">italic_β = 0.1</annotation></semantics></math> denotes the weight parameter. The distance between the target AP and the AP <math alttext="i" class="ltx_Math" display="inline" id="S4.I1.i1.p2.3.m3.1"><semantics id="S4.I1.i1.p2.3.m3.1a"><mi id="S4.I1.i1.p2.3.m3.1.1" xref="S4.I1.i1.p2.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.3.m3.1b"><ci id="S4.I1.i1.p2.3.m3.1.1.cmml" xref="S4.I1.i1.p2.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.3.m3.1d">italic_i</annotation></semantics></math> is <math alttext="d_{t,i}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.4.m4.2"><semantics id="S4.I1.i1.p2.4.m4.2a"><msub id="S4.I1.i1.p2.4.m4.2.3" xref="S4.I1.i1.p2.4.m4.2.3.cmml"><mi id="S4.I1.i1.p2.4.m4.2.3.2" xref="S4.I1.i1.p2.4.m4.2.3.2.cmml">d</mi><mrow id="S4.I1.i1.p2.4.m4.2.2.2.4" xref="S4.I1.i1.p2.4.m4.2.2.2.3.cmml"><mi id="S4.I1.i1.p2.4.m4.1.1.1.1" xref="S4.I1.i1.p2.4.m4.1.1.1.1.cmml">t</mi><mo id="S4.I1.i1.p2.4.m4.2.2.2.4.1" xref="S4.I1.i1.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.I1.i1.p2.4.m4.2.2.2.2" xref="S4.I1.i1.p2.4.m4.2.2.2.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.4.m4.2b"><apply id="S4.I1.i1.p2.4.m4.2.3.cmml" xref="S4.I1.i1.p2.4.m4.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.4.m4.2.3.1.cmml" xref="S4.I1.i1.p2.4.m4.2.3">subscript</csymbol><ci id="S4.I1.i1.p2.4.m4.2.3.2.cmml" xref="S4.I1.i1.p2.4.m4.2.3.2">𝑑</ci><list id="S4.I1.i1.p2.4.m4.2.2.2.3.cmml" xref="S4.I1.i1.p2.4.m4.2.2.2.4"><ci id="S4.I1.i1.p2.4.m4.1.1.1.1.cmml" xref="S4.I1.i1.p2.4.m4.1.1.1.1">𝑡</ci><ci id="S4.I1.i1.p2.4.m4.2.2.2.2.cmml" xref="S4.I1.i1.p2.4.m4.2.2.2.2">𝑖</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.4.m4.2c">d_{t,i}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.4.m4.2d">italic_d start_POSTSUBSCRIPT italic_t , italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1">Path loss model in urban microcell proposed by 3GPP TR 38.901 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib12" title="">12</a>]</cite>.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">The MSEs and root MSEs (RMSEs) of different cross-AP CKM inference schemes are shown in Table I. Note that the range of the channel gain from the noise floor to the maximum in RadioMapSeer Dataset is 100dB, the unit of the RMSE is dB. As shown in Table I, the accuracy of the learning-based cross-AP CKM inference is about 2.38 dB. This mean error is on the same level as that of RadioUNet(2.03dB), but without the need for the physical environment map as training data. Compared with the benchmark schemes, the accuracy of the proposed cross-AP CKM inference is 3dB higher than the distance-based weighted inference, and 33dB higher than the model-based inference. The comparisons of the cross-AP CKM inference and the benchmark schemes are presented in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#S4.F5.sf2" title="In Figure 5 ‣ IV-B Training Results ‣ IV Inference Results ‣ Generating CKM Using Others’ Data: Cross-AP CKM Inference with Deep Learning"><span class="ltx_text ltx_ref_tag">5(b)</span></a>. The inference is performed in different physical environments ranging from simple to complex. The distance-based weighted inference blurs key features of the CKM such as the AP location. In contrast, a comparison with the CKM ground-truth reveals that the proposed cross-AP CKM inference well preserves the target AP location and learns the wireless environment features and wireless propagation characteristics. Even unaware of the physical environment map, the CKM inference still exhibits the attenuation and mutation characteristics under the occlusion of buildings.</p> </div> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE I: </span>Cross-AP CKM Inference MSE</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T1.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T1.1.1.2">Scheme</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T1.1.1.1">MSE(dB<sup class="ltx_sup" id="S4.T1.1.1.1.1">2</sup>)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S4.T1.1.1.3">RMSE(dB)</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.1.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.1.2.1.1">Proposed cross-AP CKM inference</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.1.2.1.2">5.66</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.1.2.1.3">2.38</td> </tr> <tr class="ltx_tr" id="S4.T1.1.3.2"> <td class="ltx_td ltx_align_center" id="S4.T1.1.3.2.1">RadioUNet CKM generation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.17716v1#bib.bib5" title="">5</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S4.T1.1.3.2.2">4.12</td> <td class="ltx_td ltx_align_center" id="S4.T1.1.3.2.3">2.03</td> </tr> <tr class="ltx_tr" id="S4.T1.1.4.3"> <td class="ltx_td ltx_align_center" id="S4.T1.1.4.3.1">Benchmark 1: weighted CKM inference</td> <td class="ltx_td ltx_align_center" id="S4.T1.1.4.3.2">28.04</td> <td class="ltx_td ltx_align_center" id="S4.T1.1.4.3.3">5.30</td> </tr> <tr class="ltx_tr" id="S4.T1.1.5.4"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.1.5.4.1">Benchmark 2: model-based CKM inference</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.1.5.4.2">1275.58</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.1.5.4.3">35.72</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf1.g1" src="extracted/6011839/figure/441_3_target_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf1.2.1.1" style="font-size:80%;">(a)</span> </span><span class="ltx_text" id="S4.F5.sf1.3.2" style="font-size:80%;">CKM Ground-truth(simple)</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf2.g1" src="extracted/6011839/figure/441_3_predict_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf2.2.1.1" style="font-size:80%;">(b)</span> </span><span class="ltx_text" id="S4.F5.sf2.3.2" style="font-size:80%;">Proposed CKM Inference(simple)</span></figcaption> <figure class="ltx_figure" id="S4.F5.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf3.g1" src="extracted/6011839/figure/239_47_target_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf3.2.1.1" style="font-size:80%;">(c)</span> </span><span class="ltx_text" id="S4.F5.sf3.3.2" style="font-size:80%;">CKM Ground-truth(medium)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf4"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf4.g1" src="extracted/6011839/figure/239_47_predict_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf4.2.1.1" style="font-size:80%;">(d)</span> </span><span class="ltx_text" id="S4.F5.sf4.3.2" style="font-size:80%;">Proposed CKM Inference(medium)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf5"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf5.g1" src="extracted/6011839/figure/324_78_target_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf5.2.1.1" style="font-size:80%;">(e)</span> </span><span class="ltx_text" id="S4.F5.sf5.3.2" style="font-size:80%;">CKM Ground-truth(complex)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf6"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf6.g1" src="extracted/6011839/figure/324_78_predict_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf6.2.1.1" style="font-size:80%;">(f)</span> </span><span class="ltx_text" id="S4.F5.sf6.3.2" style="font-size:80%;">Proposed CKM Inference(complex)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf7"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf7.g1" src="extracted/6011839/figure/441_3_bench_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf7.2.1.1" style="font-size:80%;">(g)</span> </span><span class="ltx_text" id="S4.F5.sf7.3.2" style="font-size:80%;">Weighted CKM Inference(simple)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf8"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf8.g1" src="extracted/6011839/figure/441_3_3GPP_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf8.2.1.1" style="font-size:80%;">(h)</span> </span><span class="ltx_text" id="S4.F5.sf8.3.2" style="font-size:80%;">Model-based CKM Inference(simple)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf9"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf9.g1" src="extracted/6011839/figure/239_47_bench_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf9.2.1.1" style="font-size:80%;">(i)</span> </span><span class="ltx_text" id="S4.F5.sf9.3.2" style="font-size:80%;">Weighted CKM Inference(medium)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf10"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf10.g1" src="extracted/6011839/figure/239_47_3GPP_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf10.2.1.1" style="font-size:80%;">(j)</span> </span><span class="ltx_text" id="S4.F5.sf10.3.2" style="font-size:80%;">Model-based CKM Inference(medium)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf11"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf11.g1" src="extracted/6011839/figure/324_78_bench_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf11.2.1.1" style="font-size:80%;">(k)</span> </span><span class="ltx_text" id="S4.F5.sf11.3.2" style="font-size:80%;">Weighted CKM Inference(complex)</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5.sf12"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="180" id="S4.F5.sf12.g1" src="extracted/6011839/figure/324_78_3GPP_mask.png" width="180"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf12.2.1.1" style="font-size:80%;">(l)</span> </span><span class="ltx_text" id="S4.F5.sf12.3.2" style="font-size:80%;">Model-based CKM Inference(complex)</span></figcaption> </figure> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Comparison of the Cross-AP CKM Inference and the Benchmarks.</figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S4.F5.1"><span class="ltx_rule" style="width:2.0pt;height:71.1pt;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:2.0pt;height:71.1pt;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:2.0pt;height:71.1pt;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:2.0pt;height:71.1pt;background:black;display:inline-block;"> </span></p> </div> </div> </figure> </section> </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">Conclusion</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">This paper proposes a cross-AP CKM inference in cell-free networks. By taking advantage of the correlation of the wireless environment and the shared physical environment among APs, the trained UNet utilizes other existing APs’ CKMs to generate CKMs of potentially new APs without the need for physical environment maps or any onsite measurement. The comparisons with the CKM inference by benchmark schemes validate the feasibility and effectiveness of cross-AP CKM inference, which is significant for the construction and updating of CKMs as well as the environment-aware deployment of potentially new APs in dense networks.</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"> Y. Zeng and X. 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