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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Downsampling&redirect=no" class="mw-redirect" title="Downsampling">Downsampling</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Resampling method</div> <p>In <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, <b>downsampling</b>, <b>compression</b>, and <b>decimation</b> are terms associated with the process of <a href="/wiki/Sample_rate_conversion" class="mw-redirect" title="Sample rate conversion"><i>resampling</i></a> in a <a href="/wiki/Multi-rate_digital_signal_processing" class="mw-redirect" title="Multi-rate digital signal processing">multi-rate digital signal processing</a> system. Both <i>downsampling</i> and <i>decimation</i> can be synonymous with <i>compression</i>, or they can describe an entire process of bandwidth reduction (<a href="/wiki/Lowpass_filter" class="mw-redirect" title="Lowpass filter">filtering</a>) and sample-rate reduction.<sup id="cite_ref-Oppenheim_1-0" class="reference"><a href="#cite_note-Oppenheim-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-LiTan_2-0" class="reference"><a href="#cite_note-LiTan-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> When the process is performed on a sequence of samples of a <i>signal</i> or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or <a href="/wiki/Dots_per_inch" title="Dots per inch">density</a>, as in the case of a photograph). </p><p><i>Decimation</i> is a term that historically means the <i><a href="/wiki/Decimation_(punishment)" title="Decimation (punishment)">removal of every tenth one</a></i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> But in signal processing, <i>decimation by a factor of 10</i> actually means <i>keeping</i> only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if <a href="/wiki/Compact_disc" title="Compact disc">compact disc</a> audio at 44,100 samples/second is <i>decimated</i> by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a <i>decimator</i>. Decimation by an integer factor is also called <i>compression</i>.<sup id="cite_ref-Crochiere_4-0" class="reference"><a href="#cite_note-Crochiere-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Poularikas_5-0" class="reference"><a href="#cite_note-Poularikas-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Downsampling_by_an_integer_factor">Downsampling by an integer factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=1" title="Edit section: Downsampling by an integer factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rate reduction by an integer factor <i>M</i> can be explained as a two-step process, with an equivalent implementation that is more efficient:<sup id="cite_ref-f.harris_6-0" class="reference"><a href="#cite_note-f.harris-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Reduce high-frequency signal components with a digital <a href="/wiki/Lowpass_filter" class="mw-redirect" title="Lowpass filter">lowpass filter</a>.</li> <li><i>Decimate</i> the filtered signal by <i>M</i>; that is, keep only every <i>M</i><sup>th</sup> sample.</li></ol> <p>Step 2 alone creates undesirable <a href="/wiki/Aliasing" title="Aliasing">aliasing</a> (i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies). Step 1, when necessary, suppresses aliasing to an acceptable level. In this application, the filter is called an <a href="/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a>, and its design is discussed below. Also see <a href="/wiki/Undersampling" title="Undersampling">undersampling</a> for information about decimating <a href="/wiki/Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> functions and signals. </p><p>When the anti-aliasing filter is an <a href="/wiki/Infinite_impulse_response" title="Infinite impulse response">IIR</a> design, it relies on feedback from output to input, prior to the second step. With <a href="/wiki/FIR_filter" class="mw-redirect" title="FIR filter">FIR filtering</a>, it is an easy matter to compute only every <i>M</i><sup>th</sup> output. The calculation performed by a decimating FIR filter for the <i>n</i><sup>th</sup> output sample is a <a href="/wiki/Dot_product" title="Dot product">dot product</a>:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=\sum _{k=0}^{K-1}x[nM-k]\cdot h[k],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mi>M</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=\sum _{k=0}^{K-1}x[nM-k]\cdot h[k],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d3756569db36dcd0f869fb9acb8a24781041fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.573ex; height:7.509ex;" alt="{\displaystyle y[n]=\sum _{k=0}^{K-1}x[nM-k]\cdot h[k],}"></span></dd></dl> <p>where the <i>h</i>[•] sequence is the impulse response, and <i>K</i> is its length.  <i>x</i>[•] represents the input sequence being downsampled. In a general purpose processor, after computing <i>y</i>[<i>n</i>], the easiest way to compute <i>y</i>[<i>n</i>+1] is to advance the starting index in the <i>x</i>[•] array by <i>M</i>, and recompute the dot product. In the case <i>M</i>=2, <i>h</i>[•] can be designed as a <a href="/wiki/Half-band_filter" title="Half-band filter">half-band filter</a>, where almost half of the coefficients are zero and need not be included in the dot products. </p><p>Impulse response coefficients taken at intervals of <i>M</i> form a subsequence, and there are <i>M</i> such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with the corresponding samples of the <i>x</i>[•] sequence. Furthermore, because of downsampling by <i>M</i>, the stream of <i>x</i>[•] samples involved in any one of the <i>M</i> dot products is never involved in the other dot products. Thus <i>M</i> low-order FIR filters are each filtering one of <i>M</i> multiplexed <i>phases</i> of the input stream, and the <i>M</i> outputs are being summed. This viewpoint offers a different implementation that might be advantageous in a multi-processor architecture. In other words, the input stream is demultiplexed and sent through a bank of M filters whose outputs are summed. When implemented that way, it is called a <b>polyphase</b> filter. </p><p>For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the <i>h</i>[•] array, process the original <i>x</i>[•] sequence at the input rate (which means multiplying by zeros), and decimate the output by a factor of <i>M</i>. The equivalence of this inefficient method and the implementation described above is known as the <i>first Noble identity</i>.<sup id="cite_ref-Strang_8-0" class="reference"><a href="#cite_note-Strang-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> It is sometimes used in derivations of the polyphase method. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Spectral_effects_of_decimation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Spectral_effects_of_decimation.svg/400px-Spectral_effects_of_decimation.svg.png" decoding="async" width="400" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Spectral_effects_of_decimation.svg/600px-Spectral_effects_of_decimation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Spectral_effects_of_decimation.svg/800px-Spectral_effects_of_decimation.svg.png 2x" data-file-width="1012" data-file-height="675" /></a><figcaption>Fig 1: These graphs depict the spectral distributions of an oversampled function and the same function sampled at 1/3 the original rate. The bandwidth, B, in this example is just small enough that the slower sampling does not cause overlap (aliasing). Sometimes, a sampled function is resampled at a lower rate by keeping only every M<sup>th</sup> sample and discarding the others, commonly called "decimation". Potential aliasing is prevented by lowpass-filtering the samples before decimation. The maximum filter bandwidth is tabulated in the bandwidth units used by the common filter design applications.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Anti-aliasing_filter">Anti-aliasing filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=2" title="Edit section: Anti-aliasing filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i>(<i>f</i>) be the <a href="/wiki/Continuous_Fourier_transform" class="mw-redirect" title="Continuous Fourier transform">Fourier transform</a> of any function, <i>x</i>(<i>t</i>), whose samples at some interval, <i>T</i>, equal the <i>x</i>[<i>n</i>] sequence. Then the <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT) is a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> representation of a <a href="/wiki/Periodic_summation" title="Periodic summation">periodic summation</a> of <i>X</i>(<i>f</i>):<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ \mathrm {e} ^{-\mathrm {i} 2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </mover> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mi>T</mi> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>DTFT</mtext> </mrow> </munder> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>f</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ \mathrm {e} ^{-\mathrm {i} 2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f042baeec78e1fbce45dd46dff9e0077503392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:43.531ex; height:12.176ex;" alt="{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ \mathrm {e} ^{-\mathrm {i} 2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}"></span></dd></dl> <p>When <i>T</i> has units of seconds, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> has units of <a href="/wiki/Hertz" title="Hertz">hertz</a>. Replacing <i>T</i> with <i>MT</i> in the formulas above gives the DTFT of the decimated sequence, <i>x</i>[<i>nM</i>]: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }x(n\cdot MT)\ \mathrm {e} ^{-\mathrm {i} 2\pi fn(MT)}={\frac {1}{MT}}\sum _{k=-\infty }^{\infty }X\left(f-{\tfrac {k}{MT}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>⋅</mo> <mi>M</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>M</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>X</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mrow> <mi>M</mi> <mi>T</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }x(n\cdot MT)\ \mathrm {e} ^{-\mathrm {i} 2\pi fn(MT)}={\frac {1}{MT}}\sum _{k=-\infty }^{\infty }X\left(f-{\tfrac {k}{MT}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6e88a638f4a33511701aec6d7f0610c93abd75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.123ex; height:7.009ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }x(n\cdot MT)\ \mathrm {e} ^{-\mathrm {i} 2\pi fn(MT)}={\frac {1}{MT}}\sum _{k=-\infty }^{\infty }X\left(f-{\tfrac {k}{MT}}\right).}"></span></dd></dl> <p>The periodic summation has been reduced in amplitude and periodicity by a factor of <i>M</i>.  An example of both these distributions is depicted in the two traces of Fig 1.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup> Aliasing occurs when adjacent copies of <i>X</i>(<i>f</i>) overlap. The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap. The condition that ensures the copies of <i>X</i>(<i>f</i>) do not overlap each other is: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B<{\tfrac {0.5}{T}}\cdot {\tfrac {1}{M}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>0.5</mn> <mi>T</mi> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B<{\tfrac {0.5}{T}}\cdot {\tfrac {1}{M}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2896494dd21f04ec1c421fbd5447165d2745a547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.689ex; height:3.509ex;" alt="{\displaystyle B<{\tfrac {0.5}{T}}\cdot {\tfrac {1}{M}},}"></span> so that is the maximum <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a> of an <i>ideal</i> anti-aliasing filter.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>A<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="By_a_rational_factor">By a rational factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=3" title="Edit section: By a rational factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>M/L</i> denote the decimation factor,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>B<span class="cite-bracket">]</span></a></sup> where: <span class="nowrap">M, L ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>; M > L.</span> </p> <ol><li>Increase (resample) the sequence by a factor of <i>L</i>. This is called <a href="/wiki/Upsampling" title="Upsampling">Upsampling</a>, or <i>interpolation</i>.</li> <li>Decimate by a factor of <i>M</i></li></ol> <p>Step 1 requires a lowpass filter after increasing (<i>expanding</i>) the data rate, and step 2 requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the <i>M</i> > <i>L</i> case, the anti-aliasing filter cutoff, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {0.5}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>0.5</mn> <mi>M</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {0.5}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24f180c8a2ed12c0d454a2c8bfb33af1b6a44c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.937ex; height:3.509ex;" alt="{\displaystyle {\tfrac {0.5}{M}}}"></span> <i>cycles per intermediate sample</i>, is the lower frequency. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Upsampling" title="Upsampling">Upsampling</a></li> <li><a href="/wiki/Posterization" title="Posterization">Posterization</a></li> <li><a href="/wiki/Sample-rate_conversion" title="Sample-rate conversion">Sample-rate conversion</a></li> <li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Visvalingam%E2%80%93Whyatt_algorithm" title="Visvalingam–Whyatt algorithm">Visvalingam–Whyatt algorithm</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-upper-alpha"> <ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Realizable low-pass filters have a "skirt", where the response diminishes from near one to near zero.  In practice the cutoff frequency is placed far enough below the theoretical cutoff that the filter's skirt is contained below the theoretical cutoff.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">General techniques for sample-rate conversion by factor R ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{+}}"></span> include <a href="/wiki/Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a> and the Farrow structure.<sup id="cite_ref-Milic_15-0" class="reference"><a href="#cite_note-Milic-15"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Page_citations">Page citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=6" title="Edit section: Page citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width reflist-lower-alpha"> <ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#f.harris">Harris 2004</a>. "6.1". p 128.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#Crochiere">Crochiere and Rabiner</a> "2". p 32. eq 2.55a.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#f.harris">Harris 2004</a>. "2.2.1". p 25.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#Oppenheim">Oppenheim and Schafer</a>. "4.2". p 143. eq 4.6, where<b>:</b>  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \triangleq 2\pi f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>≜</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \triangleq 2\pi f,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668996ec825cc28bb3b2de0485ebb6ef790c99aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.196ex; height:2.843ex;" alt="{\displaystyle \Omega \triangleq 2\pi f,}"></span>  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}(i\Omega )\triangleq \sum _{n=-\infty }^{\infty }x(nT)\ \mathrm {e} ^{-\mathrm {i} \Omega nT},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi mathvariant="normal">Ω</mi> <mi>n</mi> <mi>T</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{s}(i\Omega )\triangleq \sum _{n=-\infty }^{\infty }x(nT)\ \mathrm {e} ^{-\mathrm {i} \Omega nT},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a443ceed6fe7d2c36216db41088741b0bfbeca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.617ex; height:6.843ex;" alt="{\displaystyle X_{s}(i\Omega )\triangleq \sum _{n=-\infty }^{\infty }x(nT)\ \mathrm {e} ^{-\mathrm {i} \Omega nT},}"></span>  and  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{c}(i2\pi f)\triangleq X(f).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{c}(i2\pi f)\triangleq X(f).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebf5a19bd65c238aece852df45d5353a2eb8864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.066ex; height:3.009ex;" alt="{\displaystyle X_{c}(i2\pi f)\triangleq X(f).}"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#f.harris">Harris 2004</a>. "2.2". p 22. fig 2.10.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#Oppenheim">Oppenheim and Schafer</a>. "4.6". p 171. fig 4.22.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#LiTan">Tan 2008</a>. "1.2.1". fig 12.2.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Oppenheim-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Oppenheim_1-0">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFOppenheimSchaferBuck1999" class="citation book cs1"><a href="/wiki/Alan_V_Oppenheim" class="mw-redirect" title="Alan V Oppenheim">Oppenheim, Alan V.</a>; Schafer, Ronald W.; Buck, John R. (1999). "4". <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan"><i>Discrete-Time Signal Processing</i></a></span> (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. p. <a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan/page/168">168</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-754920-2" title="Special:BookSources/0-13-754920-2"><bdi>0-13-754920-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4&rft.btitle=Discrete-Time+Signal+Processing&rft.place=Upper+Saddle+River%2C+N.J.&rft.pages=168&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=0-13-754920-2&rft.aulast=Oppenheim&rft.aufirst=Alan+V.&rft.au=Schafer%2C+Ronald+W.&rft.au=Buck%2C+John+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscretetimesign00alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-LiTan-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-LiTan_2-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTan2008" class="citation web cs1">Tan, Li (2008-04-21). <a rel="nofollow" class="external text" href="http://www.eetimes.com/document.asp?doc_id=1275556">"Upsampling and downsampling"</a>. <i>eetimes.com</i>. EE Times<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-04-10</span></span>. <q>The process of reducing a sampling rate by an integer factor is referred to as <i>downsampling</i> of a data sequence. We also refer to downsampling as <i>decimation</i>. The term <i>decimation</i> used for the downsampling process has been accepted and used in many textbooks and fields.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=eetimes.com&rft.atitle=Upsampling+and+downsampling&rft.date=2008-04-21&rft.aulast=Tan&rft.aufirst=Li&rft_id=http%3A%2F%2Fwww.eetimes.com%2Fdocument.asp%3Fdoc_id%3D1275556&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-Crochiere-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Crochiere_4-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrochiereRabiner1983" class="citation book cs1">Crochiere, R.E.; <a href="/wiki/Lawrence_Rabiner" title="Lawrence Rabiner">Rabiner, L.R.</a> (1983). "2". <a rel="nofollow" class="external text" href="https://kupdf.net/download/multirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf"><i>Multirate Digital Signal Processing</i></a>. Englewood Cliffs, NJ: Prentice-Hall. p. 32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0136051626" title="Special:BookSources/0136051626"><bdi>0136051626</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2&rft.btitle=Multirate+Digital+Signal+Processing&rft.place=Englewood+Cliffs%2C+NJ&rft.pages=32&rft.pub=Prentice-Hall&rft.date=1983&rft.isbn=0136051626&rft.aulast=Crochiere&rft.aufirst=R.E.&rft.au=Rabiner%2C+L.R.&rft_id=https%3A%2F%2Fkupdf.net%2Fdownload%2Fmultirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-Poularikas-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Poularikas_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoularikas1998" class="citation book cs1">Poularikas, Alexander D. (September 1998). <i>Handbook of Formulas and Tables for Signal Processing</i> (1 ed.). CRC Press. pp. 42–48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0849385792" title="Special:BookSources/0849385792"><bdi>0849385792</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Formulas+and+Tables+for+Signal+Processing&rft.pages=42-48&rft.edition=1&rft.pub=CRC+Press&rft.date=1998-09&rft.isbn=0849385792&rft.aulast=Poularikas&rft.aufirst=Alexander+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-f.harris-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-f.harris_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris2004" class="citation book cs1"><a href="/wiki/Fredric_J._Harris" title="Fredric J. Harris">Harris, Frederic J.</a> (2004-05-24). "2.2". <i>Multirate Signal Processing for Communication Systems</i>. Upper Saddle River, NJ: Prentice Hall PTR. pp. 20–21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0131465112" title="Special:BookSources/0131465112"><bdi>0131465112</bdi></a>. <q>The process of down sampling can be visualized as a two-step progression. The process starts as an input series x(n) that is processed by a filter h(n) to obtain the output sequence y(n) with reduced bandwidth. The sample rate of the output sequence is then reduced Q-to-1 to a rate commensurate with the reduced signal bandwidth. In reality the processes of bandwidth reduction and sample rate reduction are merged in a single process called a multirate filter.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.2&rft.btitle=Multirate+Signal+Processing+for+Communication+Systems&rft.place=Upper+Saddle+River%2C+NJ&rft.pages=20-21&rft.pub=Prentice+Hall+PTR&rft.date=2004-05-24&rft.isbn=0131465112&rft.aulast=Harris&rft.aufirst=Frederic+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-Strang-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Strang_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrangNguyen1996" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a>; Nguyen, Truong (1996-10-01). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/waveletsfilterba00stra"><i>Wavelets and Filter Banks</i></a></span> (2 ed.). Wellesley, MA: Wellesley-Cambridge Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/waveletsfilterba00stra/page/100">100</a>–101. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0961408871" title="Special:BookSources/0961408871"><bdi>0961408871</bdi></a>. <q>No sensible engineer would do that.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelets+and+Filter+Banks&rft.place=Wellesley%2C+MA&rft.pages=100-101&rft.edition=2&rft.pub=Wellesley-Cambridge+Press&rft.date=1996-10-01&rft.isbn=0961408871&rft.aulast=Strang&rft.aufirst=Gilbert&rft.au=Nguyen%2C+Truong&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fwaveletsfilterba00stra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> <li id="cite_note-Milic-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Milic_15-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilić2009" class="citation book cs1">Milić, Ljiljana (2009). <i>Multirate Filtering for Digital Signal Processing</i>. New York: Hershey. p. 192. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60566-178-0" title="Special:BookSources/978-1-60566-178-0"><bdi>978-1-60566-178-0</bdi></a>. <q>Generally, this approach is applicable when the ratio Fy/Fx is a rational, or an irrational number, and is suitable for the sampling rate increase and for the sampling rate decrease.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multirate+Filtering+for+Digital+Signal+Processing&rft.place=New+York&rft.pages=192&rft.pub=Hershey&rft.date=2009&rft.isbn=978-1-60566-178-0&rft.aulast=Mili%C4%87&rft.aufirst=Ljiljana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Downsampling_(signal_processing)&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProakis2000" class="citation book cs1">Proakis, John G. (2000). <i>Digital Signal Processing: Principles, Algorithms and Applications</i> (3rd ed.). India: Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/8120311299" title="Special:BookSources/8120311299"><bdi>8120311299</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing%3A+Principles%2C+Algorithms+and+Applications&rft.place=India&rft.edition=3rd&rft.pub=Prentice-Hall&rft.date=2000&rft.isbn=8120311299&rft.aulast=Proakis&rft.aufirst=John+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLyons2001" class="citation book cs1">Lyons, Richard (2001). <i>Understanding Digital Signal Processing</i>. Prentice Hall. p. 304. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-63467-8" title="Special:BookSources/0-201-63467-8"><bdi>0-201-63467-8</bdi></a>. <q>Decreasing the sampling rate is known as decimation.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Digital+Signal+Processing&rft.pages=304&rft.pub=Prentice+Hall&rft.date=2001&rft.isbn=0-201-63467-8&rft.aulast=Lyons&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntoniou2006" class="citation book cs1">Antoniou, Andreas (2006). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/digitalsignalpro00anto_617"><i>Digital Signal Processing</i></a></span>. McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/digitalsignalpro00anto_617/page/n855">830</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-145424-1" title="Special:BookSources/0-07-145424-1"><bdi>0-07-145424-1</bdi></a>. <q>Decimators can be used to reduce the sampling frequency, whereas interpolators can be used to increase it.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing&rft.pages=830&rft.pub=McGraw-Hill&rft.date=2006&rft.isbn=0-07-145424-1&rft.aulast=Antoniou&rft.aufirst=Andreas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdigitalsignalpro00anto_617&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilic2009" class="citation book cs1">Milic, Ljiljana (2009). <i>Multirate Filtering for Digital Signal Processing</i>. New York: Hershey. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60566-178-0" title="Special:BookSources/978-1-60566-178-0"><bdi>978-1-60566-178-0</bdi></a>. <q>Sampling rate conversion systems are used to change the sampling rate of a signal. The process of sampling rate decrease is called decimation, and the process of sampling rate increase is called interpolation.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multirate+Filtering+for+Digital+Signal+Processing&rft.place=New+York&rft.pages=35&rft.pub=Hershey&rft.date=2009&rft.isbn=978-1-60566-178-0&rft.aulast=Milic&rft.aufirst=Ljiljana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADownsampling+%28signal+processing%29" class="Z3988"></span></li> <li>T. Schilcher. RF applications in digital signal processing//" Digital signal processing". Proceedings, CERN Accelerator School, Sigtuna, Sweden, May 31-June 9, 2007. - Geneva, Switzerland: CERN (2008). - P. 258. - DOI: 10.5170/CERN-2008-003. <a rel="nofollow" class="external autonumber" href="https://cds.cern.ch/record/1100538/files/p249.pdf">[1]</a></li> <li>Sliusar I.I., Slyusar V.I., Voloshko S.V., Smolyar V.G. Next Generation Optical Access based on N-OFDM with decimation.// Third International Scientific-Practical Conference "Problems of Infocommunications. Science and Technology (PIC S&T'2016)". – Kharkiv. - October 3 –6, 2016. <a rel="nofollow" class="external autonumber" href="http://slyusar.kiev.ua/Slyusar_PIC_ST_2016.pdf">[2]</a></li> <li>Saska Lindfors, Aarno Pärssinen, Kari A. I. Halonen. A 3-V 230-MHz CMOS Decimation Subsampler.// IEEE transactions on circuits and systems— Vol. 52, No. 2, February 2005. – P. 110.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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href="/wiki/Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Detection_theory" title="Detection theory">Detection theory</a></li> <li><a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete signal</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sub-fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li> <li><a href="/wiki/Digital_image_processing" title="Digital image processing">Digital image processing</a></li> <li><a href="/wiki/Speech_processing" title="Speech processing">Speech processing</a></li> <li><a href="/wiki/Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">Statistical signal processing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> <ul><li><a href="/wiki/Advanced_z-transform" title="Advanced z-transform">Advanced z-transform</a></li> <li><a href="/wiki/Matched_Z-transform_method" title="Matched Z-transform method">Matched Z-transform method</a></li></ul></li> <li><a href="/wiki/Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li> <li><a href="/wiki/Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> (DCT)</li> <li><a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li> <li><a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li> <li><a href="/wiki/Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Post%27s_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li> <li><a href="/wiki/Starred_transform" title="Starred transform">Starred transform</a></li> <li><a href="/wiki/Zak_transform" title="Zak transform">Zak transform</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a></li> <li><a class="mw-selflink selflink">Downsampling</a></li> <li><a href="/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a> / <a href="/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></li> <li><a href="/wiki/Oversampling" title="Oversampling">Oversampling</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Sampling_rate" class="mw-redirect" title="Sampling rate">Sampling rate</a></li> <li><a href="/wiki/Undersampling" title="Undersampling">Undersampling</a></li> <li><a href="/wiki/Upsampling" title="Upsampling">Upsampling</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Downsampling_(signal_processing)&oldid=1231461663">https://en.wikipedia.org/w/index.php?title=Downsampling_(signal_processing)&oldid=1231461663</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a 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