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Comb filter
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<h2 id="applications">Applications</h2> <p>Comb filters are employed in a variety of signal processing applications, including: </p> <ul><li><a href="/facts/Cascaded_integrator%E2%80%93comb_filter/DQjgAThJ">Cascaded integrator–comb</a> (CIC) filters, commonly used for <a href="/facts/Anti-aliasing_filter/PqhNr0FQ">anti-aliasing</a> during <a href="/facts/Interpolation/8PyetX4f">interpolation</a> and <a href="/facts/Decimation_(signal_processing)/gN5Wwrhv">decimation</a> operations that change the <a href="/facts/Sample_rate/xxck7aLN">sample rate</a> of a discrete-time system.</li> <li>2D and 3D comb filters implemented in hardware (and occasionally software) in <a href="/facts/PAL/t8TugcxK">PAL</a> and <a href="/facts/NTSC/HYWIvxCR">NTSC</a> analog television decoders reduce artifacts such as <a href="/facts/Dot_crawl/wa3TN3Um">dot crawl</a>.</li> <li><a href="/facts/Audio_signal_processing/HXvBpspA">Audio signal processing</a>, including <a href="/facts/Delay_(audio_effect)/t2UR7gHI">delay</a>, <a href="/facts/Flanging/JU9Kj7WC">flanging</a>, <a href="/facts/Physical_modelling_synthesis/vkEbmbKm">physical modelling synthesis</a> and <a href="/facts/Digital_waveguide_synthesis/yYsV8FdX">digital waveguide synthesis</a>. If the delay is set to a few milliseconds, a comb filter can model the effect of <a href="/facts/Acoustics/2GpQUymy">acoustic</a> <a href="/facts/Standing_waves/n134E8TC">standing waves</a> in a cylindrical cavity or <a href="/facts/Karplus-Strong_string_synthesis/nV3kHJbA">in a vibrating string</a>.</li> <li>In astronomy the <a href="/facts/Astro-comb/SLWdzAdX">astro-comb</a> promises to increase the precision of existing <a href="/facts/Spectrograph/BjshQbA8">spectrographs</a> by nearly a hundredfold.</li></ul> <p>In <a href="/facts/Acoustics/2GpQUymy">acoustics</a>, comb filtering can arise as an unwanted artifact. For instance, two <a href="/facts/Loudspeakers/tqEjnJ2z">loudspeakers</a> playing the same signal at different distances from the listener create a comb filtering effect on the audio.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> In any enclosed space, listeners hear a mixture of direct sound and reflected sound. The reflected sound takes a longer, delayed path compared to the direct sound, and a comb filter is created where the two mix at the listener.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1 <a href="/facts/Rule_of_thumb/RopfGsJ0">rule of thumb</a> that neighboring mics should be separated at least three times the distance from its source to the mic.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="discrete-time-implementation">Discrete time implementation</h2> <h3>Feedforward form</h3> <p>The general structure of a feedforward comb filter is described by the <a href="/facts/Difference_equation/JolXC71M">difference equation</a>: </p> y [ n ] = x [ n ] + α x [ n − K ] {\displaystyle y[n]=x[n]+\alpha x[n-K]} <p>where K {\displaystyle K} is the delay length (measured in samples), and <i>α</i> is a scaling factor applied to the delayed signal. The <a href="/facts/Z_transform/ERNvDJqt"><i>z</i> transform</a> of both sides of the equation yields: </p> Y ( z ) = ( 1 + α z − K ) X ( z ) {\displaystyle Y(z)=\left(1+\alpha z^{-K}\right)X(z)} <p>The <a href="/facts/Z_transform/ERNvDJqt">transfer function</a> is defined as: </p> H ( z ) = Y ( z ) X ( z ) = 1 + α z − K = z K + α z K {\displaystyle H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}} <h4>Frequency response</h4> <p>The frequency response of a discrete-time system expressed in the <i>z</i>-domain is obtained by substitution z = e j ω , {\displaystyle z=e^{j\omega },} where j {\displaystyle j} is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> and ω {\displaystyle \omega } is <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a>. Therefore, for the feedforward comb filter: </p> H ( e j ω ) = 1 + α e − j ω K {\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}} <p>Using <a href="/facts/Euler%27s_formula/Wq7VXIV0">Euler's formula</a>, the frequency response is also given by </p> H ( e j ω ) = [ 1 + α cos ( ω K ) ] − j α sin ( ω K ) {\displaystyle H\left(e^{j\omega }\right)={\bigl [}1+\alpha \cos(\omega K){\bigr ]}-j\alpha \sin(\omega K)} <p>Often of interest is the <i>magnitude</i> response, which ignores phase. This is defined as: </p> | H ( e j ω ) | = ℜ { H ( e j ω ) } 2 + ℑ { H ( e j ω ) } 2 {\displaystyle \left|H\left(e^{j\omega }\right)\right|={\sqrt {\Re \left\{H\left(e^{j\omega }\right)\right\}^{2}+\Im \left\{H\left(e^{j\omega }\right)\right\}^{2}}}} <p>In the case of the feedforward comb filter, this is: </p> | H ( e j ω ) | = ( 1 + α cos ( ω K ) ) 2 + ( α sin ( ω K ) ) 2 = ( 1 + α 2 ) + 2 α cos ( ω K ) {\displaystyle {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega K)}}\end{aligned}}} <p>The ( 1 + α 2 ) {\displaystyle (1+\alpha ^{2})} term is constant, whereas the 2 α cos ( ω K ) {\displaystyle 2\alpha \cos(\omega K)} term varies <a href="/facts/Periodic_function/Sm8lJQsF">periodically</a>. Hence the magnitude response of the comb filter is periodic. </p><p>The graphs show the periodic magnitude response for various values of α . {\displaystyle \alpha .} Some important properties: </p> <ul><li>The response periodically drops to a <a href="/facts/Local_minimum/ADlgnyzV">local minimum</a> (sometimes known as a <i>notch</i>), and periodically rises to a <a href="/facts/Local_maximum/ADlgnyzV">local maximum</a> (sometimes known as a <i>peak</i> or a <i>tooth</i>).</li> <li>For positive values of α , {\displaystyle \alpha ,} the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter:</li></ul> f = 1 2 K , 3 2 K , 5 2 K ⋯ ω = π K , 3 π K , 5 π K ⋯ {\displaystyle {\begin{aligned}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\end{aligned}}} <ul><li>The levels of the maxima and minima are always equidistant from 1.</li> <li>When α = ± 1 , {\displaystyle \alpha =\pm 1,} the minima have zero amplitude. In this case, the minima are sometimes known as <i>nulls</i>.</li> <li>The maxima for positive values of α {\displaystyle \alpha } coincide with the minima for negative values of α {\displaystyle \alpha } , and vice versa.</li></ul> <h4>Impulse response</h4> <p>The feedforward comb filter is one of the simplest <a href="/facts/Finite_impulse_response/5u4Gaqcb">finite impulse response</a> filters.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Its response is simply the initial impulse with a second impulse after the delay. </p> <h4>Pole–zero interpretation</h4> <p>Looking again at the <i>z</i>-domain transfer function of the feedforward comb filter: </p> H ( z ) = z K + α z K {\displaystyle H(z)={\frac {z^{K}+\alpha }{z^{K}}}} <p>the numerator is equal to zero whenever <i>zK</i> = −<i>α</i>. This has <i>K</i> solutions, equally spaced around a circle in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>; these are the <a href="/facts/Zero_(complex_analysis)/gnA3U8VP">zeros</a> of the transfer function. The denominator is zero at <i>zK</i> = 0, giving <i>K</i> <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> at <i>z</i> = 0. This leads to a <a href="/facts/Pole%E2%80%93zero_plot/msm3M41X">pole–zero plot</a> like the ones shown. </p> <table><tbody><tr><td></td><td></td></tr></tbody></table> <h3>Feedback form</h3> <p>Similarly, the general structure of a feedback comb filter is described by the <a href="/facts/Difference_equation/JolXC71M">difference equation</a>: </p> y [ n ] = x [ n ] + α y [ n − K ] {\displaystyle y[n]=x[n]+\alpha y[n-K]} <p>This equation can be rearranged so that all terms in y {\displaystyle y} are on the left-hand side, and then taking the <i>z</i> transform: </p> ( 1 − α z − K ) Y ( z ) = X ( z ) {\displaystyle \left(1-\alpha z^{-K}\right)Y(z)=X(z)} <p>The transfer function is therefore: </p> H ( z ) = Y ( z ) X ( z ) = 1 1 − α z − K = z K z K − α {\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}} <h4>Frequency response</h4> <p>By substituting z = e j ω {\displaystyle z=e^{j\omega }} into the feedback comb filter's <i>z</i>-domain expression: </p> H ( e j ω ) = 1 1 − α e − j ω K , {\displaystyle H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,} <p>the magnitude response becomes: </p> | H ( e j ω ) | = 1 ( 1 + α 2 ) − 2 α cos ( ω K ) . {\displaystyle \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.} <p>Again, the response is periodic, as the graphs demonstrate. The feedback comb filter has some properties in common with the feedforward form: </p> <ul><li>The response periodically drops to a local minimum and rises to a local maximum.</li> <li>The maxima for positive values of α {\displaystyle \alpha } coincide with the minima for negative values of α , {\displaystyle \alpha ,} and vice versa.</li> <li>For positive values of α , {\displaystyle \alpha ,} the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:</li></ul> f = 0 , 1 K , 2 K , 3 K ⋯ ω = 0 , 2 π K , 4 π K , 6 π K ⋯ {\displaystyle {\begin{aligned}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \end{aligned}}} <p>However, there are also some important differences because the magnitude response has a term in the <a href="/facts/Denominator/ghnQE9B3">denominator</a>: </p> <ul><li>The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of 1/1 − <i>α</i>.</li> <li>The filter is only <a href="/facts/BIBO_stability/RxD8h2f5">stable</a> if |<i>α</i>| is strictly less than 1. As can be seen from the graphs, as |<i>α</i>| increases, the amplitude of the maxima rises increasingly rapidly.</li></ul> <h4>Impulse response</h4> <p>The feedback comb filter is a simple type of <a href="/facts/Infinite_impulse_response/0KLkxylJ">infinite impulse response</a> filter.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time. </p> <h4>Pole–zero interpretation</h4> <p>Looking again at the <i>z</i>-domain transfer function of the feedback comb filter: </p> H ( z ) = z K z K − α {\displaystyle H(z)={\frac {z^{K}}{z^{K}-\alpha }}} <p>This time, the numerator is zero at <i>zK</i> = 0, giving <i>K</i> zeros at <i>z</i> = 0. The denominator is equal to zero whenever <i>zK</i> = <i>α</i>. This has <i>K</i> solutions, equally spaced around a circle in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>; these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below. </p> <table><tbody><tr><td></td><td></td></tr></tbody></table> <h2 id="continuous-time-implementation">Continuous time implementation</h2> <p>Comb filters may also be implemented in <a href="/facts/Continuous_time/RR8wpS9e">continuous time</a> which can be expressed in the <a href="/facts/Laplace_domain/0Ge4MIc9">Laplace domain</a> as a function of the <a href="/facts/Complex_number/2w7mqI7e">complex</a> frequency domain parameter s = σ + j ω {\displaystyle s=\sigma +j\omega } analogous to the z domain. <a href="/facts/Analog_circuits/Bnlnfwwv">Analog circuits</a> use some form of <a href="/facts/Analog_delay_line/Lg6MQ47b">analog delay line</a> for the delay element. Continuous-time implementations share all the properties of the respective discrete-time implementations. </p> <h3>Feedforward form</h3> <p>The feedforward form may be described by the equation: </p> y ( t ) = x ( t ) + α x ( t − τ ) {\displaystyle y(t)=x(t)+\alpha x(t-\tau )} <p>where <i>τ</i> is the delay (measured in seconds). This has the following transfer function: </p> H ( s ) = 1 + α e − s τ {\displaystyle H(s)=1+\alpha e^{-s\tau }} <p>The feedforward form consists of an infinite number of zeros spaced along the jω axis (<a href="/facts/Laplace_transform/0Ge4MIc9">which corresponds to the Fourier domain</a>). </p> <h3>Feedback form</h3> <p>The feedback form has the equation: </p> y ( t ) = x ( t ) + α y ( t − τ ) {\displaystyle y(t)=x(t)+\alpha y(t-\tau )} <p>and the following transfer function: </p> H ( s ) = 1 1 − α e − s τ {\displaystyle H(s)={\frac {1}{1-\alpha e^{-s\tau }}}} <p>The feedback form consists of an infinite number of poles spaced along the jω axis. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dirac_comb/0tx3xslS">Dirac comb</a></li> <li><a href="/facts/Fabry%E2%80%93P%C3%A9rot_interferometer/n949LwQD">Fabry–Pérot interferometer</a></li></ul> <h2 id="external-links">External links</h2> <ul><li> Media related to Comb filters at Wikimedia Commons</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Roger Russell. "Hearing, Columns and Comb Filtering". Retrieved 2010-04-22. <a href="http://www.roger-russell.com/columns/combfilter2.htm" target="_blank">http://www.roger-russell.com/columns/combfilter2.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Acoustic Basics". Acoustic Sciences Corporation. Archived from the original on 2010-05-07. <a href="https://web.archive.org/web/20100507124237/http://www.asc-hifi.com/acoustic_basics.htm" target="_blank">https://web.archive.org/web/20100507124237/http://www.asc-hifi.com/acoustic_basics.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Burroughs, Lou (1974). Microphones: Design and Application. Sagamore Publishing Company. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Smith, J. O. "Feedforward Comb Filters". Archived from the original on 2011-06-06. <a href="https://web.archive.org/web/20110606210608/https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html" target="_blank">https://web.archive.org/web/20110606210608/https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Smith, J.O. "Feedback Comb Filters". Archived from the original on 2011-06-06. <a href="https://web.archive.org/web/20110606203008/https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html" target="_blank">https://web.archive.org/web/20110606203008/https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>