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| In [[signal processing]], a '''comb filter''' adds a delayed version of a [[signal processing|signal]] to itself, causing [[Destructive interference#Constructive and destructive interference|constructive and destructive interference]]. The [[frequency response]] of a comb filter consists of a series of regularly spaced spikes, giving the appearance of a [[comb]].
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| ==Applications==
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| Comb filters are used in a variety of signal processing applications. These include:
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| *[[Cascaded Integrator-Comb Filter|Cascaded Integrator-Comb]] (CIC) filters, commonly used for [[anti-aliasing]]{{dn|date=September 2012}} during [[interpolation]] and [[decimation (signal processing)|decimation]] operations that change the [[sample rate]] of a discrete-time system.
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| *2D and 3D comb filters implemented in hardware (and occasionally software) for [[PAL]] and [[NTSC]] television decoders. The filters work to reduce artifacts such as [[dot crawl]].
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| *[[Audio effect]]s, including [[Echo (phenomenon)|echo]], [[flanging]], and [[digital waveguide synthesis]]. For instance, if the delay is set to a few milliseconds, a comb filter can be used to model the effect of [[Acoustics|acoustic]] [[standing waves]] in a cylindrical cavity or [[Karplus-Strong string synthesis|in a vibrating string]].
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| *In astronomy the [[astro-comb]] promises to increase the precision of existing [[spectrograph]]s by nearly a hundredfold.
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| In [[acoustics]], comb filtering can arise in some unwanted ways. For instance, when two [[loudspeakers]] are playing the same signal at different distances from the listener, there is a comb filtering effect on the signal.<ref>{{cite web |url=http://www.roger-russell.com/columns/combfilter2.htm |title=Hearing, Columns and Comb Filtering |author=Roger Russell |accessdate=2010-04-22}}</ref> In any enclosed space, listeners hear a mixture of direct sound and reflected sound. Because the reflected sound takes a longer path, it constitutes a delayed version of the direct sound and a comb filter is created where the two combine at the listener.<ref>{{cite web |url=http://www.asc-hifi.com/acoustic_basics.htm#2 |title=Acoustic Basics |publisher=Acoustic Sciences Corportation |archivedate=2010-04-22 |archiveurl=http://www.webcitation.org/5pBNPaijA}}</ref>
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| ==Technical discussion==
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| Comb filters exist in two different forms, ''feedforward'' and ''[[feedback]]''; the names refer to the direction in which signals are delayed before they are added to the input.
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| Comb filters may be implemented in [[discrete time]] or [[continuous time]]; this article will focus on discrete-time implementations; the properties of the continuous-time comb filter are very similar.
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| ===Feedforward form===
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| [[Image:Comb filter feedforward.svg|thumb|right|400px|Feedforward comb filter structure]]
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| The general structure of a feedforward comb filter is shown on the right. It may be described by the following [[difference equation]]:
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| :<math>\ y[n] = x[n] + \alpha x[n-K] \,
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| </math>
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| where <math>K</math> is the delay length (measured in samples), and <math>\alpha</math> is a scaling factor applied to the delayed signal. If we take the [[Z transform]] of both sides of the equation, we obtain:
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| :<math>
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| \ Y(z) = (1 + \alpha z^{-K}) X(z) \,
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| </math>
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| We define the [[Z transform#Transfer function|transfer function]] as:
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| :<math>
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| \ H(z) = \frac{Y(z)}{X(z)} = 1 + \alpha z^{-K} = \frac{z^K + \alpha}{z^K} \,
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| </math>
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| ====Frequency response====
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| [[Image:Comb filter response ff pos.svg|thumb|right|400px|Feedforward magnitude response for various ''positive'' values of <math>\alpha</math>]]
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| [[Image:Comb filter response ff neg.svg|thumb|right|400px|Feedforward magnitude response for various ''negative'' values of <math>\alpha</math>]]
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| To obtain the frequency response of a discrete-time system expressed in the Z domain, we make the substitution <math>z = e^{j \omega}</math>. Therefore, for our feedforward comb filter, we get:
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| :<math>
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| \ H(e^{j \omega}) = 1 + \alpha e^{-j \omega K} \,
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| </math>
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| Using [[Euler's formula]], we find that the frequency response is also given by
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| :<math>
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| \ H(e^{j \omega}) = \left[1 + \alpha \cos(\omega K)\right] - j \alpha \sin(\omega K) \,
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| </math>
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| Often of interest is the ''magnitude'' response, which ignores phase. This is defined as:
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| :<math>
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| \ | H(e^{j \omega}) | = \sqrt{\Re\{H(e^{j \omega})\}^2 + \Im\{H(e^{j \omega})\}^2} \,
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| </math>
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| In the case of the feedforward comb filter, this is:
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| :<math>
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| \ | H(e^{j \omega}) | = \sqrt{(1 + \alpha^2) + 2 \alpha \cos(\omega K)} \,
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| </math>
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| Notice that the <math>(1 + \alpha^2)</math> term is constant, whereas the <math>2 \alpha \cos(\omega K)</math> term varies [[periodic function|periodically]]. Hence the magnitude response of the comb filter is periodic.
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| The graphs to the right show the magnitude response for various values of <math>\alpha</math>, demonstrating this periodicity. Some important properties:
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| *The response periodically drops to a [[local minimum]] (sometimes known as a ''notch''), and periodically rises to a [[local maximum]] (sometimes known as a ''peak'').
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| *For positive values of <math>\alpha</math>, the first minimum occurs at half the delay period and repeat at even multiples of the delay frequency thereafter: <math>f = \frac{1}{2 K}, \frac{3}{2 K}, \frac{5}{2 K} ...</math>.
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| *The levels of the maxima and minima are always equidistant from 1.
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| *When <math>\alpha = \pm 1</math>, the minima have zero amplitude. In this case, the minima are sometimes known as ''nulls''.
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| *The maxima for positive values of <math>\alpha</math> coincide with the minima for negative values of <math>\alpha</math>, and vice versa.
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| ====Impulse response====
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| The feedforward comb filter is one of the simplest [[finite impulse response]] filters.<ref>{{cite web |url=https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html |first=J. O. |last=Smith |title=Feedforward Comb Filters |archivedate=2010-04-22 |archiveurl=http://www.webcitation.org/5pBO1R7MI}}</ref> Its response is simply the initial impulse with a second impulse after the delay.
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| ====Pole-zero interpretation====
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| Looking again at the Z-domain transfer function of the feedforward comb filter:
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| :<math>
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| \ H(z) = \frac{z^K + \alpha}{z^K} \,
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| </math>
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| we see that the numerator is equal to zero whenever <math>z^K = -\alpha</math>. This has <math>K</math> solutions, equally spaced around a circle in the [[complex plane]]; these are the [[zero (complex analysis)|zeros]] of the transfer function. The denominator is zero at <math>z^K = 0</math>, giving <math>K</math> [[pole (complex analysis)|poles]] at <math>z = 0</math>. This leads to a [[pole-zero plot]] like the ones shown below.
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| {|
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| | [[Image:Comb filter pz ff pos.svg|left|thumb|200px|Pole-zero plot of feedfoward comb filter with <math>K = 8</math> and <math>\alpha = 0.5</math>]]
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| | [[Image:Comb filter pz ff neg.svg|left|thumb|200px|Pole-zero plot of feedfoward comb filter with <math>K = 8</math> and <math>\alpha = -0.5</math>]]
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| |}
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| ===Feedback form===
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| [[Image:Comb filter feedback.svg|thumb|right|400px|Feedback comb filter structure]]
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| Similarly, the general structure of a feedback comb filter is shown on the right. It may be described by the following [[difference equation]]:
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| :<math>
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| \ y[n] = x[n] + \alpha y[n-K] \,
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| </math>
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| If we rearrange this equation so that all terms in <math>y</math> are on the left-hand side, and then take the Z transform, we obtain:
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| :<math>
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| \ (1 - \alpha z^{-K}) Y(z) = X(z) \,
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| </math>
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| The transfer function is therefore:
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| :<math>
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| \ H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 - \alpha z^{-K}} = \frac{z^K}{z^K - \alpha} \,
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| </math>
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| ====Frequency response====
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| [[Image:Comb filter response fb pos.svg|thumb|right|400px|Feedback magnitude response for various ''positive'' values of <math>\alpha</math>]]
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| [[Image:Comb filter response fb neg.svg|thumb|right|400px|Feedback magnitude response for various ''negative'' values of <math>\alpha</math>]]
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| If we make the substitution <math>z = e^{j \omega}</math> into the Z-domain expression for the feedback comb filter, we get:
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| :<math>
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| \ H(e^{j \omega}) = \frac{1}{1 - \alpha e^{-j \omega K}} \,
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| </math>
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| The magnitude response is as follows:
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| :<math>
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| \ | H(e^{j \omega}) | = \frac{1}{\sqrt{(1 + \alpha^2) - 2 \alpha \cos(\omega K)}} \,
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| </math>
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| Again, the response is periodic, as the graphs to the right demonstrate. The feedback comb filter has some properties in common with the feedforward form:
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| *The response periodically drops to a local minimum and rises to a local maximum.
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| *The maxima for positive values of <math>\alpha</math> coincide with the minima for negative values of <math>\alpha</math>, and vice versa.
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| *For positive values of <math>\alpha</math>, the first minimum occurs at 0 and repeats at even multiples of the delay frequency thereafter: <math>f = 0, \frac{1}{K}, \frac{2}{K} ...</math>.
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| However, there are also some important differences because the magnitude response has a term in the [[denominator]]:
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| *The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of <math>1 \over 1 - \alpha</math>.
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| *The filter is only [[BIBO stability|stable]] if <math>|\alpha|</math> is strictly less than 1. As can be seen from the graphs, as <math>|\alpha|</math> increases, the amplitude of the maxima rises increasingly rapidly.
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| ====Impulse response====
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| The feedback comb filter is a simple type of [[infinite impulse response]] filter.<ref>{{cite web |url=https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html |first=J.O. |last=Smith |title=Feedback Comb Filters |archivedate=2010-04-22 |archiveurl=http://www.webcitation.org/5pBO6Nubb}}</ref> If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.
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| ====Pole-zero interpretation====
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| Looking again at the Z-domain transfer function of the feedback comb filter:
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| :<math>
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| \ H(z) = \frac{z^K}{z^K - \alpha} \,
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| </math>
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| This time, the numerator is zero at <math>z^K = 0</math>, giving <math>K</math> zeros at <math>z = 0</math>. The denominator is equal to zero whenever <math>z^K = \alpha</math>. This has <math>K</math> solutions, equally spaced around a circle in the [[complex plane]]; these are the poles of the transfer function. This leads to a pole-zero plot like the ones shown below.
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| {|
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| | [[Image:Comb filter pz fb pos.svg|left|thumb|200px|Pole-zero plot of feedback comb filter with <math>K = 8</math> and <math>\alpha = 0.5</math>]]
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| | [[Image:Comb filter pz fb neg.svg|left|thumb|200px|Pole-zero plot of feedback comb filter with <math>K = 8</math> and <math>\alpha = -0.5</math>]]
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| |}
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| ===Continuous-time comb filters===
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| Comb filters may also be implemented in [[continuous time]]. The feedforward form may be described by the following equation:
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| :<math>
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| \ y(t) = x(t) + \alpha x(t - \tau) \,
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| </math>
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| and the feedback form by:
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| :<math>
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| \ y(t) = x(t) + \alpha y(t - \tau) \,
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| </math>
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| where <math>\tau</math> is the delay (measured in seconds).
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| They have the following frequency responses, respectively:
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| :<math>
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| \ H(\omega) = 1 + \alpha e^{-j \omega \tau} \,
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| </math>
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| :<math>
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| \ H(\omega) = \frac{1}{1 - \alpha e^{-j \omega \tau}} \,
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| </math>
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| Continuous-time implementations share all the properties of the respective discrete-time implementations.
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| ==See also==
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| *[[Filter (signal processing)]]
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| *[[Digital filter]]
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| * [[Fabry-Pérot interferometer]]
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| == References==
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| {{reflist}}
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| {{DEFAULTSORT:Comb Filter}}
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| [[Category:Signal processing]]
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| [[Category:Filter theory]]
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