Editing Comb filter

{{Short description|Signal processing filter}}

[[Image:Comb filter feedforward.svg|thumb|right|400px|Feedforward comb filter structure]]

In [[signal processing]], a '''comb filter''' is a [[Filter (signal processing)|filter]] implemented by adding a delayed version of a [[signal processing|signal]] to itself, causing constructive and destructive [[Interference (wave propagation)|interference]]. The [[frequency response]] of a comb filter consists of a series of regularly spaced notches in between regularly spaced ''peaks'' (sometimes called ''teeth'') giving the appearance of a [[comb]].

Comb filters exist in two forms, ''feedforward'' and ''[[feedback]]''; which refer to the direction in which signals are delayed before they are added to the input.

Comb filters may be implemented in [[discrete time]] or [[continuous time]] forms which are very similar.

==Applications==
[[File:Sony VPL-HS1 - EP-GW 1-682-352-12 - Motorola MC141627FT-1.jpg|thumb|Advanced PAL Comb Filter-II (APCF-II, Motorola MC141627FT)]]
Comb filters are employed in a variety of signal processing applications, including:

* [[Cascaded integrator–comb filter|Cascaded integrator–comb]] (CIC) filters, commonly used for [[anti-aliasing filter|anti-aliasing]] during [[interpolation]] and [[decimation (signal processing)|decimation]] operations that change the [[sample rate]] of a discrete-time system.
* 2D and 3D comb filters implemented in hardware (and occasionally software) in [[PAL]] and [[NTSC]] analog television decoders reduce artifacts such as [[dot crawl]].
* [[Audio signal processing]], including [[Delay (audio effect)|delay]], [[flanging]], [[physical modelling synthesis]] and [[digital waveguide synthesis]]. If the delay is set to a few milliseconds, a comb filter can model the effect of [[Acoustics|acoustic]] [[standing waves]] in a cylindrical cavity or [[Karplus-Strong string synthesis|in a vibrating string]].
* In astronomy the [[astro-comb]] promises to increase the precision of existing [[spectrograph]]s by nearly a hundredfold.

In [[acoustics]], comb filtering can arise as an unwanted artifact. For instance, two [[loudspeakers]] playing the same signal at different distances from the listener create a comb filtering effect on the audio.<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. 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.<ref>{{cite web|url=http://www.asc-hifi.com/acoustic_basics.htm#2 |title=Acoustic Basics |publisher=Acoustic Sciences Corporation|archivedate=2010-05-07 |archiveurl=https://web.archive.org/web/20100507124237/http://www.asc-hifi.com/acoustic_basics.htm |url-status=dead }}</ref> Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1 [[rule of thumb]] that neighboring mics should be separated at least three times the distance from its source to the mic.<ref>{{Cite book |last=Burroughs |first=Lou |title=Microphones: Design and Application |publisher=Sagamore Publishing Company |year=1974 |language=English}}</ref>

==Discrete time implementation==
===Feedforward form===
[[Image:Comb filter feedforward.svg|thumb|right|400px|Feedforward comb filter structure in discrete time]]

The general structure of a feedforward comb filter is described by the [[difference equation]]: 
:<math>y[n] = x[n] + \alpha x[n-K] </math>

where <math>K</math> is the delay length (measured in samples), and {{math|''α''}} is a scaling factor applied to the delayed signal. The [[Z transform|{{math|''z''}} transform]] of both sides of the equation yields:
:<math>Y(z) = \left(1 + \alpha z^{-K}\right) X(z) </math>

The [[Z transform#Transfer function|transfer function]] is defined as:
:<math>H(z) = \frac{Y(z)}{X(z)} = 1 + \alpha z^{-K} = \frac{z^K + \alpha}{z^K} </math>

====Frequency response====
[[Image:Comb filter response ff pos.svg|thumb|Feedforward magnitude response for various ''positive'' values of {{math|''α''}} and {{math|''K'' {{=}} 1}} in discrete time]]
[[Image:Comb filter response ff neg.svg|thumb|Feedforward magnitude response for various ''negative'' values of {{math|''α''}} and {{math|''K'' {{=}} 1}} in discrete time]]

The frequency response of a discrete-time system expressed in the {{math|''z''}}-domain is obtained by substitution <math>z = e^{j\omega},</math> where <math>j </math> is the [[imaginary unit]] and <math>\omega </math> is [[angular frequency]]. Therefore, for the feedforward comb filter:
:<math>H\left(e^{j \omega}\right) = 1 + \alpha e^{-j \omega K} </math>

Using [[Euler's formula]], the frequency response is also given by
:<math>H\left(e^{j \omega}\right) = \bigl[1 + \alpha \cos(\omega K)\bigr] - j \alpha \sin(\omega K) </math>

Often of interest is the ''magnitude'' response, which ignores phase. This is defined as:
:<math>\left| H\left(e^{j \omega}\right) \right| = \sqrt{\text{Re}\left\{H\left(e^{j \omega}\right)\right\}^2 + \text{Im}\left\{H\left(e^{j \omega}\right)\right\}^2} 
</math>

In the case of the feedforward comb filter, this is:
:<math>\begin{align}
\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{align} </math>

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.

The graphs show the periodic magnitude response for various values of <math>\alpha .</math> Some important properties:
*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'' or a ''tooth'').
*For positive values of <math>\alpha,</math> the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter: 
::<math display="block">\begin{align}
f &= \frac{1}{2 K}, \frac{3}{2 K}, \frac{5}{2 K} \cdots \\
\omega &= \frac{\pi}{K}, \frac{3\pi}{K}, \frac{5\pi}{K} \cdots \,

\end{align}</math>
*The levels of the maxima and minima are always equidistant from 1.
*When <math>\alpha = \pm 1 , </math> the minima have zero amplitude. In this case, the minima are sometimes known as ''nulls''.
*The maxima for positive values of <math>\alpha </math> coincide with the minima for negative values of <math>\alpha</math>, and vice versa.

====Impulse response====
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=2011-06-06 |archiveurl=https://web.archive.org/web/20110606210608/https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html |url-status=dead }}</ref> Its response is simply the initial impulse with a second impulse after the delay.

====Pole–zero interpretation====
Looking again at the {{math|''z''}}-domain transfer function of the feedforward comb filter:
:<math>H(z) = \frac{z^K + \alpha}{z^K} </math>

the numerator is equal to zero whenever {{math|''z<sup>K</sup>'' {{=}} −''α''}}. This has {{math|''K''}} 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<sup>K</sup>'' {{=}} 0}}, giving {{math|''K''}} [[pole (complex analysis)|poles]] at {{math|''z'' {{=}} 0}}. This leads to a [[pole–zero plot]] like the ones shown.

{|
| [[Image:Comb filter pz ff pos.svg|left|thumb|200px|Pole–zero plot of feedforward comb filter with {{math|''K'' {{=}} 8}} and {{math|''α'' {{=}} 0.5}} in discrete time]]
| [[Image:Comb filter pz ff neg.svg|left|thumb|200px|Pole–zero plot of feedforward comb filter with {{math|''K'' {{=}} 8}} and {{math|''α'' {{=}} −0.5}} in discrete time]]
|}

===Feedback form===
[[Image:Comb filter feedback.svg|thumb|right|400px|Feedback comb filter structure in discrete time]]

Similarly, the general structure of a feedback comb filter is  described by the [[difference equation]]:
:<math>y[n] = x[n] + \alpha y[n-K] </math>

This equation can be rearranged so that all terms in <math>y</math> are on the left-hand side, and then taking the {{math|''z''}} transform:
:<math>\left(1 - \alpha z^{-K}\right) Y(z) = X(z) </math>

The transfer function is therefore:
:<math>H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 - \alpha z^{-K}} = \frac{z^K}{z^K - \alpha} </math>

====Frequency response====
[[Image:Comb filter response fb pos.svg|thumb|right|400px|Feedback magnitude response for various ''positive'' values of {{math|''α''}} and {{math|''K'' {{=}} 2}} in discrete time]]
[[Image:Comb filter response fb neg.svg|thumb|right|400px|Feedback magnitude response for various ''negative'' values of {{math|''α''}} and {{math|''K'' {{=}} 2}} in discrete time]]

By substituting <math>z = e^{j\omega}</math> into the feedback comb filter's {{math|''z''}}-domain expression:
:<math>H\left(e^{j \omega}\right) = \frac{1}{1 - \alpha e^{-j \omega K}} \, , </math>

the magnitude response becomes:
:<math>\left| H\left(e^{j \omega}\right) \right| = \frac{1}{\sqrt{\left(1 + \alpha^2\right) - 2 \alpha \cos(\omega K)}} \, . </math>

Again, the response is periodic, as the graphs demonstrate. The feedback comb filter has some properties in common with the feedforward form:
*The response periodically drops to a local minimum and rises to a local maximum.
*The maxima for positive values of <math>\alpha</math> coincide with the minima for negative values of <math>\alpha,</math> and vice versa.
*For positive values of <math>\alpha,</math> the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:
::<math>\begin{align}
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{align}</math>

However, there are also some important differences because the magnitude response has a term in the [[denominator]]:
*The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of {{math|{{sfrac|1|1 − ''α''}}}}.
*The filter is only [[BIBO stability|stable]] if {{math|{{abs|''α''}}}} is strictly less than 1. As can be seen from the graphs, as {{math|{{abs|''α''}}}} increases, the amplitude of the maxima rises increasingly rapidly.

====Impulse response====
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=2011-06-06 |archiveurl=https://web.archive.org/web/20110606203008/https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html |url-status=dead }}</ref> If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.

====Pole–zero interpretation====
Looking again at the {{math|''z''}}-domain transfer function of the feedback comb filter:
:<math>H(z) = \frac{z^K}{z^K - \alpha} </math>

This time, the numerator is zero at {{math|''z<sup>K</sup>'' {{=}} 0}}, giving {{math|''K''}} zeros at {{math|''z'' {{=}} 0}}. The denominator is equal to zero whenever {{math|''z<sup>K</sup>'' {{=}} ''α''}}. This has {{math|''K''}} 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.

{|
| [[Image:Comb filter pz fb pos.svg|left|thumb|Pole–zero plot of feedback comb filter with {{math|''K'' {{=}} 8}} and {{math|''α'' {{=}} 0.5}} in discrete time]]
| [[Image:Comb filter pz fb neg.svg|left|thumb|Pole–zero plot of feedback comb filter with {{math|''K'' {{=}} 8}} and {{math|''α'' {{=}} −0.5}} in discrete time]]
|}

== Continuous time implementation ==
Comb filters may also be implemented in [[continuous time]] which can be expressed in the [[Laplace domain]] as a function of the [[Complex number|complex]] frequency domain parameter <math>s = \sigma + j \omega</math> analogous to the z domain. [[Analog circuits]] use some form of [[analog delay line]] for the delay element. Continuous-time implementations share all the properties of the respective discrete-time implementations.

=== Feedforward form ===
The feedforward form may be described by the equation:

:<math>y(t) = x(t) + \alpha x(t - \tau) </math>

where {{math|''τ''}} is the delay (measured in seconds). This has the following transfer function:

:<math>H(s) = 1 + \alpha e^{-s \tau} </math>

The feedforward form consists of an infinite number of zeros spaced along the jω axis ([[Laplace transform#Fourier transform|which corresponds to the Fourier domain]]).

=== Feedback form ===
The feedback form has the equation:

:<math>y(t) = x(t) + \alpha y(t - \tau) </math>

and the following transfer function:

:<math>H(s) = \frac{1}{1 - \alpha e^{-s \tau}} </math>

The feedback form consists of an infinite number of poles spaced along the jω axis.

==See also==
*[[Dirac comb]]
*[[Fabry–Pérot interferometer]]

== References==
{{Reflist}}

==External links==
*{{Commons category-inline}}

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{{DEFAULTSORT:Comb Filter}}
[[Category:Signal processing]]
[[Category:Filter theory]]