Comb filter

Insignal processing,acomb filteris afilterimplemented by adding a delayed version of asignalto itself, causing constructive and destructiveinterference.Thefrequency responseof a comb filter consists of a series of regularly spaced notches in between regularly spacedpeaks(sometimes calledteeth) giving the appearance of acomb.

Comb filters exist in two forms,feedforwardandfeedback;which refer to the direction in which signals are delayed before they are added to the input.

Comb filters may be implemented indiscrete timeorcontinuous timeforms which are very similar.

Applications

Comb filters are employed in a variety of signal processing applications, including:

Cascaded integrator–comb(CIC) filters, commonly used foranti-aliasingduringinterpolationanddecimationoperations that change thesample rateof a discrete-time system.
2D and 3D comb filters implemented in hardware (and occasionally software) inPALandNTSCanalog television decoders, reduce artifacts such asdot crawl.
Audio signal processing,includingdelay,flanging,physical modelling synthesisanddigital waveguide synthesis.If the delay is set to a few milliseconds, a comb filter can model the effect ofacoustic standing wavesin a cylindrical cavity orin a vibrating string.
In astronomy theastro-combpromises to increase the precision of existingspectrographsby nearly a hundredfold.

Inacoustics,comb filtering can arise as an unwanted artifact. For instance, twoloudspeakersplaying the same signal at different distances from the listener, create a comb filtering effect on the audio.^[1]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.^[2]Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1rule of thumbthat neighboring mics should be separated at least three times the distance from its source to the mic.^{[citation needed]}

Discrete time implementation

Feedforward form

The general structure of a feedforward comb filter is described by thedifference equation:

y[n]=x[n]+\alpha x[n-K]

where $K$ is the delay length (measured in samples), and $α$ is a scaling factor applied to the delayed signal. The $z$ transformof both sides of the equation yields:

Y(z)=\left(1+\alpha z^{-K}\right)X(z)

Thetransfer functionis defined as:

H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}

Frequency response

The frequency response of a discrete-time system expressed in the $z$ -domain is obtained by substitution $z=e^{j\omega },$ where $j$ is theimaginary unitand $\omega$ isangular frequency.Therefore, for the feedforward comb filter:

H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}

UsingEuler's formula,the frequency response is also given by

H\left(e^{j\omega }\right)={\bigl [}1+\alpha \cos(\omega K){\bigr ]}-j\alpha \sin(\omega K)

Often of interest is themagnituderesponse, which ignores phase. This is defined as:

\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}}}

In the case of the feedforward comb filter, this is:

{\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}}

The $(1+\alpha ^{2})$ term is constant, whereas the $2\alpha \cos(\omega K)$ term variesperiodically.Hence the magnitude response of the comb filter is periodic.

The graphs show the periodic magnitude response for various values of $\alpha.$ Some important properties:

The response periodically drops to alocal minimum(sometimes known as anotch), and periodically rises to alocal maximum(sometimes known as apeakor atooth).
For positive values of $\alpha,$ the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter:

{\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}}

The levels of the maxima and minima are always equidistant from 1.
When $\alpha =\pm 1,$ the minima have zero amplitude. In this case, the minima are sometimes known asnulls.
The maxima for positive values of $\alpha$ coincide with the minima for negative values of $\alpha$ ,and vice versa.

Impulse response

The feedforward comb filter is one of the simplestfinite impulse responsefilters.^[3]Its response is simply the initial impulse with a second impulse after the delay.

Pole–zero interpretation

Looking again at the $z$ -domain transfer function of the feedforward comb filter:

H(z)={\frac {z^{K}+\alpha }{z^{K}}}

the numerator is equal to zero whenever $z K = - α$ .This has $K$ solutions, equally spaced around a circle in thecomplex plane;these are thezerosof the transfer function. The denominator is zero at $z K = 0$ ,giving $K$ polesat $z = 0$ .This leads to apole–zero plotlike the ones shown.

Feedback form

Similarly, the general structure of a feedback comb filter is described by thedifference equation:

y[n]=x[n]+\alpha y[n-K]

This equation can be rearranged so that all terms in $y$ are on the left-hand side, and then taking the $z$ transform:

\left(1-\alpha z^{-K}\right)Y(z)=X(z)

The transfer function is therefore:

H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}

Frequency response

By substituting $z=e^{j\omega }$ into the feedback comb filter's $z$ -domain expression:

H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,

the magnitude response becomes:

\left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.

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 $\alpha$ coincide with the minima for negative values of $\alpha,$ and vice versa.
For positive values of $\alpha,$ the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:

{\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}}

However, there are also some important differences because the magnitude response has a term in thedenominator:

The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/1 −α⁠$ .
The filter is onlystableif $| α |$ is strictly less than 1. As can be seen from the graphs, as $| α |$ increases, the amplitude of the maxima rises increasingly rapidly.

Impulse response

The feedback comb filter is a simple type ofinfinite impulse responsefilter.^[4]If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.

Pole–zero interpretation

Looking again at the $z$ -domain transfer function of the feedback comb filter:

H(z)={\frac {z^{K}}{z^{K}-\alpha }}

This time, the numerator is zero at $z K = 0$ ,giving $K$ zeros at $z = 0$ .The denominator is equal to zero whenever $z K = α$ .This has $K$ solutions, equally spaced around a circle in thecomplex plane;these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below.

Continuous time implementation

Comb filters may also be implemented incontinuous timewhich can be expressed in theLaplace domainas a function of thecomplexfrequency domain parameter $s=\sigma +j\omega$ analogous to the z domain.Analog circuitsuse some form ofanalog delay linefor 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:

y(t)=x(t)+\alpha x(t-\tau )

where $τ$ is the delay (measured in seconds). This has the following transfer function:

H(s)=1+\alpha e^{-s\tau }

The feedforward form consists of an infinite number of zeros spaced along the jω axis (which corresponds to the Fourier domain).

Feedback form

The feedback form has the equation:

y(t)=x(t)+\alpha y(t-\tau )

and the following transfer function:

H(s)={\frac {1}{1-\alpha e^{-s\tau }}}

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

References

^Roger Russell."Hearing, Columns and Comb Filtering".Retrieved2010-04-22.
^"Acoustic Basics".Acoustic Sciences Corporation. Archived fromthe originalon 2010-05-07.
^Smith, J. O."Feedforward Comb Filters".Archived fromthe originalon 2011-06-06.
^Smith, J.O."Feedback Comb Filters".Archived fromthe originalon 2011-06-06.

External links

Media related toComb filtersat Wikimedia Commons

[1] Roger Russell."Hearing, Columns and Comb Filtering".Retrieved2010-04-22.

[2] "Acoustic Basics".Acoustic Sciences Corporation. Archived fromthe originalon 2010-05-07.

[3] Smith, J. O."Feedforward Comb Filters".Archived fromthe originalon 2011-06-06.

[4] Smith, J.O."Feedback Comb Filters".Archived fromthe originalon 2011-06-06.

[1]

[2]

[3]

[4]

Applications

Discrete time implementation

Feedforward form

Frequency response

Impulse response

Pole–zero interpretation

Feedback form

Frequency response

Impulse response

Pole–zero interpretation

Continuous time implementation

Feedforward form

Feedback form

See also

References

External links