What Is A Comb Filter

Comb Filters

Comb filters are basic building blocks for digital audio furnishings. The audio-visual echo simulation in Fig.2.9 is 1 case of a comb filter. This section presents the two basic comb-filter types, feedforward and feedback, and gives a frequency-response analysis.

Feedforward Comb Filters

The feedforward comb filter is shown in Fig.ii.23. The directly bespeak ``feeds forwards'' around the filibuster line. The output is a linear combination of the straight and delayed signal.

**Effigy 2.23:** The feedforward comb filter.
$\includegraphics{eps/ffcf}$

The ``difference equation'' [449] for the feedforward comb filter is

$\displaystyle y(n) = b_0 x(n) + b_M x(n-M). \protect$

(iii.ii)

We see that the feedforward comb filter is a particular type of FIR filter. Information technology is also a special instance of a TDL.

Notation that the feedforward comb filter can implement the echo simulator of Fig.2.9 by setting and . Thus, it is is a computational physical model of a unmarried discrete echo. This is one of the simplest examples of audio-visual modeling using signal processing elements. The feedforward rummage filter models the superposition of a ``direct signal'' plus an attenuated, delayed betoken , where the attenuation (by $\vert b_M\vert<1$ ) is due to ``air assimilation'' and/or spherical spreading losses, and the delay is due to acoustic propagation over the distance meters, where is the sampling period in seconds, and is sound speed. In cases where the fake propagation delay needs to exist more accurate than the nearest integer number of samples , some kind of delay-line interpolation needs to exist used (the subject of §4.1). Similarly, when air assimilation needs to be simulated more than accurately, the constant attenuation factor can be replaced by a linear, time-invariant filter giving a different attenuation at every frequency. Due to the physics of air absorption, is generally lowpass in graphic symbol [349, p. 560], [47,318].

Feedback Comb Filters

The feedback comb filter uses feedback instead of a feedforward signal, equally shown in Fig.two.24 (drawn in ``direct form 2'' [449]).

**Figure 2.24:** The feedback comb filter.
$\includegraphics{eps/fbcf}$

A deviation equation describing the feedback rummage filter can be written in ``direct form 1'' [449] as^iii.9

$\displaystyle y(n) = b_0 x(n) - a_M y(n-M).$

The feedback comb filter is a special case of an Infinite Impulse Response (IIR) (``recursive'') digital filter, since there is feedback from the delayed output to the input [449]. The feedback rummage filter can be regarded as a computational physical model of a serial of echoes, exponentially decaying and uniformly spaced in fourth dimension. For example, the special case

$\displaystyle y(n) = x(n) + g\, y(n-M)$

is a computational model of an platonic plane wave bouncing dorsum and along between two parallel walls; in such a model, represents the total circular-trip attenuation (two wall-to-wall traversals, including ii reflections).

For stability, the feedback coefficient must be less than in magnitude, i.due east., $\left\vert a_M\correct\vert<1$ . Otherwise, if $\left\vert a_M\right\vert>1$ , each echo will be louder than the previous echo, producing a never-catastrophe, growing series of echoes.

Sometimes the output signal is taken from the cease of the delay line instead of the first, in which case the difference equation becomes

$\displaystyle y(n) = b_M x(n-M) - a_M y(n-M) .$

This pick of output merely delays the output signal by samples.

Feedforward Comb Filter Amplitude Response

Comb filters get their proper name from the ``comb-like'' appearance of their aamplitude response (gain versus frequency), equally shown in Figures ii.25, ii.26, and ii.27. For a review of frequency-domain assay of digital filters, see, eastward.g., [449].

The transfer role of the feedforward comb filter Eq. $\,$ (2.2) is

$\displaystyle H(z) = b_0+b_M\,z^{-M},$

(3.iii)

so that the amplitude response (gain versus frequency) is

$\displaystyle G(\omega) \isdef \left\vert H(e^{j\omega})\right\vert = \left\vert b_0 + b_M e^{-j\omega M}\right\vert, \quad -\pi \leq \omega \leq \pi. \protect$

(3.iv)

This is plotted in Fig.2.25 for , , and , , and . When , we get the simplified event

$\displaystyle G(\omega) = \left\vert 1 + e^{-j\omega M}\right\vert = \left\vert... ...ga M/2}\right\vert = 2\left\vert\cos\left(\omega\frac{M}{2}\right)\right\vert.$

In this case, we obtain nulls, which are points (frequencies) of nothing proceeds in the amplitude response. Note that in flangers, these nulls are moved slowly over time by modulating the filibuster length . Doing this smoothly requires interpolated delay lines (see Affiliate 4 and Chapter 5).

Feedback Comb Filter Amplitude Response

Figure 2.26 shows a family of feedback-rummage-filter amplitude responses, obtained using a choice of feedback coefficients.

Figure ii.27 shows a similar family unit obtained using negated feedback coefficients; the opposite sign of the feedback exchanges the peaks and valleys in the amplitude response.

Every bit introduced in §2.6.2 to a higher place, a class of feedback comb filters can be defined as any departure equation of the form

$\displaystyle y(n) = x(n) + g\,y(n-M).$

Taking the z transform of both sides and solving for

$H(z)\isdef Y(z)/X(z)$ , the transfer function of the feedback comb filter is found to be

$\displaystyle H(z) = \frac{1}{1-g\,z^{-M}}, \protect$

(3.five)

so that the amplitude response is

$\displaystyle G(\omega) \isdef \left\vert H(e^{j\omega})\right\vert = \frac{1}{\left\vert 1 - g e^{-j\omega M}\right\vert}, \quad -\pi \leq \omega \leq \pi .$

This is plotted in Fig.two.26 for and , , and . Effigy ii.27 shows the aforementioned case simply with the feedback sign-inverted.

For , the feedback-comb amplitude response reduces to

$\displaystyle G(\omega) = \frac{1}{2\left\vert\sin(\omega M/2)\right\vert},$

and for to

$\displaystyle G(\omega) = \frac{1}{2\left\vert\cos(\omega M/2)\right\vert},$

which exactly inverts the amplitude response of the feedforward comb filter with gain (Eq. $\,$ (two.iv)).

Note that produces resonant peaks at

$\displaystyle \omega_k = 2\pi\frac{k}{M}, \quad k=0,1,2,\dots,M-1,$

while for , the peaks occur midway betwixt these values.

Filtered-Feedback Comb Filters

The filtered-feedback comb filter (FFBCF) uses filtered feedback instead of but a feedback gain.

Denoting the feedback-filter transfer function by , the transfer function of the filtered-feedback rummage filter can exist written equally

$\displaystyle H(z) = \frac{b_0}{1 - H_l(z)z^{-M}}.$

Note that when is a causal filter, the FFBCF can be considered mathematically a special example of the general allpole transfer office in which the first denominator coefficients are constrained to exist zero:

$\displaystyle H(z) = \frac{b_0}{1 + 0 + \cdots + 0 + a_M z^{-M} + a_{M+1} z^{-{M+1}} + \cdots}$

It is this ``sparseness'' of the filter coefficients that makes the FFBCF more computationally efficient than other, more than full general-purpose, IIR filter structures.

In §2.6.2 higher up, nosotros mentioned the physical estimation of a feedback-comb-filter equally simulating a plane-wave bouncing back and along between ii walls. Inserting a lowpass filter in the feedback loop further simulates frequency dependent losses incurred during a propagation round-trip, as naturally occurs in real rooms.

The master physical sources of airplane-wave attenuation are air absorption (§B.7.15) and the coefficient of absorption at each wall [349]. Boosted ``losses'' for plane waves in existent rooms occur due to handful. (The plane wave hits something other than a wall and reflects off in many unlike directions.) A item scatterer used in concert halls is textured wall surfaces. In ray-tracing simulations, reflections from such walls are typically modeled as having a specular and lengthened component. More often than not speaking, wavelengths that are large compared with the ``grain size'' of the wall texture reverberate specularly (with some attenuation due to whatsoever wall motility), while wavelengths on the lodge of or smaller than the texture grain size are scattered in various directions, contributing to the diffuse component of reflection.

The filtered-feedback comb filter has many applications in computer music. Information technology was plainly first suggested for bogus reverberation past Schroeder [412, p. 223], and kickoff implemented by Moorer [314]. (Reverberation applications are discussed further in §3.6.) In the physical interpretation [428,207] of the Karplus-Strong algorithm [236,233], the FFBCF can be regarded equally a transfer-function physical-model of a vibrating string. In digital waveguide modeling of string and wind instruments, FFBCFs are typically derived routinely every bit a computationally optimized equivalent forms based on some initial waveguide model adult in terms of bidirectional delay-lines (``digital waveguides'') (see §6.10.i for an case).

For stability, the aamplitude-response of the feedback-filter must be less than in magnitude at all frequencies, i.e., $\left\vert H_l(e^{j\omega T})\right\vert<1,\,\forall\omega T\in[-\pi,\pi)$ .

Equivalence of Parallel Combs to TDLs

It is easy to prove that the TDL of Fig.ii.xix is equivalent to a parallel combination of three feedforward comb filters, each as in Fig.2.23. To see this, we but add together the three comb-filter transfer functions of Eq. $\,$ (2.3) and equate coefficients:

$\begin{eqnarray*} H(z) &=& \left(1+g_1 z^{-M_1}\right) + \left(1+g_2 z^{-M_2}\... ...\right) \\ &=& 3 + g_1 z^{-M_1} + g_2 z^{-M_2} + g_3 z^{-M_3} \end{eqnarray*}$

which implies

$\displaystyle b_0 = 3,\; b_{M_1} = g_1,\; b_{M_2} = g_2,\; b_{M_3} = g_3 .$

We run into that parallel comb filters require more delay retentivity ( elements) than the corresponding TDL, which just requires $\max(M_1,M_2,M_3)$ elements.

Equivalence of Series Combs to TDLs

It is besides straightforward to show that a series combination of feedforward rummage filters produces a sparsely tapped delay line as well. Considering the instance of two sections, we take

$\begin{eqnarray*} H(z) &=& \left(1+g_1 z^{-M_1}\right) \left(1+g_2 z^{-M_2}\right)\\ &=& 1 + g_1 z^{-M_1} + g_2 z^{-M_2} + g_1 g_2 z^{-(M_1+M_2)} \end{eqnarray*}$

which yields

$\displaystyle b_0 = 1,\; b_{M_1} = g_1,\; b_{M_2} = g_2,\; M_3=M_1+M_2,\;b_{M_3}=g_1 g_2.$

Thus, the TDL of Fig.ii.xix is equivalent too to the serial combination of two feedforward comb filters. Note that the aforementioned TDL structure results irrespective of the series ordering of the component comb filters.

Time Varying Comb Filters

Comb filters can be changed slowly over time to produce the following digital audio ``effects'', among others:

Phasing
Flanging
Chorus
Leslie

Since all of these furnishings involve modulating delay length over fourth dimension, and since time-varying filibuster lines typically require interpolation, these applications will be discussed later on Chapter 5 which covers variable delay lines. For at present, nosotros volition pursue what tin exist accomplished using stock-still (fourth dimension-invariant) delay lines. Perhaps the most important application is artificial reverberation, addressed in Chapter 3.

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Feedback Delay Networks (FDN)
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Tapped Delay Line (TDL)