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Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Comb Filters
          Feedback Comb Filter Amplitude Response

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Feedback Comb Filter Amplitude Response

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

Figure: Amplitude response of the feedback comb-filter $ H(z) = 1/(1-g z^{-M})$ (Fig.2.24 with $ b_0=1$ and $ -a_M=g$) with $ M=5$ and $ g=0.1$, $ 0.5$, and $ 0.9$. a) Linear amplitude scale. b) Decibel scale.
\includegraphics[width=\twidth ]{eps/fbcfar}

Figure 2.27 shows a similar family obtained using negated feedback coefficients; the opposite sign of the feedback exchanges the peaks and valleys in the amplitude response.

Figure: Amplitude response of the phase-inverted feedback comb-filter, i.e., as in Fig.2.26 with negated $ g=-0.1$, $ -0.5$, and $ -0.9$. a) Linear amplitude scale. b) Decibel scale.
\includegraphics[width=\twidth ]{eps/fbcfiar}

As introduced in §2.6.2 above, a class of feedback comb filters can be defined as any difference 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

$\displaystyle H(z) = \frac{1}{1-g\,z^{-M}}, \protect$ (3.5)

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.2.26 for $ M=5$ and $ g=0.1$, $ 0.5$, and $ 0.9$. Figure 2.27 shows the same case but with the feedback sign-inverted.

For $ g=1$, the feedback-comb amplitude response reduces to

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

and for $ g=-1$ 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 $ g=1$ (Eq.$ \,$(2.4)).

Note that $ g>0$ produces resonant peaks at

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

while for $ g<0$, the peaks occur midway between these values.

Previous: Feedforward Comb Filter Amplitude Response
Next: Filtered-Feedback Comb Filters

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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