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


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Feedforward Comb Filter Amplitude Response

Comb filters get their name from the ``comb-like'' appearance of their amplitude response (gain versus frequency), as shown in Figures 2.25, 2.26, and 2.27. For a review of frequency-domain analysis of digital filters, see, e.g., [449].

Figure: Amplitude responses of the feed forward comb-filter $ H(z) = x(n) + g x(n-M)$ (diagrammed in Fig.2.23) with $ M=5$ and $ g=0.1$, $ 0.5$, and $ 0.9$. a) Linear amplitude scale. b) Decibel scale. The frequency axis goes from 0 to the sampling rate (instead of only half the sampling rate, which is more typical for real filters) in order to display the fact that the number of notches is exactly $ M=5$ (as opposed to ``$ 2.5$'').
\includegraphics[width=\twidth ]{eps/ffcfar}

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

$\displaystyle H(z) = b_0+b_M\,z^{-M},$ (3.3)

so that the amplitude response (gain versus frequency) is

$\displaystyle G(\omega) \isdef \left\vert H(e^{j\omega})\right\vert = \left\vert b_0 + b_M e^{-j\omega M}\right\vert, \quad -\pi \leq \omega \leq \pi. \protect$ (3.4)

This is plotted in Fig.2.25 for $ M=5$, $ b_0=1$, and $ b_M=0.1$, $ 0.5$, and $ 0.9$. When $ b_0=b_M=1$, we get the simplified result

$\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 $ M$ nulls, which are points (frequencies) of zero gain in the amplitude response. Note that in flangers, these nulls are moved slowly over time by modulating the delay length $ M$. Doing this smoothly requires interpolated delay lines (see Chapter 4 and Chapter 5).

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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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