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Feedback Comb Filters

The feedback comb filter uses feedback instead of a feedforward signal, as shown in Fig.2.24 (drawn in ``direct form 2'' [452]).

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

A difference equation describing the feedback comb filter can be written in ``direct form 1'' [452] as3.9

$\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 [452]. The feedback comb filter can be regarded as a computational physical model of a series of echoes, exponentially decaying and uniformly spaced in time. For example, the special case

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

is a computational model of an ideal plane wave bouncing back and forth between two parallel walls; in such a model, $ g$ represents the total round-trip attenuation (two wall-to-wall traversals, including two reflections).

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

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

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

This choice of output merely delays the output signal by $ M$ samples.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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