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Equivalence of Series Combs to TDLs

It is also straightforward to show that a series combination of feedforward comb filters produces a sparsely tapped delay line as well. Considering the case of two sections, we have

\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. 1.13 is equivalent also to the series combination of two feedforward comb filters. Note that the same TDL structure results irrespective of the series ordering of the component comb filters.


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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