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*}](http://www.dsprelated.com/josimages_new/pasp/img527.png)
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.
$](http://www.dsprelated.com/josimages_new/pasp/img528.png)
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Equivalence of Parallel Combs to TDLs