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Equivalence of Parallel Combs to TDLs

It is easy to show that the TDL of Fig.2.19 is equivalent to a parallel combination of three feedforward comb filters, each as in Fig.2.23. To see this, we simply add the three comb-filter transfer functions of Eq.$ \,$(2.3) and equate coefficients:

\begin{eqnarray*} H(z) &=& \left(1+g_1 z^{-M_1}\right) + \left(1+g_2 z^{-M_2}\... ...\right) \\ &=& 3 + g_1 z^{-M_1} + g_2 z^{-M_2} + g_3 z^{-M_3} \end{eqnarray*}
which implies

$\displaystyle b_0 = 3,\; b_{M_1} = g_1,\; b_{M_2} = g_2,\; b_{M_3} = g_3 . $

We see that parallel comb filters require more delay memory ( $ M_1+M_2+M_3$ elements) than the corresponding TDL, which only requires $ \max(M_1,M_2,M_3)$ elements.
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