Equivalence of Comb Filters to
Tapped Delay Lines

We can easily show that a parallel combination of feedforward comb filters is equivalent to a tapped delay line:

$$\begin{eqnarray*} H(z) &=& \left(1+g_1 z^{-M_1}\right) + \left(1+g_2 z^{-M_2}\... ...) \\ [10pt] &=& 3 + g_1 z^{-M_1} + g_2 z^{-M_2} + g_3 z^{-M_3} \end{eqnarray*}$$

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

We can also show that a series combination of feedforward comb filters produces a sparse tapped delay line:

$$\begin{eqnarray*} H(z) &=& \left(1+g_1 z^{-M_1}\right) \left(1+g_2 z^{-M_2}\righ... ...t] &=& 1 + g_1 z^{-M_1} + g_2 z^{-M_2} + g_1 g_2 z^{-(M_1+M_2)} \end{eqnarray*}$$

$$\begin{eqnarray*} \Rightarrow \quad b_0 &=& 1,\quad b_{M_1} = g_1,\quad b_{M_2} = g_2, b_{M_3}=g_1 g_2 \\ [10pt] M_3 &=& M_1+M_2 \end{eqnarray*}$$