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Series Case

Figure 6.1: Series combination of transfer functions $ H_1(z)$ and $ H_2(z)$ to produce the combined transfer function $ H(z)=H_1(z)H_2(z)$.
\begin{figure}\input fig/series.pstex_t

Figure 6.1 illustrates the series connection of two filters $ H_1(z)=V(z)/X(z)$ and $ H_2(z)=Y(z)/V(z)$. The output $ v(n)$ from filter 1 is used as the input to filter 2. Therefore, the overall transfer function is

$\displaystyle H(z) \isdefs \frac{Y(z)}{X(z)}
\eqsp \frac{H_2(z)V(z)}{X(z)}
\eqsp H_2(z)H_1(z).

In summary, if the output of filter $ H_1(z)$ is given as input to filter $ H_2(z)$ (a series combination), as shown in Fig.6.1, the overall transfer function is $ H(z)=H_1(z)H_2(z)$--transfer functions of filters connected in series multiply together.

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Convolution Theorem