Series Case

Figure 6.1: Series combination of transfer functions $H_1(z)$ and $H_2(z)$ to produce the combined transfer function $H(z)=H_1(z)H_2(z)$ .

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

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

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.