Filter Combinations

• Filters can be cascaded in series or parallel.

  Figure 6: Cascaded digital filters.

  

• The time-domain operation of a filter is referred to as convolution (and typically denoted with the symbol $\ast$). From the properties of linear systems, convolution in the time domain corresponds to multiplication in the frequency domain.

• Thus, the series combination of filter responses $H_1(z)$ and $H_2(z)$ in Fig. 6 above would produce a new response given in the frequency domain by $H_1 H_2$ or in the time domain by $h_1 \ast h_2$.

• The response of filters in parallel is given by simple addition (in both the time- and frequency-domains).

• In this way, the filter combination shown in Fig. 6 could be replaced by a single filter with frequency-domain response $(H_1 H_2) + H_3$ or time-domain response $(h_1 \ast h_2) + h_3$.