What happens when we cascade digital filters in series or parallel?
Cascaded digital filters.
The time-domain operation of a filter is referred to as convolution (and typically denoted with the symbol ). 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 H1(z) and H2(z) in Fig. 4 above would produce a new response given in the frequency domain by H1H2 or in the time domain by .
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. 4 could be replaced by a single filter with frequency-domain response
(H1H2) + H3 or time-domain response