Filter Combinations & Linearity

• What happens when we cascade digital filters in series or parallel?

  Figure 12: Cascaded digital filters.

  

• The time-domain operation of a filter is referred to as convolution (as described above). From the properties of linear systems, convolution in the time domain corresponds to multiplication in the frequency domain.

• Thus, the series combination of filter frequency responses $H_1(f)$ and $H_2(f)$ in Fig. 12 above would produce a new response given in the frequency domain by $H_1(f) H_2(f)$ or in the time domain by $h_1(t) \ast h_2(t)$.

• 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. 12 could be replaced by a single filter with frequency-domain response $(H_1 H_2) + H_3$ or time-domain response $(h_1 \ast h_2) + h_3$.

• Filters are linear systems, which satisfy two particular properties:
  • Additivity: $h(x + y) = h(x) + h(y)$ (mixing two tracks and filtering them is the same as filtering both tracks and mixing)
  • Homogeneity: $h(\alpha x) = \alpha h(x)$ (you will get the same result if you amplify a track before or after filtering)

• Linear systems are commutative, meaning you can change the order of filters $H_1$ and $H_2$ above without changing the result ( $h_1(h_2(x)) = h_2(h_1(x))$)

• The output of a linear system will not contain frequency components that did not exist in the input. In other words, a linear system can only change the gain and phase of existing frequency components.

• If a system produces frequency components that did not exist in the input signal, it is a nonlinear system.