Audio Signal Processing and Coding

2.6 DIFFERENCE EQUATIONS AND DIGITAL FILTERS

Digital filters are characterized by difference equations of the form

In the input-output difference equation above, the output y(n) is given as the linear combination of present and past inputs minus a linear combination of past outputs (feedback term). The parameters a_i and b_i are the filter coefficients or filter taps and they control the frequency response characteristics of the digital filter. Filter coefficients are programmable and can be made adaptive (time-varying). A direct-form realization of the digital filter is shown in Figure 2.13.

The filter in the Eq. (2.26) is referred to as an infinite-length impulse response (IIR) filter. The impulse response, h(n), of the filter shown in Figure 2.13 is given by

The IIR classification stems from the fact that, when the feedback coefficients are non-zero, the impulse response is infinitely long. In a statistical signal representation, Eq. (2.26) is referred to as a time-series model. That is, if the input of this filter is white noise then y(n) is called an autoregressive moving average (ARMA) process. The feedback coefficients, a_i, are chosen such that the filter is stable, i.e., a bounded input gives a bounded output (BIBO). An input-output equation of a causal filter can also be written ...