Advanced DSP

z-Transform

Convolution is equivalent to polynomial multiplication. Thus, algebraic polynomial simplification techniques can be used to analyze convolution better. The fundamental theorem of algebra says that an order N polynomial can be written as a multiplication of N complex roots. Each of these roots can be thought of as a simple first order filter, whose cascade is equivalent to the filter in question.

The MA part of a filter is always stable. The questionable part is the AR part. If we have an AR filter, it will be unstable if and only if it is unstable for the impulse input. The response of each pole can be calculated, and if each pole is stable, the system is stable. A pole outside the unit circle is stable, inside is not.