Definition of Minimum Phase Filters

In Chapter 10 we looked at linear-phase and zero-phase digital filters. While such filters preserve waveshape to a maximum extent in some sense, there are times when phase linearity is not important. In such cases, it is valuable to allow the phase to be arbitrary, or else to set it in such a way that the amplitude response is easier to match. In many cases, this means specifying minimum phase:

$\parbox{0.8\textwidth}{An LTI filter $H(z)=B(z)/A(z)$\ is said to be \emph{minimum phase} if all its poles and zeros are inside the unit circle $\left\vert z\right\vert=1$\ (excluding the unit circle itself).}$

Note that minimum-phase filters are stable by definition since the poles must be inside the unit circle. In addition, because the zeros must also be inside the unit circle, the inverse filter $1/H(z)$ is also stable when $H(z)$ is minimum phase. One can say that minimum-phase filters form an algebraic group in which the group elements are impulse-responses and the group operation is convolution (or, alternatively, the elements are minimum-phase transfer functions, and the group operation is multiplication).

A minimum phase filter is also causal since noncausal terms in the impulse response correspond to poles at infinity. The simplest example of this would be the unit-sample advance, $H(z) = z$ , which consists of a zero at $z=0$ and a pole at $z=\infty$ .^12.1