Definition of Minimum Phase

In Chapter 11 we looked at linear-phase and zero-phase digital filters. While such filters preserve waveshape, there are times when the linearity of the phase response 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.

Definition. An LTI filter $H(z)=B(z)/A(z)$ is said to be 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, more specifically, the elements may be transfer functions of a given order, and the group operation 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 $z = \infty$.^13.1

A filter is minimum phase if both the numerator and denominator of its transfer function are minimum-phase polynomials in $z^{-1}$:

Definition. A polynomial of the form

$$\begin{eqnarray*} B(z) &=& b_0 + b_1 z^{-1}+ b_2 z^{-2} + \cdots + b_M z^{-M}\\ &=& b_0(1-\xi_1z^{-1})(1 - \xi_2 z^{-1})\cdots(1-\xi_Mz^{-1}) \end{eqnarray*}$$

is said to be minimum phase if all of its roots $\xi_i$ are inside the unit circle, i.e., $\left\vert\xi_i\right\vert<1$.

We may also define a minimum-phase signal (or sequence) as the inverse z transform of a minimum-phase polynomial:

Definition. A signal $h(n)$, $n\in{\bf Z}$, is said to be minimum phase if its z transform $H(z)$ is minimum phase.

Note that every stable, all-pole, (causal) filter $H(z)=b_0/A(z)$ is minimum phase, because stability implies that $A(z)$ is minimum phase, and there are ``no zeros'' (all are at $z=0$). This is an indication that minimum phase is in some sense the most ``natural'' phase for a digital filter to have, since it is the only phase available to a stable, causal, all-pole filter.

The effect of non-minimum-phase zeros on the complex cepstrum was described in §8.4.

The opposite of minimum phase is maximum phase:

Definition. An LTI filter $H(z)=B(z)/A(z)$ is said to be maximum phase if all zeros of the polynomial $B(z)$ are outside the unit circle.

If zeros of $B(z)$ occur both inside and outside the unit circle, the filter is said to be a mixed-phase filter. Note that zeros on the unit circle are neither minimum nor maximum phase according to our definitions. Since poles on the unit circle are sometimes called ``marginally stable,'' we could say that zeros on the unit circle are ``marginally minimum and/or maximum phase'' for consistency. However, such a term does not appear to be very useful. When pursuing minimum-phase filter design (see §12.4), we will find that zeros on the unit circle must be treated separately.

If $H(z)$ is minimum phase, then $H(z^{-1})$ is maximum phase, and vice versa.

By the flip theorem for z transforms,

$${\cal Z}_z\{$$FLIP$$(h)\} = H(z^{-1}).$$

Each zero of $H(z)$ inside the unit circle becomes a zero of $H(z^{-1})$ outside the unit circle. It follows that if $h$ is a minimum-phase impulse response, FLIP$(h)$ is maximum-phase impulse response. In other words, time reversal inverts the locations of all zeros, thereby ``reflecting'' them across the unit circle.

Example

An easy case to classify is the set of all first-order FIR filters

$$H(z) = 1 + h_1 z^{-1}$$

where we have normalized $h_0$ to 1 for simplicity. We have a single zero at $z=-h_1$. If $\left\vert h_1\right\vert< 1$, the filter is minimum phase. If $\left\vert h_1\right\vert>1$, it is maximum phase. Note that the minimum phase case is the one in which the impulse response $[1,h_1,0,\ldots]$ decays instead of grows. It can be shown that this is a general property of minimum-phase sequences:

Among all signals $h_i(n)$ having the identical magnitude spectra, the minimum-phase signal $h_{\hbox{\tiny mp}}(n)$ has the fastest decay in the sense that

$$\sum_{n=0}^K \left\vert h_{\hbox{\tiny mp}}(n)\right\vert^2 \geq \sum_{n=0}^K \left\vert h_i(n)\right\vert^2, \qquad K=0,1,2,\ldots$$

That is, the signal energy in the first $K+1$ samples of the minimum-phase case is at least as large as any other causal signal having the same magnitude spectrum. (See [60] for a proof outline.) Thus, minimum-phase signals are maximally concentrated toward time 0 among the space of causal signals having a given magnitude spectrum. As a result of this property, minimum-phase signals are sometimes called minimum-delay signals.