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Maximum Phase Filters

The opposite of minimum phase is maximum phase:

$\textstyle \parbox{0.8\textwidth}{% A stable LTI filter $H(z)=B(z)/A(z)$\ is said to be \emph{maximum phase} if all its zeros are outside the unit circle.}$
For example, every stable allpass filter (§B.2) is a maximum-phase filter, because its transfer function can be written as

$\displaystyle H(z)=\frac{z^{-N}A(z^{-1})}{A(z)}, $

where $ A(z)=1+a_1z^{-1}+a_2z^{-2}+\cdots+a_N z^{-N}$ is an $ N$th-order minimum-phase polynomial in $ z^{-1}$ (all roots inside the unit circle). As another example of a maximum-phase filter (a special case of allpass filters, in fact), a pure delay of $ N$ samples has the transfer function $ z^{-N}$, which is $ N$ poles at $ z=0$ and $ N$ zeros at $ z=\infty$.

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 §11.7), we will find that zeros on the unit circle must be treated separately.

If $ B(z)$ is order $ M$ and minimum phase, then $ z^{-M}B(z^{-1})$ is maximum phase, and vice versa. To restate this in the time domain, if $ b=[b_0,b_1,\ldots,b_M,0,\ldots]$ is a minimum-phase FIR sequence of length $ M+1$, then SHIFT$ _M($FLIP$ (b))$ is a maximum-phase sequence. In other words, time reversal inverts the locations of all zeros, thereby ``reflecting'' them across the unit circle in a manner that does not affect spectral magnitude. Time reversal is followed by a shift in order to obtain a causal result, but this is not required: Adding a pure delay to a maximum-phase filter ( $ B(z) \to z^{-1}B(z)$) gives a new maximum-phase filter with the same amplitude response (and order increased by 1).

Example

It is easy to classify completely all first-order FIR filters:

$\displaystyle 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, as elaborated in the next section.

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Minimum Phase Means Fastest Decay
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Minimum-Phase Polynomials