## Maximum Phase Filters

The opposite of minimum phase is maximum phase:

For example, every stable allpass filterB.2) is a maximum-phase filter, because its transfer function can be written as

where is an th-order minimum-phase polynomial in (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 samples has the transfer function , which is poles at and zeros at .

If zeros of 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 is order and minimum phase, then is maximum phase, and vice versa. To restate this in the time domain, if is a minimum-phase FIR sequence of length , then SHIFTFLIP 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 ( ) 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:

where we have normalized to 1 for simplicity. We have a single zero at . If , the filter is minimum phase. If , it is maximum phase. Note that the minimum-phase case is the one in which the impulse response 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