Maximum Phase Filters

The opposite of minimum phase is maximum phase:

For example, every stable allpass filter (§B.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).

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Subsections

• Example

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.