Matlab listing: mps.m

function [sm] = mps(s)
% [sm] = mps(s)
% create minimum-phase spectrum sm from complex spectrum s
   sm = exp( fft( fold( ifft( log( clipdb(s,-100) )))));

The clipdb and fold utilities are listed and described in §J.10 and §J.9, respectively.

Note that mps.m must be given a whole spectrum in ``FFT buffer format''. That is, it must contain dc and positive-frequency values followed by negative frequency values and be a power of 2 in length.

The mps function works well as long as the desired frequency response is smooth. If there are any zeros on the frequency axis (``notches''), the corresponding minimum-phase impulse response will be time aliased because the corresponding exponentials in the cepstrum never decay. To suppress time-aliasing to some extent, the desired frequency response magnitude is clipped to 100 dB below its maximum. Time aliasing can be reduced by interpolating the desired frequency response s to a higher sampling density (thereby increasing the time available for exponential decay in the cepstral domain). However, for pure notches (zeros right on the unit circle), no amount of oversampling will eliminate the time aliasing completely. To avoid time aliasing in the cepstrum, such a desired spectrum must be smoothed before taking the log and inverse FFT. Zero-phase smoothing of the spectral magnitude is a typical choice for this purpose. When greater accuracy is required, all notch frequencies can be estimated so that terms of the form can be effectively ``divided out'' of the desired spectrum and carried along as separate factors.

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About the Author: 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.