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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University