Minimum-Phase and Causal Cepstra
To show that a frequency response is minimum phase if and only if the
corresponding cepstrum is causal, we may take the log of the
corresponding transfer function, obtaining a sum of terms of the form
for the zeros and
for the poles.
Since all poles and zeros of a minimum-phase system must be inside the
unit circle of the
plane, the Laurent expansion of all such terms
(the cepstrum) must be causal. In practice, as discussed in
[263], we may compute an approximate cepstrum as an inverse FFT
of the log spectrum, and make it causal by ``flipping'' the
negative-time cepstral coefficients around to positive time (adding
them to the positive-time coefficients). That is
, for
and
for
.
This effectively inverts all unstable poles and all non-minimum-phase
zeros with respect to the unit circle. In other terms,
(if unstable), and
(if
non-minimum phase).
The Laurent expansion of a differentiable function of a complex
variable can be thought of as a two-sided Taylor expansion,
i.e., it includes both positive and negative powers of
, e.g.,
![]() |
(5.26) |
In digital signal processing, a Laurent series is typically expanded about points on the unit circle in the




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Minimum-Phase Filter Design