Poles and Zeros of the Cepstrum

The *complex cepstrum* of a sequence
is typically defined
as the inverse Fourier transform of its log spectrum
[60]

where denotes the natural logarithm (base ) of , and denotes the Fourier transform (DTFT) of ,

(The

From Eq.
(8.2), the log *z* transform can be written in terms of the
factored form as

where denotes the th zero and denotes the th pole of the

we have that each pole and zero term in Eq. (8.9) may be expanded as

Since the region of convergence of the *z* transform must include the unit
circle (where the spectrum (DTFT) is defined), we see that the
Maclaurin expansion gives us the inverse *z* transform of all terms of
Eq.
(8.9) corresponding to poles and zeros *inside* the unit
circle of the
plane. Since the poles must be inside the unit
circle anyway for stability, this restriction is normally not binding
for the poles. However, zeros outside the unit circle--so-called
``non-minimum-phase zeros''--are used quite often in practice.

For a zero (or pole) outside the unit circle, we may rewrite the corresponding term of Eq. (8.9) as

where we used the Maclaurin series expansion for
once
again with the region of convergence including the unit circle. The
infinite sum in this expansion is now the bilateral *z* transform of an
*anticausal* sequence, as discussed in §8.7. That is,
the time-domain sequence is zero for nonnegative times (
) and
the sequence decays in the direction of time minus-infinity. The
factored-out terms
and
, for all poles and zeros
outside the unit circle, can be collected together and associated with
the overall gain factor
in Eq.
(8.9), resulting in a modified
scaling and time-shift for the original sequence
which can be
dealt with separately [60].

When all poles and zeros are inside the unit circle, the complex
cepstrum is *causal* and can be expressed simply in terms of the
filter poles and zeros as

where is the number of poles and is the number of zeros. Note that when , there are really additional zeros at , but these contribute zero to the complex cepstrum (since ). Similarly, when , there are additional poles at which also contribute zero (since ).

In summary, each stable pole contributes a *positive* decaying
exponential (weighted by
) to the complex cepstrum, while each
zero inside the unit circle contributes a *negative*
weighted-exponential of the same type. The decaying exponentials
start at time 1 and have unit amplitude (ignoring the
weighting)
in the sense that extrapolating them to time 0 (without the
weighting) would use the values
and
. The
decay rates are faster when the poles and zeros are well inside the
unit circle, but cannot decay slower than
.

On the other hand, poles and zeros *outside* the unit circle
contribute *anticausal* exponentials to the complex cepstrum,
negative for the poles and positive for the zeros.

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