Poles and Zeros of the Cepstrum
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 z transform . Applying the Maclaurin series expansion
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 .
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.
Conversion to Minimum Phase
Unstable Poles--Unit Circle Viewpoint