Poles and Zeros of the Cepstrum

The complex cepstrum of a sequence $h(n)$ is typically defined as the inverse Fourier transform of its log spectrum [60]

$${\tilde h}(n)\isdef \frac{1}{2\pi}\int_{-\pi}^\pi \ln[H(e^{j\omega})] e^{j\omega n}d\omega,$$

where $\ln(x)$ denotes the natural logarithm (base $e$) of $x$, and $H(e^{j\omega})$ denotes the Fourier transform (DTFT) of $h(n)$,

$$H(e^{j\omega})\isdef \sum_{n=-\infty}^\infty h(n)e^{-j\omega n}.$$

(The real cepstrum, in contrast, is the inverse Fourier transform of the log-magnitude spectrum.) An equivalent definition (when the DTFT exists) is to define the complex cepstrum of $h$ as the inverse z transform of $\ln H(z)$. The cepstrum has numerous applications in digital signal processing including speech modeling [60] and pitch detection [34]. In §12.4, we use the cepstrum to compute minimum-phase spectra corresponding to a given spectral magnitude--an important tool in digital filter design.

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

$$\ln H(z) = \ln(g)$$

$$+ \ln(1-q_1z^{-1}) + \ln(1-q_2z^{-1}) + \cdots + \ln (1-q_Mz^{-1})$$

$$- \ln(1-p_1z^{-1}) - \ln(1-p_2z^{-1}) - \cdots - \ln(1-p_Nz^{-1})$$

$$= \ln(g) + \sum_{m=1}^M\ln(1-q_mz^{-1}) - \sum_{n=1}^N\ln(1-p_nz^{-1})$$

$$= \ln(g) - \sum_{m=1}^M\ln\left(\frac{1}{1-q_mz^{-1}}\right) + \sum... ...eft(\frac{1}{1-p_nz^{-1}}\right),\qquad{} % force eqn no. to next line \protect$$ (9.9)

where $q_m$ denotes the $m$th zero and $p_n$ denotes the $n$th pole of the z transform $H(z)$. Applying the Maclaurin series expansion

$$\ln\left(\frac{1}{1-x}\right) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^k}{k} + \cdots,\qquad \left\vert x\right\vert<1,$$

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

$$\begin{eqnarray*} \ln\left(\frac{1}{1-q_iz^{-1}}\right) &=& \sum_{n=1}^\infty \f... ...^{-n}, \qquad \left\vert z\right\vert>\left\vert p_i\right\vert. \end{eqnarray*}$$

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 $z$ 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

$$\begin{eqnarray*} \ln\left(\frac{1}{1-q_iz^{-1}}\right) &=& \ln\left(\frac{q_i^{... ...n,\qquad \left\vert z\right\vert<\left\vert q_i^{-1}\right\vert, \end{eqnarray*}$$

where we used a Maclaurin series expansion for $\ln(1/(1-x))$ 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, i.e., one that is zero for nonnegative times ($n\geq 0$) and which decays in the direction of time minus-infinity. The factored-out terms $-z/q_i$ and $-z/p_i$, for all poles and zeros outside the unit circle, can be collected together and associated with the overall gain factor $g$ in Eq. (8.9), resulting in a modified scaling and time-shift for the original sequence $h(n)$ 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

$${\tilde h}(n) = \left\{\begin{array}{ll} \ln(g), & n=0 \\ [5pt] \... ...ystyle\sum_{k=1}^M \frac{q_k^n}{n}, & n=1,2,3,\ldots\,, \\ \end{array}\right.$$

where $N$ is the number of poles and $M$ is the number of zeros. Note that when $N > M$, there are really $N - M$ additional zeros at $z=0$, but these contribute zero to the complex cepstrum (since $q_i=0$). Similarly, when $M>N$, there are $M - N$ additional poles at $z=0$ which also contribute zero (since $p_i=0$).

In summary, each stable pole contributes a positive decaying exponential (weighted by $1/n$) 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 $1/n$ weighting) in the sense that extrapolating them to time 0 (without the $1/n$ weighting) would use the values $p_i^0 = 1$ and $-q_i^0 = -1$. The decay rates are faster when the poles and zeros are well inside the unit circle, but cannot decay slower than $1/n$.

Poles and zeros outside the unit circle contribute anticausal exponentials to the complex cepstrum, negative for the poles and positive for the zeros.

As discussed in §12.4, any spectrum can be converted to minimum-phase form (without affecting the spectral magnitude) by computing its cepstrum and replacing any anticausal components with corresponding causal components. In other words, the anticausal part cepstrum, if any, is ``flipped'' about time zero so that it adds to the causal part. Doing this corresponds to reflecting non-minimum phase zeros (and any unstable poles) inside the unit circle in a manner that preserves spectral magnitude. The original spectral phase is then replaced by the unique minimum phase corresponding to the given spectral magnitude.

It can be remarked that each stable pole in the z transform gives rise to a pole in the differentiated log $z$-transform with residue $+1$, while each minimum-phase zero maps to a pole with residue $-1$.^9.5

For more about the cepstrum and the so-called Hilbert transform relations which relate the real and imaginary parts of the spectra of causal signals (which includes the phase and log-magnitude of minimum-phase spectra as a special case), see [60]. A matlab listing for computing a minimum-phase spectrum from the magnitude spectrum is given in §H.11.