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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 $ \ln(1-\xi_iz^{-1})$ for the zeros and $ -\ln(1-\rho _iz^{-1})$ for the poles. Since all poles and zeros of a minimum-phase system must be inside the unit circle of the $ z$ 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 $ c(n) \leftarrow c(n) + c(-n)$ , for $ n=1,2,\ldots\,,$ and $ c(n)\leftarrow 0$ for $ n<0$ . This effectively inverts all unstable poles and all non-minimum-phase zeros with respect to the unit circle. In other terms, $ p_i \leftarrow 1/p_i$ (if unstable), and $ \xi_i\leftarrow 1/\xi_i$ (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 $ z$ , e.g.,

$\displaystyle H(z) = \cdots + h(-2)z^2 + h(-1)z + h(0) + h(1)z^{-1}+ h(2)z^{-2}+ \cdots\,.$ (5.26)

In digital signal processing, a Laurent series is typically expanded about points on the unit circle in the $ z$ plane, because the unit circle--our frequency axis--must lie within the annulus of convergence of the series expansion in most applications. The power-of-$ z$ terms are the noncausal terms, while the power-of-$ z^{-1}$ terms are considered causal. The term $ h(0)$ in the above general example is associated with time 0, and is included with the causal terms.

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