## 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 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- terms are the noncausal terms, while the power-of- terms are considered causal. The term in the above general example is associated with time 0, and is included with the causal terms.

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