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 , 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.,