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 [242], we may compute an approximate cepstrum as the 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.,

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.

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Minimum Phase Filter Design
Next: The Hilbert Transform

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.