Minimum Phase Filter Design

Above we used the Hilbert transform to find the imaginary part of an analytic signal from its real part. A closely related application of the Hilbert transform is contructing a minimum phase frequency response from an amplitude response.

Let denote a desired complex, minimum-phase frequency response,

and suppose we have only the amplitude response

Then the phase response can be computed as the Hilbert transform of . This can be seen by inspecting the log frequency response:

If is computed from by the Hilbert transform, then is an ``analytic signal'' in the frequency domain. Therefore, it has no ``negative times''. The time domain signal corresponding to a log spectrum is called the cepstrum. It is shown in §B.8 below that a frequency response is minimum phase if and only if the corresponding cepstrum is causal [179, Ch. 10], [242, Ch. 11].

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Generalized Window Method
Next: Minimum Phase and Causal Cepstra

---

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.