## 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 constructing a*minimum phase*[263] frequency response from an amplitude response.

Let denote a desired complex, minimum-phase frequency response in the digital domain ( plane):

(5.23) |

and suppose we have only the amplitude response

(5.24) |

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

(5.25) |

If is computed from by the Hilbert transform, then is an ``analytic signal'' in the frequency domain. Therefore, it has no ``negative times,''

*i.e.*, it is causal. The time domain signal corresponding to a log spectrum is called the

*cepstrum*[263]. It is reviewed in the next section that a frequency response is minimum phase if and only if the corresponding cepstrum is causal [198, Ch. 10], [263, Ch. 11].

**Next Section:**

Minimum-Phase and Causal Cepstra

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Generalized Window Method