## 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].

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Minimum-Phase and Causal Cepstra
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