## 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 §E.9 below that a frequency response is minimum phase if and only if the corresponding cepstrum is causal [187, Ch. 10], [247, Ch. 11].

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