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,''

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