
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
CepstraPrevious Section: Generalized Window Method