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 $H(\ejo )$ denote a desired complex, minimum-phase frequency response in the digital domain ($z$ plane):

$$H(\ejo ) \isdefs G(\omega)e^{j\Theta(\omega)},$$ (5.23)

and suppose we have only the amplitude response

$$G(\omega) \isdefs \left\vert H(\ejo )\right\vert.$$ (5.24)

Then the phase response $\Theta(\omega)$ can be computed as the Hilbert transform of $\ln G(\omega)$ . This can be seen by inspecting the log frequency response:

$$\ln H(\ejo ) \eqsp \ln G(\omega) + j\Theta(\omega)$$ (5.25)

If $\Theta$ is computed from $G$ by the Hilbert transform, then $\ln H(\ejo )$ 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].