Creating Minimum Phase Filters
and Signals

Minimum-phase filter design often requires creating a minimum-phase desired frequency response $H(e^{j\omega})$ (usually starting from a real amplitude response $\left\vert H(e^{j\omega})\right\vert$). As is clear from §12.2, any filter transfer function $H(z)$ can be made minimum-phase, in principle, by completely factoring $H(z)$ and ``reflecting'' all zeros $z_i$ for which $\vert z_i\vert>1$ inside the unit circle, i.e., replacing $z_i$ by $1/z_i$. However, factoring a polynomial this large can be impractical. An approximate ``non-parametric'' method is based on the property of the complex cepstrum (see §8.4) that each minimum-phase zero in the spectrum gives rise to a causal exponential in the cepstrum, while each non-minimum-phase zero corresponds to an anti-causal exponential in the cepstrum [60]. Therefore, by computing the cepstrum and converting anti-causal exponentials to causal exponentials, the corresponding spectrum is converted to minimum-phase form.

A matlab function mps.m which carries out this method is listed in §H.11.^13.3 It works well for smooth desired frequency response curves. Specifically, the inverse DFT of the log magnitude frequency response should not be longer than the number of samples in the frequency response (no ``time aliasing'').