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Introduction to Digital Filters
    Minimum-Phase Filters
       Minimum-Phase/Allpass Decomposition

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Minimum-Phase/Allpass Decomposition

Every causal stable filter $ H(z)$ with no zeros on the unit circle can be factored into a minimum-phase filter in cascade with a causal stable allpass filter:

$\displaystyle H(z) \eqsp H_{\hbox{mp}}(z)\,S(z) \qquad\hbox{(Minimum-Phase/Allpass Decomposition)} $

where $ H_{\hbox{mp}}(z)$ is minimum phase, $ S(z)$ is a stable allpass filter:

$\displaystyle S(z) \eqsp \frac{s_L + s_{L-1}z^{-1}+ \cdots + s_1 z^{-(L-1)} + z^{-L}} {1 + s_1z^{-1}+ s_2 z^{-2}+ \cdots + s_L z^{-L}}, $

and $ L$ is the number of maximum-phase zeros of $ H(z)$.

This result is easy to show by induction. Consider a single maximum-phase zero $ \xi$ of $ H(z)$. Then $ \left\vert\xi\right\vert>1$, and $ H(z)$ can be written with the maximum-phase zero factored out as

$\displaystyle H(z) \eqsp H_1(z) (1-\xi z^{-1}). $

Now multiply by $ 1=(1-\xi^{-1}z^{-1})/(1-\xi^{-1}z^{-1})$ to get

$\displaystyle H(z) \eqsp \underbrace{H_1(z) (1-\xi^{-1}z^{-1})}_{\displaystyle\... ...nderbrace{\frac{1-\xi z^{-1}}{1-\xi^{-1}z^{-1}}}_{\displaystyle\isdef S_1(z)}. $

We have thus factored $ H(z)$ into the product of $ H_2(z)$, in which the maximum-phase zero has been reflected inside the unit circle to become minimum-phase (from $ z=\xi$ to $ z=1/\xi$), times a stable allpass filter $ S_1(z)$ consisting of the original maximum-phase zero $ \xi$ and a new pole at $ z=1/\xi$ (which cancels the reflected zero at $ z=1/\xi$ given to $ H_2(z)$). This procedure can now be repeated for each maximum-phase zero in $ H(z)$.

In summary, we may factor maximum-phase zeros out of the transfer function and replace them with their minimum-phase counterparts without altering the amplitude response. This modification is equivalent to placing a stable allpass filter in series with the original filter, where the allpass filter cancels the maximum-phase zero and introduces the minimum-phase zero.

A procedure for computing the minimum phase for a given spectral magnitude is discussed in §11.7 below. More theory pertaining to minimum phase sequences may be found in [60].

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Previous: Minimum Phase Means Fastest Decay
Next: Is Linear Phase Really Ideal for Audio?
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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