Minimum-Phase/Allpass Decomposition
Every causal stable filter 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)}
$](http://www.dsprelated.com/josimages_new/filters/img1265.png)
![$ H_{\hbox{mp}}(z)$](http://www.dsprelated.com/josimages_new/filters/img1266.png)
![$ S(z)$](http://www.dsprelated.com/josimages_new/filters/img1267.png)
![$\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}},
$](http://www.dsprelated.com/josimages_new/filters/img1268.png)
![$ L$](http://www.dsprelated.com/josimages_new/filters/img1269.png)
![$ H(z)$](http://www.dsprelated.com/josimages_new/filters/img308.png)
This result is easy to show by induction. Consider a single
maximum-phase zero of
. Then
, and
can be written with the maximum-phase zero factored out as
![$\displaystyle H(z) \eqsp H_1(z) (1-\xi z^{-1}).
$](http://www.dsprelated.com/josimages_new/filters/img1272.png)
![$ 1=(1-\xi^{-1}z^{-1})/(1-\xi^{-1}z^{-1})$](http://www.dsprelated.com/josimages_new/filters/img1273.png)
![$\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)}.
$](http://www.dsprelated.com/josimages_new/filters/img1274.png)
![$ H(z)$](http://www.dsprelated.com/josimages_new/filters/img308.png)
![$ H_2(z)$](http://www.dsprelated.com/josimages_new/filters/img35.png)
![$ z=\xi$](http://www.dsprelated.com/josimages_new/filters/img1275.png)
![$ z=1/\xi$](http://www.dsprelated.com/josimages_new/filters/img1276.png)
![$ S_1(z)$](http://www.dsprelated.com/josimages_new/filters/img1277.png)
![$ \xi$](http://www.dsprelated.com/josimages_new/filters/img1270.png)
![$ z=1/\xi$](http://www.dsprelated.com/josimages_new/filters/img1276.png)
![$ z=1/\xi$](http://www.dsprelated.com/josimages_new/filters/img1276.png)
![$ H_2(z)$](http://www.dsprelated.com/josimages_new/filters/img35.png)
![$ H(z)$](http://www.dsprelated.com/josimages_new/filters/img308.png)
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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Minimum Phase Means Fastest Decay