## 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:

where is minimum phase, is a stable allpass filter:

and is the number of maximum-phase zeros of . 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

Now multiply by to get

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