# Minimum phase All-pass

Why is it that reflecting any poles or zeros of a rational function gives a minimum phase system? And why is doing that make a unique minimum phase system?

I understand the all-pass function absorbs the poles or zeros when you reflect it, not sure if that has something to do with it. Also, wouldn't reflecting any poles or zeros potentially cause some poles and zeros that were before in the stability region to be out of it?

An LTI system is said to be minimum-phase if the system and its inverse system are causal and stable. That's implying that all poles and zeros must be strictly inside the unit circle.

An all-pass filter's zero-pole pairs are reflections of each other across the unit circle so that an all-pass filter absorbs the poles or zeros. For a stable all-pass filter, its zeros are outside the unit circle and its poles are inside the unit circles.

A causal and stable mixed-phase system has all poles inside the unit circle and zeros which could be inside or outside the unit circle. Therefore, reflecting zeros outside the unit circle by cascading an all-pass system makes all zeros inside the unit circle, which means you get a minimum-phase system. (As Dan pointed out in the comment, cascading an all pass with a mixed-phase system is incorrect. Extracting an all pass from the mixed phase results in a minimum phase, or in the contrary cascading an all pass with a minimum phase results in a mixed-phase system.)

Once you get a minimum-phase system, the only thing you can do to prevent changing its magnitude response is to cascade an all-pass system. That brings additional zeros outside the unit circle and the system is no longer minimum-phase. Therefore minimum-phase system with a specific magnitude response is unique.

• a pure delay is also an all-pass filter. it's not an IIR APF. Commented Apr 2, 2021 at 5:02
• @robertbristow-johnson yes i've edited the answer Commented Apr 2, 2021 at 5:13
• So lets say I have a rational function H, if I just reflect all the poles and zeros of that H, that gives me a minimum phase system? What if H has poles and zeroes in the unit circle? Or maybe I'm missing something... Commented Apr 2, 2021 at 5:38
• @d4898ty You have a rational function H, with all poles and some of zeros inside the unit circle and other zeros outside the unit circle. Reflect all the zeros outside the unit circle and you get a minimum-phase system. If H already has all the poles and zeros inside the unit circle, it is a minimum-phase system. Commented Apr 2, 2021 at 6:11
• @DanBoschen Yes, you are right. I meant not cascading, but extracting an all-pass from the mixed-phase system and geting a minimum-phase system, which in another word means cascading a minimum-phase with an all-pass results in a mixed-phase. Commented Sep 26, 2022 at 1:44