# Minimum Phase & maximum phase

i have 2 questions: 1-If zeros on the unite circle that will be min or max phase?and this system has low-pass frequency response or not ? 2-Is the min phase has the shortest impulse response?

• that's 3 questions. dunno the answer to the zeros on the unit circle. i guess it's as minimum phase as you're gonna get because normally reflecting them "outside" the unit circle (with the conjugate-reciprocal operation $\frac{1}{z^*}$), leaves the zeros at the same place. so you can't reflect them and get another filter that has identical magnitude response and more phase shift. Commented Jan 12, 2016 at 4:53

## 1 Answer

Let me first state the commonly used definitions of minimum-phase and maximum-phase for discrete-time systems:

A causal and stable discrete-time system is said to be strictly minimum-phase when all its zeros are inside the unit circle.

Of course, all its poles are inside the circle as well, otherwise it wouldn't be causal and stable.

A causal and stable discrete-time system is said to be strictly maximum-phase when all its zeros are outside the unit circle.

Again, causality and stability requires that all the poles are inside the unit circle as well.

If you have zeros on the unit circle (and all others either inside or outside the circle), the system is usually referred to as loosely minimum-phase or loosely maximum-phase. A system with all its zeros on the unit circle is a degenerate case of that definition, in the sense that it is loosely minimum-phase and loosely maximum-phase at the same time. But in that case these definitions do not make much sense. These definitions are meant for systems that have some zeros inside (or outside) the circle and some on the circle.

Your question about the low pass frequency response can't be answered, because the concepts of minimum-phase and maximum-phase have absolutely nothing to do with the magnitude response of the filter. You can have a minimum-phase low pass filter, and a maximum-phase low pass filter with exactly the same magnitude response. The same is true for all other frequency selective filter characteristics (high pass, band pass, etc.).

Concerning your last question, it depends what you mean by shortest. If you mean the number of samples of the impulse response, then the answer is no. As an example, take a length $N$ FIR minimum-phase impulse response. If you flip the coefficients (i.e., time reverse them) you obtain a maximum-phase filter of the same length with the same magnitude response.

The minimum-phase property means that of all filters with the same magnitude response, the impulse response of the minimum-phase filter is maximally concentrated near time index $n=0$ in a mean square sense. I.e., if $h[n]$ is a minimum-phase impulse response, then

$$\sum_{n=0}^{N}|h[n]|^2\ge \sum_{n=0}^{N}|g[n]|^2\tag{1}$$

for all $N\ge 0$, and for any $g[n]$ with the same magnitude response as $h[n]$.