# Definition of minimum-phase system

I saw a couple of definitions for minimum-phase in different textbooks and I'm trying to understand what the implication of each of them. The first definition I saw was: An invertible system which both it and it's inverse are causal and (BIBO) stable. The second definition is: An invertible system which both it and it's inverse are causal and have finite energy. It was written in one textbook that had the second definition as an alternative to the first one that these constraints allow the system to have zeros and poles on the unit circle and I don't understand why (as opposed to the first def). just to make sure, I'm not restricting myself to rational systems. Another thing I saw in some textbook and didn't understand is that the second definition implied that the transfer function of the system and it's inverse are analytic in the exterior of the unit circle. So, to conclude, my questions are:

1. why we have both of the definitions of minimum-phase?
2. why the 2nd def implies that the system can have zeros and poles on the unit circle as opposed to the first def?
3. why the 2nd def implies that the system and it's inverse have analytic transfer functions in the exterior of the unit circle?

Sorry for any grammar mistakes and thanks for any clarifications.

• Not a complete answer so i will comment in case this wasn't clear to you: A minimum phase system has all of its zeros inside or on the unit circle (and as a stable system any system MUST have all of its poles inside or on the unit circle)-- for this reason it's inverse will have all of its poles inside or on the unit circle and therefore also be stable. It is "minimum phase" as given it's magnitude response it will have the least amount of group delay. (Compared to a maximum phase system that has the same zeros outside the unit circle at locations 1/z, which will have the same mag response!) – Dan Boschen Jun 22 '17 at 11:29

one thing about a non-minimum phase system (with a rational transfer function), is that it can be thought of as the series concatenation (or cascade) of a minimum-phase system, having identical magnitude response as the given non-min-phase filter, with an all-pass filter. the APF will have a poles that cancels specific zeros of the min-phase system that are inside the unit circle, and puts in new zeros ("reflected zeros") outside the unit circle. so the new system always has more phase shift than the min-phase system.

so here's another definition of a minimum phase system:

An LTI system or filter

$$H(f) = |H(f)| e^{j \arg\{H(f)\}} = |H(f)| e^{j \phi(f)}$$

is minimum phase if and only if the natural phase response, in radians, is the negative of the Hilbert transform of the natural logarithm of the magnitude response:

$$\phi(f) \triangleq \arg\{ H(f) \} = -\mathscr{H}\big\{ \ln( |H(f)| ) \big\}$$

since

\begin{align} H(f) & = |H(f)| e^{j \phi(f)} \\ & = e^{\ln(|H(f)|)} e^{j \phi(f)} \\ & = e^{\ln(|H(f)|) + j \phi(f)} \\ & = e^{\ln(H(f))} \\ \end{align}

this is relating the real and imaginary parts of the complex natural logarithm of the frequency response.

$$\Im\{ \ln(H(f)) \} = -\mathscr{H}\big\{ \Re\{\ln(H(f)) \} \big\}$$

so a question to ask, why is the pole/zero definition stated in the question equivalent to this Hilbert transform definition??

• not exactly an answer, but too long for a comment. – robert bristow-johnson Jun 22 '17 at 22:27