What is the true meaning of a minimum phase system? Reading the Wikipedia article and Oppenheim is some help, in that, we understand that for an LTI system, minimum phase means the inverse is causal and stable. (So that means zeros and poles are inside the unit circle), but what does "phase" and "minimum" have to do with it? Can we tell a system is minimum phase by looking at the phase response of the DFT somehow?
The relation of "minimum" to "phase" in a minimum phase system or filter can be seen if you plot the unwrapped phase against frequency. You can use a pole zero diagram of the system response to help do a incremental graphical plot of the frequency response and phase angle. This method helps in doing a phase plot without phase wrapping discontinuities.
Put all the zeros inside the unit circle (or in left half plane in the continuous-time case), where all the poles have to be as well for system stability. Add up the angles from all the poles, and the negative of the angles from all the zeros, to calculate total phase to a point on the unit circle, as that frequency response reference point moves around the unit circle. Plot phase vs. frequency. Now compare this plot with a similar plot for a pole-zero diagram with any of the zeros swapped outside the unit circle (non-minimum phase). The overall average slope of the line with all the zeros inside will be lower than the average slope of any other line representing the same LTI system response (e.g. with a zero reflected outside the unit circle). This is because the "wind ups" in phase angle are all mostly cancelled by the "wind downs" in phase angle only when both the poles and zeros are on the same side of the unit circle line. Otherwise, for each zero outside, there will be an extra "wind up" of increasing phase angle that will remain mostly uncancelled as the plot reference point "winds" around the unit circle from 0 to PI. (...or up the vertical axis in the continuous-time case.)
This arrangement, all the zeros inside the unit circle, thus corresponds to the minimum total increase in phase, which corresponds to minimum average total phase delay, which corresponds to maximum compactness in time, for any given (stable) set of poles and zeros with the exact same frequency magnitude response. Thus the relationship between "minimum" and "phase" for this particular arrangement of poles and zeros.
Also see my old word picture with strange crank handles in the ancient usenet comp.dsp archives: https://groups.google.com/d/msg/comp.dsp/ulAX0_Tn65c/Fgqph7gqd3kJ
As you've already seen, the minimum phase has many physical meanings and implications. Where the phase comes from is that, for a given magnitude of frequency response, it corresponds to the filter that has the least amount of group delay. That is, you can have several filters with the same magnitude of frequency response, but one of them can be realized with the smallest amount of filter delay. In this sense, it's highly desired in control systems where filtering delay can be critical to stability. I'm abusing some notation here, as the phase "delay" can have many meanings, but the gist is there (and for group delay, it's a fact).
In other realms, if a system is a minimum phase, its inverse will have all of its poles inside the unit circle and be causal. So a minimum phase system has a stable inverse. This is important in many other applications for obvious reasons. If you must solve a linear system of equations, knowing the system is minimum phase guarantees its inverse will be minimum phase, and so stability is guaranteed (outside of any quantization effects).
It may not be obvious if a system is a minimum phase by looking at the DFT. There is a relationship between the magnitude of a minimum phase system and its phase, but it may not be visually obvious. However, adaptive lattice filters have the neat feature in that minimum phase filters are easily identified if all of the reflection coefficients are less than or equal to one in magnitude. That way, filters adaptively calculated can be determined if they're stable on the fly with little logic.
This paper seems to have some wisdom on the subject of minimum phase systems:
- John Bechhoefer, "Kramers-Kronig, Bode, and the meaning of zero", American Journal of Physics 79, 10 (2011).