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?
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asked May 2, 2012 at 2:21
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1$\begingroup$ Welcome do Signal Processing! This is a great question. Make sure to read our FAQ which contains a lot of useful info about the site. $\endgroup$– PhononCommented May 2, 2012 at 13:19
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5 Answers
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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
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$\begingroup$ Hmm, interesting - so we CAN tell that a system is min-phase by looking at the phase response from its DFT then it looks like, correct? $\endgroup$– SpaceyCommented May 2, 2012 at 15:26
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$\begingroup$ @Mohammad : One issue with using a DFT for phase response is unwrapping phase, which may or may not have a unique or closed form solution. (Especially a problem if there are "discontinuities" in the impulse response.) $\endgroup$– hotpaw2Commented May 2, 2012 at 15:28
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$\begingroup$ @hotpaw2 With unwrapping we are undoing the modulo 2*pi or -2*pi, (two ways of doing it), but even then I didnt think it would be an issue. $\endgroup$– SpaceyCommented May 2, 2012 at 15:33
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2$\begingroup$ hotpaw- Great analogy. I have a book that uses the Argument Principle from complex analysis instead. It's an elegant proof, but not for non-mathematicians. $\endgroup$– BryanCommented May 3, 2012 at 15:18
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1$\begingroup$ @Bryan This seems very interesting. What is the title of the book? $\endgroup$– shamisenCommented Oct 1, 2018 at 14:00
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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.
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4$\begingroup$ I would add that the "poles inside the unit circle" stability criterion is valid for discrete-time systems, while for continuous-time systems, you would want poles to be in the left half of the $s$-plane. $\endgroup$– Jason RCommented May 2, 2012 at 2:47
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$\begingroup$ Ah yes, excellent point. For those unfamiliar with the bilinear transform (which effectively maps the left hand s-plane into the unit circle on the z-plane), that's an important distinction. Thanks. $\endgroup$– BryanCommented May 2, 2012 at 2:50
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1$\begingroup$ The "relationship" between log amplitude and minimum-phase is the Hilbert transform $\endgroup$– HilmarCommented May 2, 2012 at 2:55
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$\begingroup$ Minimum phase filter seems to be IIR, but how minimum is their phase compared to FIR? $\endgroup$ Commented May 2, 2012 at 5:24
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2$\begingroup$ There is no reason that a minimum-phase filter can't be FIR. The only condition is that all of the filter zeros must be inside the unit circle. Given a non-minimum-phase filter, you can always transform it to a minimum-phase filter that has the same magnitude response by moving any zeros outside of the unit circle to their conjugate reciprocal. That is, for all of the filter zeros $z_i$, if $|z_i| > 1$, replace $z_i$ with $\frac{1}{z_i^*}$. $\endgroup$– Jason RCommented May 2, 2012 at 13:04
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One of the most useful properties of minimum phase system is that they have an impulse response that is the most compact in time as is possible for any given amplitude function. Technically this can be expressed as $$\sum_{i=0}^{k}h[i]^2 = min, k\epsilon \mathbb{N}$$
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2$\begingroup$ Shouldn't it be max rather than min if $h[n]$ has most of its energy upfront? $\endgroup$– PhononCommented May 22, 2013 at 17:38
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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).
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This video explains this nicely and goes into more depth.
TLDR: The minimality is w.r.t. all systems which produce the same bode magnitude plot. There are infinitely many systems that can produce the same magnitude plot but only one which has minimum phase delay. Predictably, an easy way to make a minimum phase system non-minimum phase is by adding a delay, which obviously increases the phase delay but doesn't affect the magnitude plot.
