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Why is it that reflecting any poles or zeros of a rational function across the unit circle gives a minimum phase system? Here's an example, it seems reflecting any poles or zeros would result in the zeros being outside the circle? enter image description here

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Why is it that reflecting any poles or zeros of a rational function across the unit circle gives a minimum phase system?

It doesn't. You are starting with a wrong assumption.

Here's an example, it seems reflecting any poles or zeros would result in the zeros being outside the circle?

Correct. Hence it's nonsense and you shouldn't do it. Assuming that the 'x' marks the poles, the system is unstable or non-causal anyway.

Roughly speaking:

  1. A system is minimum phase if all zeros and poles are inside the unit circle.
  2. A system with zeros outside the unit circle can be made minimum phase by reflecting the zeros at the unit circle
  3. If have poles outside the unit circle, chances are something went wrong already and minimum phase is the least of your problems.
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  • $\begingroup$ Poles outside the unit circle might temporarily provide a lower system phase response than "minimum phase", depending on starting conditions, before blowing up, due to being unstable. So is the term "minimum phase" only to be applied to stable systems? $\endgroup$
    – hotpaw2
    Apr 2 at 17:05
  • $\begingroup$ Personally I think it's only "useful" to apply it to causal and stable systems. Poles outside the unit circle often just make the system non-causal and of course that creates a lower system phase response. A trivial example is a delay of -1 sample. It's obviously non-causal and it has a pole at infinity. Same for any causal allpass filter: if you invert it, all the poles end up outside the unit circle. The impulse response just time flips, i.e. it becomes entirely non-casual, but it's still well defined. $\endgroup$
    – Hilmar
    Apr 3 at 13:14
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A causal system with all poles and zeros inside the unit circle is a minimum phase system (and all poles inside the unit circle is a stable causal system).

This post will demonstrate more intuitively why a system with all zeros inside the unit circle will result in a minimum phase system (which has the minimum delay for a given magnitude response). Here we show a finite impulse response (FIR) filter and why the reverse of that filter (the coefficients in reverse order) results in a system with the same magnitude response in frequency but has a maximum phase response (maximum phase filter) resulting in the maximum delay for that given magnitude response.

Min and Max phase

To graphically create the magnitude and phase of any FIR filter, consider that each delay produces a rotate of one cycle on the complex plane. So the frequency response of the example minimum phase two-tap FIR filter with coefficients [1, -0.5] as given here can be visualized on a complex plane as the sum of a vector of magnitude 1 and angle 0 that stays fixed together with a second vector starting with magnitude 0.5 and starting angle 180 degrees (as given by the coefficient) and rotating once clockwise as the frequency extends from $0$ to $2\pi$ in normalized radian frequency corresponding to a frequency going from DC to the sampling rate. Similarly the frequency response of the example maximum phase tow-tap FIR filter with coefficients [-.5, 1] (the reverse of the minimum phase filter) can be visualized on a complex plane as the sum of a vector of magnitude 0.5 and angle 180 degrees that stays fixed together with a second vector of magnitude 1 and starting angle 0 and rotating once clockwise as the frequency extends over the same $0$ to $2\pi$ range. The plots for both cases showing both the magnitude and phase response versus frequency is shown below.

It's interesting to note that the magnitude response of the two systems above is identical but for the first case the overall phase deviation as the second vector rotates is minimized, while in the second case given the sum itself will enclose the origin is maximized. Since the phase goes through a maximum change in the same frequency range, and delay is the derivative of phase with respect to frequency, then the maximum phase system will also have the maximum delay (and the minimum phase system will have the minimum delay).

Freq Response

The key here is the result of the summation is an encirclement on the complex plane that may or may not encircle the origin. For any all zero system such as this where the origin is not encircled, the zeros must be inside the unit circle as given by the Cauchy Argument Principle of 1831 (and 1855).

For information on the Cauchy Argument Principle (and how Harry Nyquist applied it to his Nyquist Stability Criterion), and how it ultimately provides an intuitive explanation for minimum and maximum phase systems, please see

Nyquist's Stability Criterion and Cauchy's Argument Principle

and

Can I set a constraint on the first tap of an FIR filter such that its inverse is stable?

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