So, there was some confusion about what "minimum phase" means, what "excess phase" is, and why 'excess group delay' is not really in the vocabulary in regards to filter discussions,although it can be a kind of "pure delay" (aka linear phase) or a part of a filter that is "pure delay" (yes,it's possible to have a mixed-phase filter, although except for some "apodizing filters" there isn't much of that)
So, let's design an FIR filter (all zeros, no poles). We'll do that as a linear phase filter.
Now, this is a 64 tap constant delay filter. The impulse response looks like plot 1 of the next plot:
So, using Matlab I calculated the roots of the first impulse response, and replaced each one with radius larger than '1' with exactly 1/radius at the same angle. See next plots.
Top is the roots (in complex space) of the original filter. Bottom is the one with all outside roots inverted,which you may note, land EXACTLY on top of the "inside" set of roots on the right side of the graph. (why the right side has only to do with the frequency design I chose, it doesn't have to be 'right side'). The roots ON the unit circle are unchanged, and yes, they are also paired (on top of each other). Not an accident.
Now, what's that other impulse response (remember this is pure FIR filtering) on about?
Well, it's the minimum phase version of the first impulse response, with all zeros outside the unit circle moved inside. And here's the frequency response plot.
Notice, exactly (yes I mean EXACTLY) the same amplitude response, but the phase response isn't even close to similar. Surprise? No, not really.
So minimum phase filters have some particular properties.
1) their group delay is a function of the derivative of their amplitude response. This is not true of constant-delay filters.
2) For any filter, you can extract the delay part, by calculating the linear fit of phase response. If you then remove the linear part, you have the actual "phase shift" that represents non-uniform delay across frequency.
3) Minimum phase filters have no poles on or outside of the unit circle. This includes all, I mean ALL practical IIR filters. Roots outside the unit circle (or in the right half-plane for analog) are UNSTABLE.
4) IIR filters can have non-minimum phase if their zeros are not all on or inside the unit circle. Those zeros outside the unit circle can be moved inside if you want minimum phase.
5) Constant delay filters are ONLY possible exactly in FIR FILTERS,and they will always be symmetric impulse responses, or for some kinds of filters with a zero at DC, antisymmetric. There are NO OTHER OPTIONS. It's not a "good idea",it's the law.
Note: I am talking about filters with real coefficients. Having complex coefficients gets ... INTERESTING.
The math, however, remains the same, and then you have to decide what the complex output you get from the filter means. (note, yes, it can mean something, but that's a horse of another color indeed)
Now you may have heard the word "apodizing" in connection to filters. That means that SOME of the zeros outside the unit circle are moved inside, so the filter consists of a constant delay part and a minimum phase part.
Now,what is a "maximum phase" filter? Yeah, it exists. You push all the zeros OUTSIDE the unit circle. When you do this, it gives you exactly the time reverse of the minimum phase filter.
So, let's design an FIR filter (all zeros, no poles). We'll do that as a linear phase filter.
Now, this is a 64 tap constant delay filter. The impulse response looks like plot 1 of the next plot:
So, using Matlab I calculated the roots of the first impulse response, and replaced each one with radius larger than '1' with exactly 1/radius at the same angle. See next plots.
Top is the roots (in complex space) of the original filter. Bottom is the one with all outside roots inverted,which you may note, land EXACTLY on top of the "inside" set of roots on the right side of the graph. (why the right side has only to do with the frequency design I chose, it doesn't have to be 'right side'). The roots ON the unit circle are unchanged, and yes, they are also paired (on top of each other). Not an accident.
Now, what's that other impulse response (remember this is pure FIR filtering) on about?
Well, it's the minimum phase version of the first impulse response, with all zeros outside the unit circle moved inside. And here's the frequency response plot.
Notice, exactly (yes I mean EXACTLY) the same amplitude response, but the phase response isn't even close to similar. Surprise? No, not really.
So minimum phase filters have some particular properties.
1) their group delay is a function of the derivative of their amplitude response. This is not true of constant-delay filters.
2) For any filter, you can extract the delay part, by calculating the linear fit of phase response. If you then remove the linear part, you have the actual "phase shift" that represents non-uniform delay across frequency.
3) Minimum phase filters have no poles on or outside of the unit circle. This includes all, I mean ALL practical IIR filters. Roots outside the unit circle (or in the right half-plane for analog) are UNSTABLE.
4) IIR filters can have non-minimum phase if their zeros are not all on or inside the unit circle. Those zeros outside the unit circle can be moved inside if you want minimum phase.
5) Constant delay filters are ONLY possible exactly in FIR FILTERS,and they will always be symmetric impulse responses, or for some kinds of filters with a zero at DC, antisymmetric. There are NO OTHER OPTIONS. It's not a "good idea",it's the law.
Note: I am talking about filters with real coefficients. Having complex coefficients gets ... INTERESTING.
Now you may have heard the word "apodizing" in connection to filters. That means that SOME of the zeros outside the unit circle are moved inside, so the filter consists of a constant delay part and a minimum phase part.
Now,what is a "maximum phase" filter? Yeah, it exists. You push all the zeros OUTSIDE the unit circle. When you do this, it gives you exactly the time reverse of the minimum phase filter.
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