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It is expected that any person reading this article will understand why is this property called "minimum phase". Stable zeros, cauzality, and all the other properties written until now in the article do not motivate why and how comes in the "minimum phase" claim. The article must explain how is the "phase" come into discussion, how is the "minimum" established and why exactly is this particular minim phase property relevant practically.

this article is confusing to beginners. Any transfer function has amplitude and phase. Given only the amplitude response, that amplitude could be modeled with a function that does not have zeroes in the right-half-plane. IF the phase is ALSO then matched, the original function is minimum phase. IF NOT, and additional function called an 'all-pass network' is cascaded to match the phase. This implies the original function is non-minimum-phase, because an 'all-pass' just adds phase CorvetteZ51 (talk) 09:18, 16 July 2015 (UTC)[reply]

In the maximum phase section i think there is an error repeated twice. Instead of "anti-causal and stable" it should be "anti-causal and un-stable" ?

Is the following section correct?

"Minimum phase in the time domain For all causal and stable systems that have the same magnitude response, the minimum phase system has its energy concentrated near the end of the impulse response. i.e., it maximizes the following function which we can think of as the delay of energy in the impulse response."

Even if it is correct it is not clear.

Sorry, but the first sentence of the article is wrong:

  In control theory and signal processing, a linear, 
  time-invariant system is minimum-phase if the system
  and its inverse are causal and stable.

Instead, in control theory an LTI system is minimum-phase if it has stable zeros (or alternatively, if the inverse is stable). The definition is NOT coupled to "causal" nor to "stability" of the system.

MartinOtter 19:53, 3 February 2007 (UTC)[reply]

What is your source? The page is consistent with the definitions given at <http://ccrma.stanford.edu/~jos/filters/Definition_Minimum_Phase_Filters.html>. LachlanA (talk) 16:37, 18 May 2008 (UTC)[reply]

I'm confused about maximum phase definition. Quoting Manolakis, Ingle, Kogon - Statistical and Adaptive Signal Processing (2005):

  (...) In an analogous manner, we can define a maximum-phase system as one in which both the
  system and its inverse are anticausal and stable. A PZ system then is maximum-phase
  if all its poles and zeros are outside the unit circle. (...)

---

—Preceding unsigned comment added by 200.193.3.224 (talk) 04:10, 5 June 2008 (UTC)[reply]

• NB: The following discussion has been moved from User talk:Zvika (permalink) and user talk:مبتدئ (permalink). --Zvika (talk) 18:39, 10 December 2008 (UTC)[reply]

Hi zvika
can you clarify in which extent what i wrote contradicts a books that you have? may be i m wrong i dont know. As far as i know minimum phased systems means (roughly speaking) that the inverse system is stable. And as you know a system is also stable when it has a pole at 0 (2 poles at zero means instable). Now in the case of linear systems where the transfer functions are expressed in terms of polinomials this means you can write the numerator as s*(s-a)*(s-b)*..... . You can easly construct a system which is not proper and fullfill this req. It is clear that non proper systems do not make any physical sens. Concerning the zerp dynamic: it is the dynamic remaning in the system and which is not observable at the output. For a fully observable system you will see the hole dynamic. For an only partially observable system some dynamic may be hidden. As you know the concept and poles is not defined for nonlinear systems. But there is an extension which extend many of the concepts of linear systems to nonlinear ones. In the case of zeros it is the zerodynamic. The connection is that if you linearize your zero dynamic and find its poles this will be the same as the zeros of the linearized system.
best regards مبتدئ (talk) 19:53, 7 December 2008 (UTC)[reply]

Thanks for the detailed response. The book I cited (Kailath) says: "A LTI system is called minimum-phase if both it and its inverse, are stable and causal." (Kailath et al., Linear Estimation, p.193.) This is exactly the definition currently given in the article, from which you removed the "causal" part, which I then restored.

Concerning the nonlinear extension, I don't know very much about this. However, since all the rest of the article is about LTI systems, I think there should be a more detailed explanation of what is meant in the nonlinear context, as well as a reference to a book where more information about this can be found. As the sentence stands right now, it is not really clear what is meant by it, and there is no way for a reader to obtain more information.

All the best, --Zvika (talk) 07:35, 8 December 2008 (UTC)[reply]

Hi

since i dont want to question the scientific authority of the author you have mentioned i will assume the point with properness is correct if it is stated like that in the book, but may be you can check in other books. I was starting from the understanding that the inverse of such a system should be stable (which is fullfilled if you have a zero at 0 which means a pole at 0 for the inverse system as long as the multiplicity of the pole is 1. If you have a zero at 0 which means you have the Form s*(s-b)*.../polynom and this zero does not cancel out/ is not simplified this means you have non proper system.). But if properness is required in many formulations in different books than its more likely that i m wrong. Anyway minimumphasness is not (directly) or only related to the zeros of a system. The relation is rather defined pver an integral relation (see arabic wikipedia concerning this). Concerning the source for the definition in the nonlinear case: I have unfurtonately no sources (only my lecture notes). It may be noticed somewhere in some books about control or systen theory but i havent searched for them nor have i a title in mind now. Anyway if you think what i wrote is false feel free to delete it.

Best Regards مبتدئ (talk) 03:32, 9 December 2008 (UTC)[reply]

OK, thanks. I don't speak Arabic so I can't really look this up in the Arabic Wikipedia. If you do come up with a source for your statement, please do add it in. --Zvika (talk) 13:02, 9 December 2008 (UTC)[reply]

I guess that you will also be able to understand the relation in the arabic article. math Formula looks the same in all languages :-). Just follow the interwiki :-) مبتدئ (talk) 05:11, 10 December 2008 (UTC)[reply]

Well, I took a look, but I wasn't really able to figure out much from the formulas. In any case, to the best of my understanding, a minimum phase system cannot contain a zero at 0 (see minimum phase#Continuous-time frequency analysis). Concerning the extension to nonlinear systems: as far as I can see, the condition you wrote does not take into account the causality requirement which is a requirement for LTI minimum phase systems, so that the nonlinear definition does not currently reduce to the linear definition. So I think it would be best to remove this sentence until we can find a reliable source for it. --Zvika (talk) 06:03, 10 December 2008 (UTC)[reply]

As you like. It doesnt hurt me to delete the information :-) although i can guarentee that it is true since i have it black on white in my lecture notes. The formulas you saw were giving a bound for the phase of a system at a given frequency. Systems which dont obey to this Formula are called non minimum phsed because the phase is then more negative than allowed by this bounds. Best Regards مبتدئ (talk) 17:01, 10 December 2008 (UTC)[reply]

by the way, look in the discussion page of the article, there is a person which also claims properness is not needed an who seems to give a source. مبتدئ (talk) 17:04, 10 December 2008 (UTC)[reply]

On the contrary, it appears that User:MartinOtter claimed that causality is not required, but was unable to provide a source, whereas User:LachlanA provided an additional source reiterating the definition in the article. This does create the impression that perhaps there are different definitions of minimum phase, but it's not clear to me whether this is a real difference in nomenclature between control theory and signal processing, or whether it is just a common but incorrect usage. Until we have a reliable source saying otherwise, I think we will have to stick with the definition that we can reference. --Zvika (talk) 18:32, 10 December 2008 (UTC)[reply]

OK zvika, here is a source for my sentence:[1]. Look at page 293. Best Regards مبتدئ (talk) 05:19, 26 December 2008 (UTC)[reply]

I've never seen causality required of the inverse. Of all my controls books, none of the authors require this. Here is a sampling:

From "Feedback Control of Dynamic Systems", (Franklin, Powell, Emami-Naeini, 2002), pg 385, "A system with a zero in the right half-plane undergoes a net change in phase when evaluated for frequency inputs between zero and infinity, which, for an associated magnitude plot, is greater than if all poles and zeros were in the left half-plane. Such a system is called nonminimum-phase."

From "Multivariable Feedback Control" (Skogestad and Postlethwaite, 2015), pg 18 "For stable systems which are minimum-phase (no time delays or right-half plane zeros) there is a unique relationship between the gain and phase of the frequency response...The name *minimum-phase* refers to the fact that such a system has the minimum possible phase lag for the given magnitude response $|G(j\omega)|$.

From "Control System Design", (Goodwin, Graeb, Salgado, 2001), pg 77 "A special class of transfer functions arises when all poles and zeros lie in the left half of the complex plane $S$. Traditionally, these transfer function have been called *minimum-phase transfer functions*. In the sequel, however, we will use this name as referring simply to transfer function with no RHP zeros, irrespective of whether they have RHP poles." — Preceding unsigned comment added by 128.138.189.177 (talk) 01:56, 18 December 2018 (UTC)[reply]

The examples given (the ones that correspond to the complex plane figures) do not appear to be causal (i.e., they have more zeros than poles), which means that they aren't stable in the useful sense. For example, the noncausal system:

has a pole at . The reason why textbooks define minimum phase systems as causal/stable systems with inverses that are also causal/stable is that that guarantees that you count all of your implicit and explicit poles. —TedPavlic | (talk) 00:29, 30 December 2008 (UTC)[reply]

I think that there are too much of the same links to other wiki pages. In the section Minimum_phase_as_minimum_group_delay alone I've counted 7 links to group delay, 7 to zero(s), 4 to transfer function and another 4 to unit circle. There are a lot more duplicated links like these on the entire page. I'm still a bit new to wiki, so I'm just stating it here. --Gligoran (talk) 09:18, 13 February 2010 (UTC)[reply]

Aren't these systems the ones with sharper cut-off response? If so, I think it should be said in the article. I was reading about all-pass filters being designed to have maximum group delay in order to perform better delay equalization http://en.wikipedia.org/wiki/Delay_equalization — Preceding unsigned comment added by 87.15.157.85 (talk) 14:57, 1 September 2014 (UTC)[reply]

No. Dicklyon (talk) 15:28, 1 September 2014 (UTC)[reply]

But the maximum phase section is unsourced and wrong. I put a dubious tag. Someone should rework it from sources. Dicklyon (talk) 15:43, 1 September 2014 (UTC)[reply]

Well, I don't think "No" is valid in general. At least 2nd-order all-pass filters with maximum group delay should have the sharpest cut-off delay response. So they do the out of cut-off (delay) equalization job better than other systems with the same magnitude response. That is what I understand from http://publications.lib.chalmers.se/records/fulltext/177900/177900.pdf (see graph at page 16).

About the dubious tag, these other sources seem to clear it:

The only way to correct the non-linear effect of the delay is to introduce a system that delays some frequency components less than others, but it is important to see that this always causes more delay in the system

http://cdn.teledynelecroy.com/files/whitepapers/group_delay-designcon2006.pdf

and

Definition. An LTI filter H(z)=B(z)/A(z) is said to be minimum phase if all its poles and zeros are inside the unit circle (excluding the unit circle itself).

https://ccrma.stanford.edu/~jos/fp/Definition_Minimum_Phase_Filters.html

Definition. An LTI filter H(z)=B(z)/A(z) is said to be maximum phase if all zeros of the polynomial B(z) are outside the unit circle.

...every stable allpass filter is a maximum-phase filter...

https://ccrma.stanford.edu/~jos/fp/Maximum_Phase_Filters.html

That is, the signal energy in the first K+1 samples of the minimum-phase case is at least as large as any other causal signal having the same magnitude spectrum... Thus, minimum-phase signals are maximally concentrated toward time 0 when compared against all causal signals having the same magnitude spectrum. As a result of this property, minimum-phase signals are sometimes called minimum-delay signals.

https://ccrma.stanford.edu/~jos/fp/Minimum_Phase_Means_Fastest.html

--87.15.58.177 (talk) 15:18, 2 September 2014 (UTC)[reply]

The current version of the section "Minimum phase as minimum group delay" starts with the statement:

"For all causal and stable systems that have the same magnitude response, the minimum phase system has the minimum group delay."

After some manipulations, which in themselves I can follow, there is the statement:

"The denominator and are invariant to reflecting the zero outside of the unit circle, i.e., replacing with ."

I wonder, could anyone explain why one would like to substitute with its (complex conjugate) inverse? I fail to see how this would preserve the magnitude response (which I understand to be the modulus of the system response). Specifically:

.

Or am I mistaken? On a more fundamental level, the article could do with a(n even) more conceptual explanation about minimum-phase(d)ness, or at least I would appreciate that.Redav (talk) 10:40, 19 July 2020 (UTC)[reply]