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According to the Paley-Wiener criterion, a system is causal if satisfies:

$$\int\limits_{-\infty }^{+\infty }{\frac{\ln (|H(f)|)}{1+{{f}^{2}}}}df<\infty$$

So I want to know

  1. This equation is related to LTI system only?
  2. Only casual system is realizable?for linear and non-linear systems?
  3. Which kinds of systems can we create?(only Paley-Wiener criterion can be enough?)
Matt L.
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Ehsan Zakeri
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    Oh. Seriously. When I [said](http://dsp.stackexchange.com/questions/35698/convergence-of-fourier-transform-of-e-t-sin2-pi-ftut#comment67874_35698) "please don't ask THREE unrelated questions in one question", I did NOT mean to say "ask TWO totally unrelated questions in one question" – Marcus Müller Nov 20 '16 at 15:41
  • I'm voting to close this question as off-topic because OP asks two completely unrelated questions in one question. – Marcus Müller Nov 20 '16 at 15:42
  • also, this is not even just two questions; in fact, the first two "?" seem to belong to two totally unrelated topics. – Marcus Müller Nov 20 '16 at 15:43
  • Sorry I did mistakes. I apologize – Ehsan Zakeri Nov 20 '16 at 15:55
  • So, now that we have the number of questions reduced to 2, which seem to be related (I'm not quite sure how), could you please try to formulate your question more mathematically answerable? The Paley-Wiener theorem is a theorem; the term "Paley-Wiener criterion" (by the way, Wiener, not Weiner) does appear in the literature, but I couldn't find a common definition of it in short time. Would you mind explicitly stating it here, if possible? – Marcus Müller Nov 20 '16 at 18:06
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    @MarcusMüller, instead of being so agressive, you should guide the OP to conduct their concerns. It is very simple to be ironic on other mistakes. We all know you have high rep. – Brethlosze Nov 21 '16 at 04:50
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    this question should be re-opened. – robert bristow-johnson Nov 21 '16 at 06:19
  • @robertbristow-johnson aahrgh! my fault! I forgot to retract my close vote after OP removed the unrelated questions. I just casted my reopen vote – this is a very precise question now. – Marcus Müller Nov 21 '16 at 11:01
  • @hypfco I hope I *was* guiding OP; that's why I followed up with him in my first comment; and as you can see, my comment regarding the mathematical nature of the criterion was absolutely focused on figuring out how to answer :) – Marcus Müller Nov 21 '16 at 11:02
  • Thanks for your comments and your contribution. Actually the first question was my fault that I didn't ask precise and related question. So I apologize and appreciate for helping me in this new question. – Ehsan Zakeri Nov 21 '16 at 11:18
  • The last two questions are not relevant to the first one. The first one can be answered by investigating http://dsp.stackexchange.com/questions/35702/causal-system-and-physical-systems the other two depends on modeling paradigms. – percusse Nov 30 '16 at 01:08

1 Answers1

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The Paley-Wiener criterion defines a condition on the magnitude spectrum of a causal time-domain function. So if the Paley-Wiener criterion is satisfied for a given $A(\omega)=|H(\omega)|$, we know that there is a causal function with magnitude spectrum $A(\omega)$. It should be noted that the Paley-Wiener criterion is only applicable to square-integrable functions $A(\omega)$. If $A(\omega)$ is not square-integrable then the criterion is neither necessary nor sufficient.

A frequently occurring misunderstanding is that, given a frequency response $H(\omega)$, we can check the causality of the corresponding system using the Paley-Wiener criterion. This is generally not the case. If the criterion is not satisfied we know for sure that $H(\omega)$ is not the frequency response of a causal system. But if it is satisfied we only know that there must be a causal function with magnitude spectrum $A(\omega)=|H(\omega)|$, but we still don't know if the given $H(\omega)$ corresponds to a causal system. This depends on the phase of $H(\omega)$, and the only thing we know is that it is possible to find a phase response $\phi(\omega)$ such that $H(\omega)=A(\omega)e^{j\phi(\omega)}$ corresponds to a causal system.

To answer your questions:

  1. Yes, it only applies to LTI systems because only LTI systems are fully characterized by a frequency response.
  2. Yes, only causal systems can be realized because we cannot look into the future.
  3. This question is a bit broad. There are realizability conditions for certain restricted sets of systems (e.g., passive LTI systems), but in general, causality is definitely a necessary condition. However, for the above mentioned reason we cannot just take a given $H(\omega)$, check the Paley-Wiener criterion and conclude that we can realize $H(\omega)$ as a causal system, because we didn't take into account the system's phase.
Matt L.
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