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If the Fourier Transform of a signal is enough to determine its spectral density, why do we complicate our lives by defining the autocorrelation function?

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    $\begingroup$ Wiener-Kinchin doesn't apply to all possible signals; the Fourier transform isn't even defined for all possible stochastic signals. $\endgroup$ – Marcus Müller Dec 21 '17 at 14:16
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    $\begingroup$ Besides, the autocorrelation gives you information that the PSD doesn't. $\endgroup$ – MBaz Dec 21 '17 at 15:10
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    $\begingroup$ @MBaz Huh? What information is there in the autocorrelation function that cannot be determined from the inverse Fourier transform of its power spectral density? $\endgroup$ – Dilip Sarwate Dec 21 '17 at 15:43
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    $\begingroup$ @DilipSarwate Well, what I was thinking is that WK is true only for stationary processes whose autocorrelation has a FT. I'll grant you that this is a bit pedantic, but the question is quite open-ended. $\endgroup$ – MBaz Dec 21 '17 at 16:31
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    $\begingroup$ @MBaz No pedantism in your comments. Words and theorems are important, esp. when one wants to "not complicate one's life" $\endgroup$ – Laurent Duval Dec 21 '17 at 17:02
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Signals can be classified in many ways, and one of them is according to their nature such as being deterministic or random (stochastic). When a signal is deterministic, its spectral content is given by the Fourier transform provided that the signal is absolutely (or square o.w.) integrable; i.e., its Fourier transform exists.

On the other hand when an signal is stochastic, then ideally by definition its Fourier transform does not exist. (a random signal will always have infinite energy as it cannot decay to zero when time goes to infinity because then it will not be random otherwise.)

Therefore for random signals, the Fourier transform will not exist and their spectral density cannot be computed that way. Nevertheless, for such signals an indirect description, based on an autocorrelation measure, can also be defined, whose advantage being that the Fourier transform of it may exist. Therefore the PSD of a random process is defined as the Fourier transform of its ACF.

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  • $\begingroup$ Fat32: Is it correct to say that what you described is exactly why there is the whole field called spectral estimation which focuses on methods for estimating the spectrum when the signal is stochastic. This area is really just a branch of statistics ( see Kundu's "statistical signal processing") and how I got into finding out about DSP in the first place. Thanks. $\endgroup$ – mark leeds Dec 21 '17 at 18:30
  • $\begingroup$ @mark leeds. Yes indeed ! However please note that, electrical engineering borrows subject of stochastic signals from its physical applications such as brownian motion, thermal noise, Boltzman statistics etc. Which is at least different in notation from a purely statistical treatment which may not have any physical concerns. $\endgroup$ – Fat32 Dec 21 '17 at 19:02
  • $\begingroup$ Thanks for the wonderful explanation. yes, often the apps are not physical but the problem is the same: find the important frequencies !!!! $\endgroup$ – mark leeds Dec 21 '17 at 20:14
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    $\begingroup$ @markleeds When you're commenting on someone's post, they'll get notified, so no need to (or ability to) @ them. That way you can get at most two notifications generated: the author of the commented-upon post and someone else commenting on it. I don't believe it's possible to @ more than one person. $\endgroup$ – Peter K. Dec 21 '17 at 21:21
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    $\begingroup$ Thanks Fat32 and Peter K. The knowledge is appreciated. I also think that I'm not supposed to say thanks but I find it rude not to. All the best. $\endgroup$ – mark leeds Dec 22 '17 at 0:34

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