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Everywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then backtrack the thing using windows. Is inifinity necessary? Or can we do processing without this do-undo process?

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    $\begingroup$ The fast Fourier transform deals with finite samples. No infinities there. $\endgroup$ – copper.hat Jun 13 '14 at 5:54
  • $\begingroup$ @copper.hat FFT is in discrete or digital space. Do we equivalent in analog domain without windows? How about correlations and other operations that talk about infinite duration signals? $\endgroup$ – Seetha Rama Raju Sanapala Jun 13 '14 at 6:01
  • $\begingroup$ Well, its a model of reality that simplifies some aspects. Like using reals to do financial calculations, most of us will never see infinities. You could analyse using a finite time span, but will hit mostly irrelevant complexities just dealing with boundary conditions. $\endgroup$ – copper.hat Jun 13 '14 at 6:04
  • $\begingroup$ @copper.hat Thank you. $\endgroup$ – Seetha Rama Raju Sanapala Jun 13 '14 at 6:08
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Fourier analysis requires a function to be defined on a (locally compact, abelian) group. The set of real numbers is a group under addition; this is the setting of Fourier transform. So is the circle $\mathbb R/(a\mathbb Z)$ for some $a>0$; this is the setting of Fourier series. So is the cyclic group $\mathbb Z/(n\mathbb Z)$, which is the setting of the discrete Fourier transform.

An interval $[a,b]$ is not a group. So, even though we may only have a function defined on such an interval, from the mathematical point of view the Fourier transform is taken on the real line. The function can be set to $0$ outside of $[a,b]$.

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  • $\begingroup$ For a long time, I puzzled over the "essence" that unites Fourier analysis in all the domains I had seen it applied. And then I saw an exposition (by a pure mathematician) that developed Fourier analysis on locally compact Abelian groups. The light went on. I wish I had been told that from the beginning. It would have saved me lots of time wondering about what unites it all. $\endgroup$ – Will Nelson Jul 19 '14 at 2:27

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