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What does the statement "simultaneously localize a function in time and frequency domain" mean when it comes to signals and systems? What does it mean to localize?

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The Fourier transform of a nonzero function with compact support does not have compact support. So if something is "local in time" (i.e. it vanishes outside a finite time interval), then it is not "local in frequency" (i.e. it does not vanish outside any finite frequency interval). Similarly, the inverse Fourier transform of a nonzero function with compact support does not have compact support. So if something is "local in frequency" (i.e. it vanishes outside a finite frequency interval) then it is not "local in time" (i.e. it does not vanish outside any finite time interval).

The above is a qualitative manifestation of this "uncertainty principle". A quantitative version can be stated as follows. Take $f$ such that $\| f \|_{L^2}=1$. Let $\xi$ be a random variable with PDF $|f(x-a)|^2$ and $\eta$ be a random variable with PDF $|\hat{f}(x-b)|^2$. (Here I have used the Plancherel theorem to note that $\| \hat{f} \|_{L^2}=1$.) Then the variance of $\xi$ is at least a constant divided by the variance of $\eta$, and vice versa. Thus the two variances cannot simultaneously tend to zero. So if the "mass" of $f$ is concentrated near some number $a$, then the "mass" of $\hat{f}$ cannot be concentrated near any number $b$, and vice versa.

A helpful document on the subject: http://www2.math.umd.edu/~begue/Expository/Uncertainty.pdf

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    $\begingroup$ what is "compact support"? $\endgroup$
    – quantum231
    Aug 11 '15 at 23:00
  • $\begingroup$ @quantum231 The support of a function is the closure of the set of points where the function is nonzero. For example, if $f(x)=\max \{ 1-|x|,0 \}$, then the support of $f$ is $[-1,1]$. In this setting, a set is compact if it is closed and bounded. $\endgroup$
    – Ian
    Aug 11 '15 at 23:07
  • $\begingroup$ @quantum231, http://mathworld.wolfram.com/CompactSupport.html ... I think in grossly oversimplified terms, a function with "compact support" tends to zero outside a particular neighborhood, i.e. it's localized? $\endgroup$
    – user253804
    Aug 11 '15 at 23:09
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    $\begingroup$ @jrodatus A function does not merely tend to zero outside of its support; a function must exactly take the value of zero at points outside of its support. $\endgroup$
    – Max
    Aug 11 '15 at 23:38

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