What does it mean to "simultaneously localize a function in time and frequency domain"?

Question

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?

Answer 1

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