short time Fourier transform

Hey, I have a question about localization a signal in time domain by windowing.

would anyone explain to me these phrases:

1) The basis functions sin(ωt) and cos(ωt) are not localized in time!(support region in frequency = 0)

2) The δ(t) is not localized in frequency! (support region in time = 0)

cheers.

Re: short time Fourier transform

1) The basis functions are nonzero for arbitrarily large times. For example, for arbitrarily large .

2) The delta distribution is nonzero for artitrarily large frequencies. Note that , and hence in the freqency domain, is the constant function 1 which does not have compact support (take nonzero values over an infinite range).

Re: short time Fourier transform

Quote:

Originally Posted by

**vincisonfire** 1) The basis functions are nonzero for arbitrarily large times. For example,

for arbitrarily large

.

2) The delta distribution is nonzero for artitrarily large frequencies. Note that

, and hence in the freqency domain,

is the constant function 1 which does not have compact support (take nonzero values over an infinite range).

Thanks for reply, but I don't get the point of view :(

what it means: support region in frequency = 0 for sin & cos?

Re: short time Fourier transform

The support is the set of times such that the function takes nonzero values . For a -function, the support is .

So what the statement above is saying is: note that even though a -function's support is a single number, its Fourier transform is completely unbounded.