How to zoom into a certain part of FFT?

As mentioned above, you are doing a band pass. The f_hat as a result
of a band pass need not 'look' much like f(t), it depends on how wide
your band is - make it tooo narrow, and all you will have is one
sinusoidal signal. Make it very wide and you get something that looks
more like f(t). This does not give you a higher 'resolution' in time.
Assuming you have the same number of samples (since in FFT, IFFT, time
doesn't really make sense. It is always in terms of samples), your
f_hat will have the same length as f i.e. a=c, b=d.
You can't better the resolution actually, since you already got your
samples, as mentioned above. you only have information of f(t) at the
samples. information about f(t) at every other instant of time is
lost. If you want to interpolate, there is no need to go to the
frequency domain

HTH
Srikanth

On Jun 27, 8:44 pm, "Vista" <a...@gmai.com> wrote:
> When I truncate/extract out F(w) for w in [-B, B] and use step
> size deltaB to sample it and then do IFFT, what is the portion
> of f(t) I see?  Say f(t) for t in [a, b]. What are a and b?

-inf to +inf, folded up into the width of your IFFT.

If you multiply by a rectangular [-B, B] filter response in the
frequency domain, that is the same as convolution with a Sinc
function in the time domain.  The narrower the rectangle [-B, B],
the wider the main lobe of your Sinc.  The wider the main lobe
of your Sinc, the "blurrier" and more spread out the result.
A finite width IFFT will also add the wrapped tails of the Sinc
convolution to your result, since a Sinc has infinite extent,
and the tails of the periodic images of the Sinc will also be
convolved into your IFFT aperture as if they were wrapped
around or folded.

As mentioned by a previous poster, Oli, if f(t) is sufficiently
time-limited, then F(w) might be smooth enough to be sampled
without aliasing given a small enough deltaB.  If not...

> Now suppose I find there is some fine structure in [c, d],
> which is shown from the visual display of f_hat(t), t in
> [a, b]. And a<c<d<b.

Given that [a, b] is -inf to +inf, then this will be true
for any [c, d]

> How to do IFFT targeting at f(t) on [c, d] with higher resolution?

You can do a sparse matrix DFT using a wider [-B, B] rectangular
window in the frequency domain.  But you can't increase deltaB
past some limit without aliasing.

Posted to comp.dsp only.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

But I can take more samples on the F(v) and refine FFT in order to zoom in , 
am I right?

"Srikanth" <skt@xdtech.com> wrote in message 
news:1183005029.716692.222720@i13g2000prf.googlegroups.com...
> As mentioned above, you are doing a band pass. The f_hat as a result
> of a band pass need not 'look' much like f(t), it depends on how wide
> your band is - make it tooo narrow, and all you will have is one
> sinusoidal signal. Make it very wide and you get something that looks
> more like f(t). This does not give you a higher 'resolution' in time.
> Assuming you have the same number of samples (since in FFT, IFFT, time
> doesn't really make sense. It is always in terms of samples), your
> f_hat will have the same length as f i.e. a=c, b=d.
> You can't better the resolution actually, since you already got your
> samples, as mentioned above. you only have information of f(t) at the
> samples. information about f(t) at every other instant of time is
> lost. If you want to interpolate, there is no need to go to the
> frequency domain
>
> HTH
> Srikanth
> 

<snip, terminolgy?...

Maybe the OP is trying to describe Zoom-FFT?  Maybe not, but worth a
punt.  This sort of well-known stuff: http://www.numerix-dsp.com/zoomfft.html