Thread Subject: How to zoom into a certain part of FFT?

Hi all,

Suppose I have a signal f(t), t is in [0, +infinity).

And I have its spectrum F(w).

Let's say I found out that its main spectrum has 99.9% in [-B, B].

So I truncate/extract out the portion of F(w), for w in [-B, B], and
discretized the interval into small grids with step size deltaB.

And I then do the inverse FFT on the above samples of F(w), let's call the
inverse FFT reconstruction f_hat.

Which part of f(t) does this inverse FFT f_hat represent?

If I want to zoom into a certain part of f(t), how can I use inverse FFT to
do that?

Let's say I only need [a, b] where 0<a<b< infinity, in the time domain,

I only need to visualize the f(t), for t in [a, b],

how do I sample F(w) and do the inverse FFT?

-------------------------

More generally, I am actually thinking of desing a "spectrum analyzer" with
zoom-in feactures:

as you can see, first do a coarse level inverse FFT to gain a big picture of
the f(t) curve, and then allow user to zoom into a particular part of the
f(t) curve and display it with higher resolution of IFFT.

How to do all of these?

Thanks a lot!

On Jun 26, 8:06 pm, "Vista" <a...@gmai.com> wrote:
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-B, B], and
> discretized the interval into small grids with step size deltaB.
>
> And I then do the inverse FFT on the above samples of F(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?

In order to meaningfully sample F(w), you need to know
something about its smoothness, else your sample points
might fall into deep gaps in F(w), or some-such.

There may also be a theorem saying that the width of
some portion of a waveform in one domain is inversely
proportional to its width in the other domain. The
smaller the wiggles in f(t) you want to see, the wider
you need to make [-B, B], which makes sense, since the
higher frequency components in F(w) are what produces
the densest small wiggles in f(t).

If you want a tiny enough window from an ifft (much
less than log(n) points), you might be better off
directly calculating a segment of the dft.


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

On Jun 27, 3:49 pm, "Ron N." <rhnlo...@yahoo.com> wrote:
> On Jun 26, 8:06 pm, "Vista" <a...@gmai.com> wrote:
>
> > Suppose I have a signal f(t), t is in [0, +infinity).
>
> > And I have its spectrum F(w).
>
> > Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> > So I truncate/extract out the portion of F(w), for w in [-B, B], and
> > discretized the interval into small grids with step size deltaB.
>
> > And I then do the inverse FFT on the above samples of F(w), let's call the
> > inverse FFT reconstruction f_hat.
>
> > Which part of f(t) does this inverse FFT f_hat represent?
>
> In order to meaningfully sample F(w), you need to know
> something about its smoothness, else your sample points
> might fall into deep gaps in F(w), or some-such.
>
> There may also be a theorem saying that the width of
> some portion of a waveform in one domain is inversely
> proportional to its width in the other domain. The
> smaller the wiggles in f(t) you want to see, the wider
> you need to make [-B, B], which makes sense, since the
> higher frequency components in F(w) are what produces
> the densest small wiggles in f(t).
>
> If you want a tiny enough window from an ifft (much
> less than log(n) points), you might be better off
> directly calculating a segment of the dft.
>
> IMHO. YMMV.
> --
> rhn A.T nicholson d.0.t C-o-M

What you're doing is "band-pass filtering".
You'll need to take the sqrt because a spectrum is energy, but you
want amplitude for the ifft.
But the spectrum contains no information on phase, so what are you
going to do about that?
Also, you must be aware that you'll have problems with Gibb's
phenomenon which causes "ringing" in the inverse Fourier transform.
To avoid that you'll need to smoothly taper the spectral cutoffs - I
use a tanh function for this.

Vista wrote:
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).

> Which part of f(t) does this inverse FFT f_hat represent?

The part with frequencies inside the bandpass you selected.

> If I want to zoom into a certain part of f(t), how can I use inverse FFT to
> do that?

If you want to zoom in on f(t), did you ever think of just zooming in
on f(t)?
No FT required! Duh!

Methinks you have no idea what you are doing.

when you say you want to zoom in on certain parts of f(t), what do you
mean? The data is already in form of samples, you really can't get new
data, since all the information is already in the samples. For example
if your samples represent the signal at T,2T etc, there is no way you
can exactly know what happened at 1.5 T - that data is lost. For your
application, the easiest way would be to display every 10th sample,
for example, to start with. Then, if the user selects a region,
display every 5th sample in that region. If he zooms in further,
display every sample. If he goes closer than this, you need to do some
interpolation.

On Jun 27, 4:06 am, "Vista" <a...@gmai.com> wrote:
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-B, B], and
> discretized the interval into small grids with step size deltaB.
>
> And I then do the inverse FFT on the above samples of F(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?

Just as sampling a non-bandlimited function in the time domain causes
time-domain aliasing, the dual occurs in your scenario. You are
sampling a non-time-limited function in the frequency domain, which
will cause frequency-domain aliasing.


--
Oli

On Jun 27, 7:14 pm, Oli Charlesworth <c...@olifilth.co.uk> wrote:
> On Jun 27, 4:06 am, "Vista" <a...@gmai.com> wrote:
>
> > Hi all,
>
> > Suppose I have a signal f(t), t is in [0, +infinity).
>
> > And I have its spectrum F(w).
>
> > Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> > So I truncate/extract out the portion of F(w), for w in [-B, B], and
> > discretized the interval into small grids with step size deltaB.
>
> > And I then do the inverse FFT on the above samples of F(w), let's call the
> > inverse FFT reconstruction f_hat.
>
> > Which part of f(t) does this inverse FFT f_hat represent?
>
> Just as sampling a non-bandlimited function in the time domain causes
> time-domain aliasing, the dual occurs in your scenario. You are
> sampling a non-time-limited function in the frequency domain, which
> will cause frequency-domain aliasing.
>
> --
> Oli

It's very interesting.
We've all got different ideas about what Vista is trying to do.
The only thing we all agree on is that he/she is going about it the
wrong way.
Perhaps Vista is a troll, deliberately posing an inane question, then
sitting back to watch the fun.

>>
>> > Suppose I have a signal f(t), t is in [0, +infinity).
>>

Your problem is completely academic. you don't need to solve it, you
just don't have time. Even if you get a solution - with the signal
going all the way to infinity you will only have just started your
data acquisition by the time you die of old age, and it will still be
going strong when the sun goes nova. Sit back and enjoy a beer is
what I would do in your place;-)

Dave Robinson

On Jun 26, 11:06 pm, "Vista" <a...@gmai.com> wrote:
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-B, B], and
> discretized the interval into small grids with step size deltaB.
>
> And I then do the inverse FFT on the above samples of F(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?
>

It depends on what you mean by "focus". If you mean "set all the
unwanted spectral components to zero" and then you IFFT the result,
f_hat(nT) represents a filtered version of the original f(nT) over the
entire interval represented by the samples of f(nT).

If, by "focus", you mean "discard the unwanted spectral components"
and then you IFFT the result (containing a smaller number of spectral
components), f_hat(nT_prime) represents a decimated and filtered
version of the original f(nT), still over the entire interval
represented by the samples of f(nT).

Hi folks,

This is not a joke post. I really have this question in my mind for long
time.

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?

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.

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

And so on and so forth?

Thanks!


"Vista" <abc@gmai.com> wrote in message
news:f5sk5g$qmm$1@news.Stanford.EDU...
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-B, B], and
> discretized the interval into small grids with step size deltaB.
>
> And I then do the inverse FFT on the above samples of F(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?
>
> If I want to zoom into a certain part of f(t), how can I use inverse FFT
> to do that?
>
> Let's say I only need [a, b] where 0<a<b< infinity, in the time domain,
>
> I only need to visualize the f(t), for t in [a, b],
>
> how do I sample F(w) and do the inverse FFT?
>
> -------------------------
>
> More generally, I am actually thinking of desing a "spectrum analyzer"
> with zoom-in feactures:
>
> as you can see, first do a coarse level inverse FFT to gain a big picture
> of the f(t) curve, and then allow user to zoom into a particular part of
> the f(t) curve and display it with higher resolution of IFFT.
>
> How to do all of these?
>
> Thanks a lot!
>

<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

Vista wrote:
>
>
> Hi folks,
>
> This is not a joke post. I really have this question in my mind for
> long
> time.
>
> 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?
>
> 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.
>
> How to do IFFT targeting at f(t) on [c, d] with higher resolution?
>
> And so on and so forth?
>
> Thanks!
>
>
If you want to know relations between F(w) and f(t), here it is:

The bandwidth of F(w), [0, B] determines the sample rate in f(t),
i.e. 2B (in Hz) = 1/T (in second); The numer of samples of F(w) in [0
B] equals the number of samples of f(t) in [0 tf] after IFFT; Then,
tf, the time range of f(t) can be determined from T and number of
samples.

hth
Yi

"Yi Cao" <y.cao@cranfield.ac.uk> wrote in message
news:ef5bd15.13@webcrossing.raydaftYaTP...
> Vista wrote:
>>
>>
>> Hi folks,
>>
>> This is not a joke post. I really have this question in my mind for
>> long
>> time.
>>
>> 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?
>>
>> 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.
>>
>> How to do IFFT targeting at f(t) on [c, d] with higher resolution?
>>
>> And so on and so forth?
>>
>> Thanks!
>>
>>
> If you want to know relations between F(w) and f(t), here it is:
>
> The bandwidth of F(w), [0, B] determines the sample rate in f(t),
> i.e. 2B (in Hz) = 1/T (in second); The numer of samples of F(w) in [0
> B] equals the number of samples of f(t) in [0 tf] after IFFT; Then,
> tf, the time range of f(t) can be determined from T and number of
> samples.
>
> hth
> Yi

Thanks. If you want [c, d] in the time domain, not [0, tf], how to do it?

Btw, I am not sure how the "0" (the left boundary) is determined?

Vista wrote:
>
>
>
> "Yi Cao" <y.cao@cranfield.ac.uk> wrote in message
> news:ef5bd15.13@webcrossing.raydaftYaTP...
>> Vista wrote:
>>>
>>>
>>> Hi folks,
>>>
>>> This is not a joke post. I really have this question in my
mind
> for
>>> long
>>> time.
>>>
>>> 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?
>>>
>>> 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.
>>>
>>> How to do IFFT targeting at f(t) on [c, d] with higher
> resolution?
>>>
>>> And so on and so forth?
>>>
>>> Thanks!
>>>
>>>
>> If you want to know relations between F(w) and f(t), here it
is:
>>
>> The bandwidth of F(w), [0, B] determines the sample rate in
f(t),
>> i.e. 2B (in Hz) = 1/T (in second); The numer of samples of F(w)
> in [0
>> B] equals the number of samples of f(t) in [0 tf] after IFFT;
> Then,
>> tf, the time range of f(t) can be determined from T and number
of
>> samples.
>>
>> hth
>> Yi
>
> Thanks. If you want [c, d] in the time domain, not [0, tf], how to
> do it?
>
> Btw, I am not sure how the "0" (the left boundary) is determined?
>
>
>
>

Depend on how you get F(w). If you get it from experiment, then c is
determined by your record starting time, but reference it as 0. If
you get F(w) from FFT of a time series f(t), then c is determined by
the original f(t) when you pick of a section [c, d] of f(t) to
perform FFT. Again, you can reference the starting time as 0. No way
you can determine c purly from F(w) without any extra information.

hth
Yi

Yopu can actually zoom using the spectrum scope provided in
matlab. Just the menu bars are hidden. To view the menu
bars type the following commands

>> set(0,'ShowHiddenHandles','On')
>> set(gcf,'menubar','figure')


Cheers,

Chris


"Vista" <abc@gmai.com> wrote in message
<f5sk5g$qmm$1@news.Stanford.EDU>...
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in
[-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-
B, B], and
> discretized the interval into small grids with step size
deltaB.
>
> And I then do the inverse FFT on the above samples of F
(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?
>
> If I want to zoom into a certain part of f(t), how can I
use inverse FFT to
> do that?
>
> Let's say I only need [a, b] where 0<a<b< infinity, in
the time domain,
>
> I only need to visualize the f(t), for t in [a, b],
>
> how do I sample F(w) and do the inverse FFT?
>
> -------------------------
>
> More generally, I am actually thinking of desing
a "spectrum analyzer" with
> zoom-in feactures:
>
> as you can see, first do a coarse level inverse FFT to
gain a big picture of
> the f(t) curve, and then allow user to zoom into a
particular part of the
> f(t) curve and display it with higher resolution of IFFT.
>
> How to do all of these?
>
> Thanks a lot!
>
>

Yopu can actually zoom using the spectrum scope provided in
matlab. Just the menu bars are hidden. To view the menu
bars type the following commands

>> set(0,'ShowHiddenHandles','On')
>> set(gcf,'menubar','figure')


Cheers,

Chris


"Vista" <abc@gmai.com> wrote in message
<f5sk5g$qmm$1@news.Stanford.EDU>...
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in
[-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-
B, B], and
> discretized the interval into small grids with step size
deltaB.
>
> And I then do the inverse FFT on the above samples of F
(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?
>
> If I want to zoom into a certain part of f(t), how can I
use inverse FFT to
> do that?
>
> Let's say I only need [a, b] where 0<a<b< infinity, in
the time domain,
>
> I only need to visualize the f(t), for t in [a, b],
>
> how do I sample F(w) and do the inverse FFT?
>
> -------------------------
>
> More generally, I am actually thinking of desing
a "spectrum analyzer" with
> zoom-in feactures:
>
> as you can see, first do a coarse level inverse FFT to
gain a big picture of
> the f(t) curve, and then allow user to zoom into a
particular part of the
> f(t) curve and display it with higher resolution of IFFT.
>
> How to do all of these?
>
> Thanks a lot!
>
>

Yopu can actually zoom using the spectrum scope provided in
matlab. Just the menu bars are hidden. To view the menu
bars type the following commands

>> set(0,'ShowHiddenHandles','On')
>> set(gcf,'menubar','figure')


Cheers,

Chris


"Vista" <abc@gmai.com> wrote in message
<f5sk5g$qmm$1@news.Stanford.EDU>...
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).
>
> And I have its spectrum F(w).
>
> Let's say I found out that its main spectrum has 99.9% in
[-B, B].
>
> So I truncate/extract out the portion of F(w), for w in [-
B, B], and
> discretized the interval into small grids with step size
deltaB.
>
> And I then do the inverse FFT on the above samples of F
(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?
>
> If I want to zoom into a certain part of f(t), how can I
use inverse FFT to
> do that?
>
> Let's say I only need [a, b] where 0<a<b< infinity, in
the time domain,
>
> I only need to visualize the f(t), for t in [a, b],
>
> how do I sample F(w) and do the inverse FFT?
>
> -------------------------
>
> More generally, I am actually thinking of desing
a "spectrum analyzer" with
> zoom-in feactures:
>
> as you can see, first do a coarse level inverse FFT to
gain a big picture of
> the f(t) curve, and then allow user to zoom into a
particular part of the
> f(t) curve and display it with higher resolution of IFFT.
>
> How to do all of these?
>
> Thanks a lot!
>
>

Corrected for the heinous crime of Top-Posting.

On Jun 27, 10:44 pm, "Vista" <a...@gmai.com> wrote:

> "Vista" <a...@gmai.com> wrote in message
> news:f5sk5g$qmm$1@news.Stanford.EDU...
>
> > Suppose I have a signal f(t), t is in [0, +infinity).

f(t) is then categorized as "causal"

> > And I have its spectrum F(w).

For causal time functions, there can be no singularities of
F(w = wr+j*wi) in the closed lower half complex plane wi <= 0.

> > Let's say I found out that its main spectrum has 99.9% in [-B, B].
>
> > So I truncate/extract out the portion of F(w), for w in [-B, B], and
> > discretized the interval into small grids with step size deltaB.
>
> > And I then do the inverse FFT on the above samples of F(w), let's call the
> > inverse FFT reconstruction f_hat.
>
> > Which part of f(t) does this inverse FFT f_hat represent?

A low pass filtered part. When 0 <= t < inf, think of f_hat as a
moving
window average of f(t). However, it is more complicated than that
because f_hat will be noncausal and nonzero in the negative time
interval -inf < t < 0.

> > If I want to zoom into a certain part of f(t), how can I use inverse FFT
> > to do that?
> > Let's say I only need [a, b] where 0<a<b< infinity, in the time domain,
>
> > I only need to visualize the f(t), for t in [a, b],
>
> > how do I sample F(w) and do the inverse FFT?

There is no prescibed way to do this. In general, the function
fab(t) = f(t) for t in [a,b] but zero elsewhere has a transform
Fab(w) with infinite support; i.e., there is no interval on (-
inf,inf)
where Fab(w) is identically zero. In addition there is no interval
on (-inf,inf) where abs(Fab(w)) is guaranteed to be small.

> > -------------------------
>
> > More generally, I am actually thinking of desing a "spectrum analyzer"
> > with zoom-in feactures:
>
> > as you can see, first do a coarse level inverse FFT to gain a big picture
> > of the f(t) curve, and then allow user to zoom into a particular part of
> > the f(t) curve and display it with higher resolution of IFFT.
>
> > How to do all of these?
>
> This is not a joke post. I really have this question in my mind for long
> time.
>
> 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?

Unfortunately, there is an inverse relationship between B and b-a.
The
extreme cases are B = 0 ==> b-a = inf and b-a = 0 ==> B = inf (in
particular,
impulse <--> sinc). Consequently, there is no finite frequency
interval that
directly corresponds to a finite time interval.

F(w) = FB(w) + FE(w), -inf < w < inf

FB(w) = F(w), |w| < B Bandlimited part of the spectrum
           = 0 , |w| > B
FE(w) = 0, |w| < B
           = F(w) , |w| > B

Inverse transforming

 f(t) = fB(t) + fE(t), 0 <= t < inf

where

FB -> fB, -inf < t < inf
FE -> fE, -inf < t < inf

and

fB(t)+fE(t) = 0 , -inf < t < 0.

1. FB(w) = 0, |w| > B ==> fB is not time limited
==> In general, there is no finite time interval [t0-d,t0+d] for
which fB(t) = 0 for |t-t0| > d.
2. Therefore fE = f-fB is also not time limited.
3. Statements 1 and 2 are true even if f is causal (i.e., f(t) = 0, t
< 0).
4. If f is causal, neither fB or fE are causal and fE = - fB for t <
0.
5. In intervals where |fB| >> |fE|, think of fB as a smooth version of
f
    and fE as the residual.

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

Fine structure in any interval can be resolved by increasing B.
Although this tends to increase the average of the ratio |fE^2| / |
fB^2|
over (-inf,inf), it is most significant in the intervals of interest
where
|fE^2| / |fB^2| is originally small.

Hope this helps.

Greg

"Vista" <abc@gmai.com> wrote in message
<f5sk5g$qmm$1@news.Stanford.EDU>...
> Hi all,
>
> Suppose I have a signal f(t), t is in [0, +infinity).

It is a shame that you only have half the signal - missing
-infinity <= t < 0. But seriously, you don't mention
sampling. You have to be aware of the works in the '70s at
Bell Labs of Nyquist and Shannon.
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

>
> And I have its spectrum F(w).

Given that your time series is infinite, you spectrum will
have infinite resolution.

>
> Let's say I found out that its main spectrum has 99.9% in
[-B, B].
>

This is low-pass filtering.

> So I truncate/extract out the portion of F(w), for w in
[-B, B], and
> discretized the interval into small grids with step size
deltaB.

Truncation of frequency spectra (or time series) is always a
BAD idea. You need to shape them with a window that reduces
the out of band components to zero but avoiding a steep
cut-off. Common windows are Hanning, Hamming and I like
Kaiser-Bessel.

>
> And I then do the inverse FFT on the above samples of
F(w), let's call the
> inverse FFT reconstruction f_hat.
>
> Which part of f(t) does this inverse FFT f_hat represent?

It represents all of f(t) - in your example (-B to B), just
low-pass filtered and and at a lower sampling frequency.

>
> If I want to zoom into a certain part of f(t), how can I
use inverse FFT to
> do that?

You don't.

> Let's say I only need [a, b] where 0<a<b< infinity, in the
time domain,
>
> I only need to visualize the f(t), for t in [a, b],
>
> how do I sample F(w) and do the inverse FFT?

You don't.
 
> -------------------------
>
> More generally, I am actually thinking of desing a
"spectrum analyzer" with
> zoom-in feactures:
>
> as you can see, first do a coarse level inverse FFT to
gain a big picture of
> the f(t) curve, and then allow user to zoom into a
particular part of the
> f(t) curve and display it with higher resolution of IFFT.
>
> How to do all of these?

Be aware that a single spectral line maps to all of time
(-infinity < t < infinity) and that a single point in time
maps to all frequencies (-infinity < f < infinity).

FFT analysis is based on a fixed number of regular time
points mapping to the same number of frequencies. This only
works when an event in time can be captured completely or
repeats itself and you capture exactly one repeat.

Outside of those conditions, windows (Hanning etc) are used
to limit the time series and to get a useful approximation
of the spectrum albeit with some 'smudging' and changes in
amplitude and/or energy.

In order to complete your FFT analyser, you should include a
flat-top window to allow an accurate estimate of the height
of a single frequency.

>
> Thanks a lot!
>
>
I suspect this reply might be too late to help with your
assignment, but I hope that it helps you understand why your
grade was so bad.