Thread Subject: how to use IFFT to reconstruct signal in a specific region t in [a, b]?

Hi all,

Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
able to reconstruct f(t), for t in [0, T].

Now I want to ask is there a way to do another IFFT to reconstruct the
specific part f(t) for t in [T, 2T], without any waste of previous
calculations?

Basically, I want to ask, if it is possible to use IFFT to reconstruct to
any slot t in [a, b] in the time domain for signal f(t)?

Thanks a lot!

"Luna Moon" <lunamoonmoon@gmail.com> wrote in message
news:f9e4pe$klc$1@news.Stanford.EDU...
> Hi all,
>
> Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I
> was able to reconstruct f(t), for t in [0, T].
>
> Now I want to ask is there a way to do another IFFT to reconstruct the
> specific part f(t) for t in [T, 2T], without any waste of previous
> calculations?
>
> Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> any slot t in [a, b] in the time domain for signal f(t)?

The answer is: "it depends".

First of all, since you're doing an IFFT, the spectral information is given
at discrete frequencies, the time series must be periodic. And, since the
time sequence resulting is also discrete, the corresponding spectral
sequence is periodic.

So, once you've done the IFFT, you have generated one period of a periodic /
infinite time series. After that, you should be able to figure out the
values for any other time period ... but it's a bit of a trivial exercise
when you know it's periodic isn't it?

In your opening description, you left out an important step:
In doing the IFFT, you generate a time sequence in [0,T] but have not yet
reconstructed it on t (i.e. have not made it continuous which is usually
what "reconstruction" means).

Proper reconstruction might use a Dirichlet kernel (which is periodic) - so
once the reconstruction is done, you have the periodic f(t) for all t.

Fred

On Aug 8, 11:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
> Hi all,
>
> Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> able to reconstruct f(t), for t in [0, T].
>
> Now I want to ask is there a way to do another IFFT to reconstruct the
> specific part f(t) for t in [T, 2T], without any waste of previous
> calculations?
>
> Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> any slot t in [a, b] in the time domain for signal f(t)?
>
> Thanks a lot!

You can use the Discrete Fourier Transform (DFT) to do it.
By definition, the FFT is restricted to the "Fast" version of
the DFT.

By the way, did you realize that relating a signal via the DFT
or FFT implicitly assume periodicity in both time and frequency?
I can't understand your notation, but if my guess is correct you
will find that x[n] is periodic in N. In your notation somehow
you are using continuous time t, which is incorrect.

I hate to nitpick, but these points can be important.
Julius

On Aug 9, 5:43 am, julius <juli...@gmail.com> wrote:
> On Aug 8, 11:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>
> > Hi all,
>
> > Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> > able to reconstruct f(t), for t in [0, T].
>
> > Now I want to ask is there a way to do another IFFT to reconstruct the
> > specific part f(t) for t in [T, 2T], without any waste of previous
> > calculations?
>
> > Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> > any slot t in [a, b] in the time domain for signal f(t)?
>
> > Thanks a lot!
>
> You can use the Discrete Fourier Transform (DFT) to do it.
> By definition, the FFT is restricted to the "Fast" version of
> the DFT.
>
> By the way, did you realize that relating a signal via the DFT
> or FFT implicitly assume periodicity in both time and frequency?
> I can't understand your notation, but if my guess is correct you
> will find that x[n] is periodic in N. In your notation somehow
> you are using continuous time t, which is incorrect.

Doesn't an ordinary infinitely periodic and bandlimited
continuous function have a finite discrete spectrum F(w),
from which it is possible to completely reconstruct
f(t) in continuous time? (and approached by several
methods).

On Aug 9, 3:37 pm, "Ron N." <rhnlo...@yahoo.com> wrote:
> On Aug 9, 5:43 am, julius <juli...@gmail.com> wrote:
>
>
>
> > On Aug 8, 11:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>
> > > Hi all,
>
> > > Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> > > able to reconstruct f(t), for t in [0, T].
>
> > > Now I want to ask is there a way to do another IFFT to reconstruct the
> > > specific part f(t) for t in [T, 2T], without any waste of previous
> > > calculations?
>
> > > Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> > > any slot t in [a, b] in the time domain for signal f(t)?
>
> > > Thanks a lot!
>
> > You can use the Discrete Fourier Transform (DFT) to do it.
> > By definition, the FFT is restricted to the "Fast" version of
> > the DFT.
>
> > By the way, did you realize that relating a signal via the DFT
> > or FFT implicitly assume periodicity in both time and frequency?
> > I can't understand your notation, but if my guess is correct you
> > will find that x[n] is periodic in N. In your notation somehow
> > you are using continuous time t, which is incorrect.
>
> Doesn't an ordinary infinitely periodic and bandlimited
> continuous function have a finite discrete spectrum F(w),
> from which it is possible to completely reconstruct
> f(t) in continuous time? (and approached by several
> methods).

I know that, but the author specifically said "iFFT". Either
the person is wrong in saying "iFFT" instead of "Fourier
series" or in using "t" versus "n". Unless there is a "fast"
Fourier series computation in continuous-time that has
been invented ...

On Aug 9, 2:34 pm, julius <juli...@gmail.com> wrote:
> On Aug 9, 3:37 pm, "Ron N." <rhnlo...@yahoo.com> wrote:
>
>
>
> > On Aug 9, 5:43 am, julius <juli...@gmail.com> wrote:
>
> > > On Aug 8, 11:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>
> > > > Hi all,
>
> > > > Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> > > > able to reconstruct f(t), for t in [0, T].
>
> > > > Now I want to ask is there a way to do another IFFT to reconstruct the
> > > > specific part f(t) for t in [T, 2T], without any waste of previous
> > > > calculations?
>
> > > > Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> > > > any slot t in [a, b] in the time domain for signal f(t)?
>
> > > > Thanks a lot!
>
> > > You can use the Discrete Fourier Transform (DFT) to do it.
> > > By definition, the FFT is restricted to the "Fast" version of
> > > the DFT.
>
> > > By the way, did you realize that relating a signal via the DFT
> > > or FFT implicitly assume periodicity in both time and frequency?
> > > I can't understand your notation, but if my guess is correct you
> > > will find that x[n] is periodic in N. In your notation somehow
> > > you are using continuous time t, which is incorrect.
>
> > Doesn't an ordinary infinitely periodic and bandlimited
> > continuous function have a finite discrete spectrum F(w),
> > from which it is possible to completely reconstruct
> > f(t) in continuous time? (and approached by several
> > methods).
>
> I know that, but the author specifically said "iFFT". Either
> the person is wrong in saying "iFFT" instead of "Fourier
> series" or in using "t" versus "n". Unless there is a "fast"
> Fourier series computation in continuous-time that has
> been invented ...

Yes, but a discrete iFFT can be used as part of method
to approximately (re)construct a continuous time function,
given some assumptions, as per above. Might not be
the most direct or efficient method... or what the OP
meant as opposed to what the OP wrote (or what the
homework question asked :)

On Aug 9, 12:19 am, "Luna Moon" <lunamoonm...@gmail.com> wrote:
> Hi all,
>
> Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> able to reconstruct f(t), for t in [0, T].
>
> Now I want to ask is there a way to do another IFFT to reconstruct the
> specific part f(t) for t in [T, 2T], without any waste of previous
> calculations?
>
> Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> any slot t in [a, b] in the time domain for signal f(t)?
>
> Thanks a lot!

You can use Chirp Z Transform methods to evaluate portions of either
the frequency or time domain. The CZT is discussed in the O&S books.

Cheers,
David

On Aug 9, 8:43 am, julius <juli...@gmail.com> wrote:
> On Aug 8, 11:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>
> > Hi all,
>
> > Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> > able to reconstruct f(t), for t in [0, T].
>
> > Now I want to ask is there a way to do another IFFT to reconstruct the
> > specific part f(t) for t in [T, 2T], without any waste of previous
> > calculations?
>
> > Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> > any slot t in [a, b] in the time domain for signal f(t)?
>
> > Thanks a lot!
>
> You can use the Discrete Fourier Transform (DFT) to do it.
> By definition, the FFT is restricted to the "Fast" version of
> the DFT.
>
> By the way, did you realize that relating a signal via the DFT
> or FFT implicitly assume periodicity in both time and frequency?
> I can't understand your notation, but if my guess is correct you
> will find that x[n] is periodic in N. In your notation somehow
> you are using continuous time t, which is incorrect.
>
> I hate to nitpick, but these points can be important.
> Julius

thanks! Of course I realize that DFT/FFT assumes the signal is
periodic. My question is related to the window of one such period. Yes
DFT/FFT has a focal window, and everything outside this window is
assumed to be periodic extension of the content within this window.
But in a reconstruction of time-domain signal from spectrum using
Inverse FFT/DFT, what is the default focal window? And how do we shift
the focal window? Eventually I want to be able to slide the window
along all the time-domain signal and focus on one part of the signal
at a time.

How to do that? Thanks a lot!

On Aug 10, 9:24 am, dspg...@netscape.net wrote:
> On Aug 9, 12:19 am, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>
> > Hi all,
>
> > Let's say by doing IFFT on F(v), which is the spectrum of signal f(t), I was
> > able to reconstruct f(t), for t in [0, T].
>
> > Now I want to ask is there a way to do another IFFT to reconstruct the
> > specific part f(t) for t in [T, 2T], without any waste of previous
> > calculations?
>
> > Basically, I want to ask, if it is possible to use IFFT to reconstruct to
> > any slot t in [a, b] in the time domain for signal f(t)?
>
> > Thanks a lot!
>
> You can use Chirp Z Transform methods to evaluate portions of either
> the frequency or time domain. The CZT is discussed in the O&S books.
>
> Cheers,
> David

Thanks David, is CZT for the following usage?
---------------------
I knew that DFT/FFT assumes the signal is periodic. My question is
related to the window of one such period. Yes DFT/FFT has a focal
window, and everything outside this window is assumed to be periodic
extension of the content within this window. But in a reconstruction
of time-domain signal from spectrum using Inverse FFT/DFT, what is the
default focal window? And how do we shift the focal window? Eventually
I want to be able to slide the window along all the time-domain signal
and focus on one part of the signal at a time.

How to do that? Thanks a lot!