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

Started by Luna Moon August 9, 2007
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, 9:19 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
> 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?
This sounds like a useful framework for asking homework questions about data dependency assumptions/requirements.
> 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)?
An IFFT is just a faster implementation of an IDFT. You can use the definition equations of a IDFT/DFT to compute the value for any single point or bin in f(t) or F(v). However, after doing this O(log(n)) times, it would probably be faster to just to do an entire IFFT and select subsets. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
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!