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From: "Mike" <meatheadIV@gmail.com>
Newsgroups: comp.soft-sys.matlab,comp.dsp
Subject: Re: DFT the same as sampled Foureir transform?
Date: Thu, 24 May 2007 09:56:38 -0700
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"Andor" <andor.bariska@gmail.com> wrote in message 
news:1179998515.753120.45840@h2g2000hsg.googlegroups.com...
> Mike wrote:
>> Hi all,
>>
>> I have the following question regarding the relation between DFT and 
>> Foureir
>> Transform.
>
> Your misspelling is too consistent to by a typo. The guy's name was
> Fourier!
>
>
>> Suppose I have a sequence of discrete time signal x0, x1, x2, ... xn, ...
>> (possibly infinite length), uniformly spaced in time, with spacing T; 
>> that's
>> to say, x0 is the signal value at time 0, x1 is the signal value at time
>> 1*T, x2 is the signal value at time 2*T, ...
>>
>> and the DFT of this sequence is F1(v).
>
> You are mixing up all kinds of objects here.
>
> For infinite length sequences that are square-summable and represent
> evenly sampled bandlimited finite-energy signals, you can use the DTFT
> (if it converges), or the z-transform. For periodic, discrete and
> infinitely long sequences the z-transform doesn't converge, and you
> need the DFS (discrete Fourier series).
>
> For finite length vectors, you can use the DFT (discrete Fourier
> transform). The DFS and the DFT are closely related.
>
> There are infinitely many distinct sequences x[n] with DTFT X(w) where
> the inverse DFT y of the unifomly sampled DTFT, namely
>
> y = IDFT{ X(2 pi k/N) }, k=0,1,...,N-1,
>
> are all equal. For some of them, you have y[n] = x[n], n=0,1,...,N-1,
> for others you don't.
>
> Did this help?
>
> Regards,
> Andor
>

Thanks Andor! It's midnight so I was too sleepy. Yes, it should be 
"Fourier". Thanks for pointing it out!

I agree my question is not well-posed. Here is a reformulation:

Given a continuous time signal x(t), infinitely long. Sample it to obtain 
discrete time sequence x0, x1, x2, ..., xn, ..., infinitely long, with 
uniform samples spaced at T apart.

Now I do two things:

(1) Truncate the above sequence to make it finite, x0, x1, ..., xn, and take 
the DFT of the truncated sequence. Call the DFT F1(v). (Capitalized letters 
denote spectrum domain)

(2) Without truncation, taking the DTFT of the infinitely long sequence x0, 
x1, ..., xn, .... Call the DTFT F2(v). And then take one period of F2(v), 
since it is periodic, and then sample F2(v) in the frequency domain to 
discretize it. Call the result F3(v), which is the discretized version of 
the one period of F2(v).

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

Both (1) and (2) yield vectors of length n in the spectrum domain, 
representing the discretized version of the spectrum.

My question is: under what conditions do these two vectors of discretized 
spectrum equate?

Thanks again!