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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 10:20:44 -0700
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"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:1180026933.108924.307690@u30g2000hsc.googlegroups.com...
> On 24 May, 18:56, "Mike" <meathea...@gmail.com> wrote:
>> "Rune Allnor" <all...@tele.ntnu.no> wrote in message
>>
>> news:1180021783.330512.237760@w5g2000hsg.googlegroups.com...
>>
>>
>>
>>
>>
>> > On 24 May, 09:45, "Mike" <meathea...@gmail.com> wrote:
>> >> Hi all,
>>
>> >> I have the following question regarding the relation between DFT and
>> >> Foureir
>> >> Transform.
>>
>> > Fourier. After Jean Baptiste Joseph Fourier. Read his biography
>>
>> >http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Fourier.html
>>
>> > and maybe you get enough respect for him to spell his name correctly.
>>
>> >> 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).
>>
>> > Wrong. The DFT is not defined for a sequence of "possibly
>> > infinite length." The DFT is defined for discrete-time sequences
>> > of *finite* length.
>>
>> >> Also, for this sequence of signal,
>>
>> > What is a "sequence of signal"?
>>
>> >> I have an ordinary Foureir Transform
>> >> F2(v), I guess it's called DTFT.
>>
>> > No, you don't. I have no idea what a "sequence of signal" is.
>> > The Dirscrete-Time Fourier transform is defined for a discrete-
>> > time sequence of *infinite* length.
>>
>> >> I plan to sample the F2(v) to obtain the discrete version of the F2(v)
>> >> and
>> >> call it F3(v).
>>
>> > No need to do that, the discrete-time signals are already
>> > "sampled". Sampling is a way to convert from a contionuous-time
>> > signal to a discrete-time signal. This can be done for signals
>> > of either finite of (formally) infinite duration in time.
>>
>> >> My question is:
>>
>> >> Under what condition and for what kind of signal x's do the DFT F1(v) 
>> >> and
>> >> sampled version of ordinary FT F3(v) equate? I want F1(v) and F3(v) to 
>> >> be
>> >> exactly the same... what conditions shall I impose?
>>
>> > There is an answer to such questions. Not the one you expect
>> > or will be happy to hear, but an answer to questions such as
>> > yours exists. Now, I took very great care to avoid "your
>> > question" in the past sentence, because you don't have
>> > the necessary basis to formulate the proper question.
>> > Before asking again, take your time to read up on, and
>> > contemplate, the different variations of the Fourier transform.
>>
>> > You will have four cases to consider:
>>
>> > 1) Countinuos time, infinite duration
>> > 2) Continuous time, finite duration
>> > 3) Discrete time, infinite duration
>> > 4) Discrete time, finite duration
>>
>> > Once you have done that, you will be able to formulate
>> > a question which makes sense and, consequently, can be
>> > answered in a meaningful way.
>>
>> > Rune
>>
>> Thanks Rune! 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?
>
> IFF your infinitely long sequence is truly periodic AND
> you happen to truncate it so that you have exactly one
> period worth of data, then -- and only then -- your DFT
> and your DTFT are equal.
>
> There are only three problems with this:
>
> 1) Real-life data series from infinite-duration processes
>   are never truly priodic.
> 2) Even if they are *almost* periodic, you can't expect
>   to sample exactly one period's worth of data.
> 3) You are stuck with the DFT as a computational tool
>   anyway, since it is highly impragtical to implement
>   infinite summations and sample infinite amounts
>   of data.
>
> Rune
>

Thanks Rune! The original signal x(t) is not periodic. I guess my next 
question is:

How to handle such a situation and get an approximation error that is as 
small as possible?