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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 10:42:15 -0700
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"Randy Yates" <yates@ieee.org> wrote in message 
news:m3lkferv9e.fsf@ieee.org...
> Randy Yates <yates@ieee.org> writes:
>
>> "Mike" <meatheadIV@gmail.com> writes:
>>> [...]
>>> 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?
>>
>> When x(t) is periodic with period n*T.
>
> PS: You probably meant to label the sequences x0, x1, ..., x(n-1), making
> n samples.
> -- 
> %  Randy Yates                  % "Remember the good old 1980's, when
> %% Fuquay-Varina, NC            %  things were so uncomplicated?"
> %%% 919-577-9882                % 'Ticket To The Moon'
> %%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
> http://home.earthlink.net/~yatescr

Thanks, Randy! You are right!