Re: Confused about DFT and Fourier Series and Fourier Transform?

From: Gordon Sande (g.sande_at_worldnet.att.net)
Date: 11/15/04 

Date: Mon, 15 Nov 2004 13:49:00 GMT

Brad Griffis wrote:
> Kiki,
>
> The DTFT takes a discrete time domain signal and gives you a continuous,
> periodic frequency domain signal.
>
> The various transforms can be summarized as follows:
>
> CTFT: continuous <--> continuous
> FS: continuous <--> discrete
> DTFT: discrete <--> continuous
> DFT: discrete <--> discrete
>

Writing things down as it it were always time and frequency there are
four kinds of FTs. Two have unbounded time and two have periodic time
as unbounded and periodic are a yes/no pair. Also two have continuous
time and two have discrete time which is another yes/no pair. So you
get the list above which should also show whether time is unbounded or
periodic. The real interest however is in the kind of frequencies that
go with the various kinds of time. If time is periodic then the
frequencies can only take on discrete values and if the time is
unbounded then the frequencies can take continuous values. If the time
is discrete then the frequencies will be periodic but if time is
continuous then the frequencies can be unbounded. So the complete
table would be

               TIME FREQUENCY
CTFT: unbounded continuous <--> continuous unbounded
FS: periodic continuous <--> discrete unbounded
DTFT: unbounded discrete <--> continuous periodic
DFT: periodic discrete <--> discrete periodic

CTFT is the classical Fourier Transform.
FS, or Fourier Series, is the Fourier Transform of rotation angles.
DTFT, or Fourier Sequences, is the Fourier Transform of sampled time.
DFT is the Discrete Fourier Transform of numerical computation.

The names Fourier Series and Fourier Sequences are not so standard
that they can assumed known by everyone. Usually it is just FT for
CTFT which has been used because of the lack of a good name for
Fourier Sequences.

Notice that unbounded and continuous go together just as periodic
and discrete go together whether time or frequency.

If you multiply by a Dirac Comb then you get discrete time and if
you convolve with a Dirac Comb then you get periodic time. If you
do both then your get discrete periodic time. A Dirac Comb is either
a good heuristic or a very subtle discussion of distribution theory
depending upon whether is shows up in Inroduction to Engineering
Calculus or Advanced Measure Theory.

> I don't have a good reference for you concerning conversions among these.
> They all are more or less the same (e.g. sinc function time is rectangle in
> frequency, etc.). The only thing "tricky" is probably the scale factor
> since all of these are orthonormal transforms and maintain the same signal
> power between time and frequency.
>
> Brad
>
>
> "kiki" <lunaliu3@yahoo.com> wrote in message
> news:cn9qg1$qbd$1@news.Stanford.EDU...
>
>>Dear all,
>>
>>I am confused by the four transforms in Signal & Systems...
>>
>>The Continuous Time Fourier Transform(CTFT) is most understandable; DFT
>>and Fourier Series alone are individually recoginizable and
>>understandable... Not sure about how does DTFT kick in...
>>
>>Anyway, remembering all of these four transforms' formulas are already
>>very headache... very easily got confuse one with another...
>>
>>Even worse, homework and test problems often asks for conversion among
>>these four transforms...
>>
>>Given a signal's CTFT, how do you get DFT for N-point? How does the DFT
>>compare to the Fourier Series(looks to me they are all discrete spectrum,
>>etc.) so on and so forth, how are they related and how to get one from
>>another?
>>
>>Are there any good resources that clearly demonstrate the relationship and
>>conversion among these 4 transforms?
>>
>>Thanks a lot,
>>
>>
>
>
>

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