Re: Confused about DFT and Fourier Series and Fourier Transform?
From: Stan Pawlukiewicz (spam_at_spam.mitre.org)
Date: 11/15/04
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Date: Mon, 15 Nov 2004 09:52:34 -0500
Gordon Sande wrote:
You can also view the DFT as a filter bank, as the Z transform evaluated
at a set of points on the unit circle, or an orthogonal matrix operation.
>
>
> 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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