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,

# Confused about DFT and Fourier Series and Fourier Transform?

Started by ●November 15, 2004

Reply by ●November 15, 20042004-11-15

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 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, > >

Reply by ●November 15, 20042004-11-15

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, >> >> > > >

Reply by ●November 15, 20042004-11-15

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, >>> >>> >> >> >>

Reply by ●November 15, 20042004-11-15

Folks, I have found these issues to be clearly discussed in: 1. "Digital Spectral Analysis with Applications", by S. Lawrence Marple, Prentice-Hall, 1987, Chapter 2. 2. "Digital Signal Processing, Principles, Algorithms, and Applications, 3'rd Edition" by John Proakis and Dimitris Manolakis, Prentice-Hall, 1996, Chapters 4 and 5. The Marple book has the shorter explanation and covers the scale factors relating the discrete forms to the continuous. Unfortunately, it is out of print (except perhaps on ebay!). The Proakis book spreads out the coverage of this material, and although somewhat pricey (I think), discusses many other very useful topics. Both books are well-worth having on your shelf. 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 > > 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, >> >> > > >

Reply by ●November 20, 20042004-11-20

"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... >I know several people have said out the point. Here's another perspective. Depending upon a signal being continuos/discrete in time and periodic/aperiodic in nature, you have CTFS/CTFT/DTFS/DTFT which represents the corresponding frequency characteristics (thumb rule: Series<=> periodicity & transform <=> aperiodic). Of specific importance is DTFT which is a continuous function of frequency. Inorder to make it available for practical processing, one does sampling of DTFT which is nothing but DFT. -KK

Reply by ●November 23, 20042004-11-23

Check http://www.ee.vt.edu/~ee4624ss/week4.pdf "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,