Hi all, I have the following question regarding the relation between DFT and Foureir Transform. 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). Also, for this sequence of signal, I have an ordinary Foureir Transform F2(v), I guess it's called DTFT. I plan to sample the F2(v) to obtain the discrete version of the F2(v) and call it F3(v). 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? Thanks!
DFT the same as sampled Foureir transform?
Started by ●May 24, 2007
Reply by ●May 24, 20072007-05-24
"Mike" <meatheadIV@gmail.com> wrote in message news:f33fqk$2f8$1@news.Stanford.EDU...> Hi all, > > I have the following question regarding the relation between DFT and > Foureir Transform. > > 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). > > Also, for this sequence of signal, I have an ordinary Foureir Transform > F2(v), I guess it's called DTFT. > > I plan to sample the F2(v) to obtain the discrete version of the F2(v) and > call it F3(v). > > 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? > > Thanks!I have this diagram below which I think will help answer your question. http://12000.org/my_notes/transforms/index.htm Nasser
Reply by ●May 24, 20072007-05-24
Mike wrote:> Hi all, > > I have the following question regarding the relation between DFT and Foureir > Transform.Your misspelling is too consistent to by a typo. The guy's name was Fourier!> 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).You are mixing up all kinds of objects here. For infinite length sequences that are square-summable and represent evenly sampled bandlimited finite-energy signals, you can use the DTFT (if it converges), or the z-transform. For periodic, discrete and infinitely long sequences the z-transform doesn't converge, and you need the DFS (discrete Fourier series). For finite length vectors, you can use the DFT (discrete Fourier transform). The DFS and the DFT are closely related. There are infinitely many distinct sequences x[n] with DTFT X(w) where the inverse DFT y of the unifomly sampled DTFT, namely y = IDFT{ X(2 pi k/N) }, k=0,1,...,N-1, are all equal. For some of them, you have y[n] = x[n], n=0,1,...,N-1, for others you don't. Did this help? Regards, Andor
Reply by ●May 24, 20072007-05-24
Andor wrote:> Mike wrote: >> Hi all, >> >> I have the following question regarding the relation between DFT and Foureir >> Transform. > > Your misspelling is too consistent to be a typo. The guy's name was > Fourier!And it's pronounced Foo-ree-yay or Foor-yay). Maybe that will help. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●May 24, 20072007-05-24
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
Reply by ●May 24, 20072007-05-24
"Rune Allnor" <allnor@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? Thanks again!
Reply by ●May 24, 20072007-05-24
"Rune Allnor" <allnor@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? Thanks again!
Reply by ●May 24, 20072007-05-24
"Rune Allnor" <allnor@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? Thanks again!
Reply by ●May 24, 20072007-05-24
"Rune Allnor" <allnor@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? Thanks again!
Reply by ●May 24, 20072007-05-24
"Rune Allnor" <allnor@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? Thanks again!
