Sirs, While reading on frequency representation, I am a little confused with discrete fourier series (as opposed to discrete fourier transform) and its application. Can I get some inputs please. Thanks, manish _____________________________ Posted through www.DSPRelated.com

# discrete fourier series and transform

Started by ●July 29, 2013

Reply by ●July 29, 20132013-07-29

On 29.07.2013 10:28, manishp wrote:> Sirs, > > While reading on frequency representation, I am a little confused with > discrete fourier series (as opposed to discrete fourier transform) and its > application.The discrete Fourier transform transforms one vector into another. The output of a discrete Fourier transform is a vector whose elements are the coefficients in the corresponding Fourier series. As far as applications go: Oh well, look, you could also ask for applications of a screw driver. Manifold! Separation of a signal into its frequency components certainly.

Reply by ●July 29, 20132013-07-29

A discrete Fourier series gives the representation of a **periodic continuous-time signal** as a sum of sinusoids or complex exponentials. For example, x(t) = sum c_n exp(j 2pi nt/T) for all t, -infty < t < infty where T is the period of the signal x(t). The continuous-time signal is effectively represented by the sequence of complex numbers .... c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2}, ... This sequence might be finite in length (e.g. c_{n} = 0 for all n such that |n| > N) or extend out to plus/minus infinity. However, x(t), being periodic, extends all the way out to plus/minus infinity and for each and every one of those times, its value is given by the expression\ above. A discrete Fourier transform gives the representation of the elements of a **vector** of N numbers as a weighted sum of complex N-th roots of unity exp(j 2pi k/N). Thus, each entry in the vector (x[0], x[1], ... , x[N-1]) can be represented as x[k] = (1/N) sum from 0 to N-1 X[i]exp(j 2pi ik/N) where each exp(j 2pi ik/N) = exp(j 2pi m/N) is a complex N-th root of unity weighted by (1/N)X[i]. The vector (X[0], X[1], ... , X[N-1]) is called the discrete Fourier transform of the vector (x[0], x[1], ... , x[N-1]). Note that the discrete Fourier transform is of finite length just as (x[0], x[1], ... , x[N-1]) is. The rb-j's of this world will tell you that actually both x and X are actually one period of **sequences** of period N that extend out to plus/minus infinity, that is, for **any** integer M, x[M] = x[M mod N] and X[M] = X[M mod N] and the mathematics supports them, but lots of other folks will equally vehemently insist that this is sheer nonsense, and the value of x[N], say, has nothing to do with the values of x[0], x[1], ... , x[N-1]. Which side you choose to believe is up to you. Dilip Sarwate

Reply by ●July 29, 20132013-07-29

On 7/29/13 6:36 AM, dvsarwate wrote:> A discrete Fourier series gives the representation of > a **periodic continuous-time signal** as a sum of sinusoids > or complex exponentials. For example, > > x(t) = sum c_n exp(j 2pi nt/T) for all t, -infty< t< infty > > where T is the period of the signal x(t). The continuous-time > signal is effectively represented by the sequence of complex > numbers .... c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2}, ... > This sequence might be finite in length (e.g. c_{n} = 0 > for all n such that |n|> N) or extend out to plus/minus > infinity.but Dilip, i think the term we have for that is simply "Fourier series" or, in the case of c_n * e^(j*2*pi*n*t/T), we might call it the "complex Fourier series". at least in my copy of O&S, the term "Discrete Fourier Series" is about "a sequence ~x[n] that is periodic with period N so that ~x[n] = ~x[n+rN] for any integer value of r." (the tilde ~ is meant to be above the "x".) that's what O&S call the "Discrete Fourier Series". it's explicitly a mapping of one discrete and periodic sequence with period N to another discrete and periodic sequence of period N. and, other than the little tilde above the x[n] (or X[k]), there is not one single equation that differentiates DFS from DFT with the exception of what they *must* do to make x[n] (without the tilde) work with the mathematically imposed circularity of the DFS (*and* the DFT, that's what the whole argument is about). with *all* of the DFT theorems that *potentially* shift x[n] or X[k] (that is *all* of the theorems except for linearity), then it is necessary to use the "~x[n] = x[n mod N]" or "~X[k] = X[k mod N]". there is no exception other than linearity (which is a trivial exception). so periodicity is imposed on us by the mathematics (unless you do nothing of substance in the transformed domain), whether you call it "DFS" or "DFT". that's why those other guys are wrong about this, Dilip. The Discrete Fourier Series is one and the same as the Discrete Fourier Transform. adding a tilde to the symbols is not a salient difference. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."

Reply by ●July 29, 20132013-07-29

On Monday, July 29, 2013 6:36:51 AM UTC-7, dvsarwate wrote:> ... > Note that the > discrete Fourier transform is of finite length > just as (x[0], x[1], ... , x[N-1]) is. The rb-j's > of this world will tell you that actually both > x and X are actually one period of **sequences** > of period N that extend out to plus/minus infinity, > that is, for **any** integer M, x[M] = x[M mod N] > and X[M] = X[M mod N] and the mathematics supports > them,Specifically this is the case of "sequences periodic-in-N" if you want to find the supporting math.> but lots of other folks will equally > vehemently insist that this is sheer nonsense, and > the value of x[N], say, has nothing to do with the > values of x[0], x[1], ... , x[N-1].A clearer statement of the other side is that there are people who believe that they can assume, construct or measure sequences of arbitrary length that have values independent of rbj's choice of N and that can be periodic-in-rbj's-N, periodic-in-integer-other-than-rbj's-N or aperiodic. Mathematics supports all these cases. (Corollaries of this are that values of the terms in their sequences need not be a function of rbj's choice of N and that it is possible to pick their own value M and think about the M-point DFT without changing the values of terms in their sequences like rbj's must when he picks or changes N.)> Which side > you choose to believe is up to you. > > Dilip SarwateDale B. Dalrymple

Reply by ●July 29, 20132013-07-29

On Monday, July 29, 2013 12:10:52 PM UTC-5, robert bristow-johnson wrote:> but Dilip, i think the term we have for that is simply "Fourier series" > > or, in the case of c_n * e^(j*2*pi*n*t/T), we might call it the > > "complex Fourier series". at least in my copy of O&S, the term > > "Discrete Fourier Series" is about "a sequence ~x[n] that is periodic > > with period N so that ~x[n] = ~x[n+rN] for any integer value of r." (the > > tilde ~ is meant to be above the "x".) that's what O&S call the > > "Discrete Fourier Series". it's explicitly a mapping of one discrete > > and periodic sequence with period N to another discrete and periodic > > sequence of period N.Robert: With genuflections and obeisances in the general direction of O&S which tome I do not have on my bookshelf, I suggest that you are misinterpreting what O&S are saying, and if not, I am willing to entertain the possibility that what O&S are saying is flat out wrong. The discrete Fourier transform maps a vector x of N complex-valued numbers to another vector X of length N of complex-valued numbers. If we extend x periodically to construct a complex-valued sequence of period N, then the discrete Fourier transform of the sequence can be defined as the the corresponding periodic extension of X. On the other hand, a discrete Fourier series (manishp's choice, not the one I would have chosen) represents a periodic continuous-time signal x(t) as the sum of complex exponentials exp(j 2pi nt/T) weighted by complex numbers c_n. There is _nothing_ that is periodic about the sequence {c_n}: indeed, if the sequence of c_n's were repeating periodically, the continuous-time signal x(t) would deliver infinite energy in each period, and we could all retire since the energy crisis will have been solved! So, I disagree with you, and possibly O&S, if you are asserting that there is no difference except for the typographical notation ~ between these two notions: the representation of a periodic sequence of complex numbers in terms of another periodic sequence of complex numbers, and the representation of a continuous-time periodic signal by an aperiodic sequence of complex numbers. There is considerable family resemblance: they are kissing cousins, maybe even first cousins or siblings, but identical twins, No.

Reply by ●July 29, 20132013-07-29

>With genuflections and obeisances in the general >direction of O&S which tome I do not have on my >bookshelf, I suggest that you are misinterpreting >what O&S are saying, and if not, I am willing to >entertain the possibility that what O&S are saying >is flat out wrong. > >The discrete Fourier transform maps a vector x of >N complex-valued numbers to another vector X of >length N of complex-valued numbers. If we extend >x periodically to construct a complex-valued >sequence of period N, then the discrete Fourier >transform of the sequence can be defined as the >the corresponding periodic extension of X. > >On the other hand, a discrete Fourier series >(manishp's choice, not the one I would have >chosen) represents a periodic continuous-time >signal x(t) as the sum of complex exponentials >exp(j 2pi nt/T) weighted by complex numbers >c_n. There is _nothing_ that is periodic about >the sequence {c_n}: indeed, if the sequence of >c_n's were repeating periodically, the >continuous-time signal x(t) would deliver infinite >energy in each period, and we could all retire >since the energy crisis will have been solved! > >So, I disagree with you, and possibly O&S, if you >are asserting that there is no difference except >for the typographical notation ~ between these >two notions: the representation of a periodic >sequence of complex numbers in terms of another >periodic sequence of complex numbers, and the >representation of a continuous-time periodic >signal by an aperiodic sequence of complex >numbers. There is considerable family resemblance: >they are kissing cousins, maybe even first cousins >or siblings, but identical twins, No. >Dilip, as far as I remember O&S (and I read it a long time ago), when O&S mentions Discrete Fourier Series, they address the representation of a *discrete-time* periodic sequence while keeping the term Fourier Series for a continuous-time periodic signal (the one you are taking as discrete Fourier series). If you think about the above choice, Robert is also right, as much as you are in clarifying the difference between the two. _____________________________ Posted through www.DSPRelated.com

Reply by ●July 29, 20132013-07-29

manishp <58525@dsprelated> wrote:> While reading on frequency representation, I am a little confused > with discrete fourier series (as opposed to discrete fourier > transform) and its application.The naming does get confusing, and maybe isn't always consistent. If you consider the possible Fourier transform pairs, among other you find that periodic data transforms to discrete points in the transform space (and vice versa, as usual). Non-periodic data (limit as the period goes to infinity) transforms to/from a continuous spectrum. Some additional transform pairs are that real data transform to symmetric (even) functions, and imaginary to asymetric (odd) functions, and again vice versa. The Fourier series, then, is used to describe a periodic continuous signal as a sum of sinusoids that are integer multiples of the source frequency, but continuous. That is, a periodic continuous function has a discrete non-periodic transform. From the symmetry of the transform, a discrete non-periodic function will have a periodic continuous transform. These allow one to consider a (periodic) square wave as an infinite sum of sines and/or cosines, or (following Nyquist) to consider a band limited source sampled at a series of discrete points. Since computers can't represent either continuous data or data of infinite extent, the FFT (or DFT) represent periodic discrete input with a discrete periodic transform. (Note the symmetry.) This is usually called the discrete Fourier transform, though it seems that discrete Fourier series would be more appropriate. -- glen

Reply by ●July 30, 20132013-07-30

On 7/29/13 2:54 PM, dvsarwate wrote:> > On the other hand, a discrete Fourier series > represents a periodic continuous-time > signal x(t) as the sum of complex exponentials > exp(j 2pi nt/T) weighted by complex numbers > c_n. There is _nothing_ that is periodic about > the sequence {c_n}:yup. c_n need not be periodic, and cannot be for an x(t) that is finite (which leaves out dirac delta). but we call that "Fourier series". continuous in time and discrete in frequency.> indeed, if the sequence of > c_n's were repeating periodically, the > continuous-time signal x(t) would deliver infinite > energy in each period,not just each period, but each sample. if the c_n's were periodic, they would be the same as the periodic X[n] of the DFT (or DFS, as one might call it). *but* what would add up in continuous time would be a periodic sequence of delta functions (and they each have infinite energy, even if they have finite area). remember: periodic in one domain means sampled (or discrete) in the reciprocal domain (and vise versa). -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."

Reply by ●August 2, 20132013-08-02

On Mon, 29 Jul 2013 14:54:08 -0700 (PDT), dvsarwate <dvsarwate@yahoo.com> wrote:>On Monday, July 29, 2013 12:10:52 PM UTC-5, robert bristow-johnson wrote: > >> but Dilip, i think the term we have for that is simply "Fourier series" >> >> or, in the case of c_n * e^(j*2*pi*n*t/T), we might call it the >> >> "complex Fourier series". at least in my copy of O&S, the term >> >> "Discrete Fourier Series" is about "a sequence ~x[n] that is periodic >> >> with period N so that ~x[n] = ~x[n+rN] for any integer value of r." (the >> >> tilde ~ is meant to be above the "x".) that's what O&S call the >> >> "Discrete Fourier Series". it's explicitly a mapping of one discrete >> >> and periodic sequence with period N to another discrete and periodic >> >> sequence of period N. > > >Robert: > >With genuflections and obeisances in the general >direction of O&S which tome I do not have on my >bookshelf, I suggest that you are misinterpreting >what O&S are saying, and if not, I am willing to >entertain the possibility that what O&S are saying >is flat out wrong. > >The discrete Fourier transform maps a vector x of >N complex-valued numbers to another vector X of >length N of complex-valued numbers. If we extend >x periodically to construct a complex-valued >sequence of period N, then the discrete Fourier >transform of the sequence can be defined as the >the corresponding periodic extension of X. > >On the other hand, a discrete Fourier series >(manishp's choice, not the one I would have >chosen) represents a periodic continuous-time >signal x(t) as the sum of complex exponentials >exp(j 2pi nt/T) weighted by complex numbers >c_n. There is _nothing_ that is periodic about >the sequence {c_n}: indeed, if the sequence of >c_n's were repeating periodically, the >continuous-time signal x(t) would deliver infinite >energy in each period, and we could all retire >since the energy crisis will have been solved! > >So, I disagree with you, and possibly O&S, if you >are asserting that there is no difference except >for the typographical notation ~ between these >two notions: the representation of a periodic >sequence of complex numbers in terms of another >periodic sequence of complex numbers, and the >representation of a continuous-time periodic >signal by an aperiodic sequence of complex >numbers. There is considerable family resemblance: >they are kissing cousins, maybe even first cousins >or siblings, but identical twins, No.Hi Dilip, for what it's worth, at the beginning of their chapter titled: "The Discrete Fourier Transform", on page 541 of their 2nd Edition, Oppenheim & Schafer state: "Although several points of view can be taken toward the derivation and interpretation of the DFT representation of a finite-duration sequence, we have chosen to base our presentation on the relationship between periodic sequences and finite-length sequences. We will begin by considering the Fourier series representation of periodic sequences. While this representation is important in its own right, we are most often interested in the application of Fourier series results to the representation of finite-length sequences. We accomplish this by constructing a periodic sequence for which each period is identical to the finite-length sequence. As we will see. the Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence. Thus, our approach is to define the Fourier series representation for periodic sequences and to study the properties of such representations. Then we repeat essentially the same derivations, assuming that the sequence to be represented is a finite-length sequence." Notice their initial words: "Although several points of view can be taken ..." Dilip, as far as I can tell, I conform to your viewpoint concerning this topic. [-Rick-]