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In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the significance of discrete Fourier series?

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  • $\begingroup$ They're not the exact same thing. The only difference is a factor, specifically, in which equation (analysis equation/forward transform or synthesis equation/inverse transform) such factor is written. $\endgroup$
    – alejnavab
    Jun 21, 2021 at 5:30

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we have had this fight many, many, many times at comp.dsp. the DFT is the same thing as the DFS. the DFT maps a discrete and periodic sequence of numbers with period length of $N$ to another discrete and periodic sequence of numbers with period length of $N$ and the iDFT (which has the same form as the DFT) maps it back.

some people don't like anthropomorphizing algorithms or procedures, but i do. the DFT "assumes" that the $N$ samples passed to it are one period of a periodic sequence. the DFT periodically extends the data passed to it.

it is clear in the math, both in the definition of the DFT (and iDFT), and in any theorem applicable to the DFT other than linearity (the periodic nature of the DFT is not evident in the linearity property, but it is evident in anything that causes shifting or convolution in one domain or multiplication by a non-constant in the other domain).

this is why, if periodicity is not assumed (a better word would be "recognized"), then people need to use this clunky modulo notation in the indices, like $x[ ((n))_N ]$ (this is the notation that O&S use) and that, in my opinion, is a pathetic confession from the periodicity deniers that, when it comes down to the bottom line, even they recognize that the DFT is inherently periodic.

to be explicit, the periodic extension of the $N$ samples of $x[n]$ passed to the DFT is:

$$ \tilde{x}[n] = x[ ((n))_N ] = x[n \bmod N] \qquad \forall n \in \mathbb{Z}, \ N \in \mathbb{Z}>0$$

where $ \qquad\qquad\qquad ((n))_N \triangleq n \bmod N = n - N \left\lfloor \frac{n}{N} \right\rfloor $

the notation $\lfloor \cdot \rfloor$ means the floor() function which is the largest integer that does not exceed the argument.

to use any of the shifting or convolution theorems of the DFT, this modulo arithmetic of the indices is absolutely required. for the scaling or superposition theorems, this modulo arithmetic is not required, but does not break those theorems in any case.

therefore, to be consistent, when using the DFT for any theorems to do any real work with the DFT, one should simply apply the modulo arithmetic all of the time. doing so explicitly periodically extends the $N$-sample sequence, $x[n]$ passed the DFT.

for me, it's just easier to drop the tilde "$\tilde{\ }$" and simply say that $\tilde{x}[n]$ is the same as $x[n]$ and that $\tilde{X}[k]$ is the same as $X[k]$ and just stop fucking around with this DFT business.

people should read this other answer i wrote a long time ago regarding the inherent periodic nature of the DFT.

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    $\begingroup$ I don't think it's necessary to ridicule the point of view of mathematicians only because you don't share it. Mathematically, the spaces that DFT and DFS map from are different, specifically in terms of their topology. For practical implementations this doesn't seem important, but it is very fundamental for certain theoretical considerations. The relevant concepts may be missing from an engineer's vocabulary, but that does not make them trivial or laughable. And yes, I was there on comp.dsp too. $\endgroup$
    – Jazzmaniac
    May 8, 2014 at 17:12
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    $\begingroup$ A subset of $\mathbb{Z}$ inherits the order Topology from $\mathbb{Z}$. The quotient ring $\mathbb{Z}/n\mathbb{Z}$ is incompatible with the order topology, because it does not possess a total order. $\endgroup$
    – Jazzmaniac
    May 8, 2014 at 20:22
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    $\begingroup$ the space that both the DFT and the DFS (since they're just different labels for the same thing) map from and to is $\mathbb{C}^N$. $\endgroup$ May 8, 2014 at 21:55
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    $\begingroup$ From an engineering point of view that's alright, but a mathematician wants to distinguish between the two function spaces $F_A := \{ f:A\subset\mathbb{Z}\to\mathbb{C}\}$ and $F_n:=\{f:\mathbb{Z}/n\mathbb{Z}\to\mathbb{C}\}$. For $\#A=n$ the two spaces are isomorphic. But there is no natural isomorphism,i.e one that makes more sense than all the rest. And there is no homeomorphism between the two. That may not be important to you, because the numbers are just the same. But to a mathematician interested in structural properties, this is relevant. $\endgroup$
    – Jazzmaniac
    May 9, 2014 at 8:41
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    $\begingroup$ Jazz, what you have skirted around is that, only by someone's chosen definition does the $F_A$ mapping apply to the DFT, when in fact, the properties of the DFT naturally lend itself to the $F_n$ mapping (where $n=N$, given the typical expressions of the definition). "DFT" and "DFS" are just different labels for the same thing. $\endgroup$ May 9, 2014 at 15:19
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I think part of the problem is an awkward and inconsistent naming convention. There are 4 flavors of Fourier Transforms depending on which domain is continuous or discrete (which maps to being aperiodic or perodic in the other domain). So we have

         Name                  Time                  Frequency
Fourier Transform    continous/aperiodic     continous/aperiodic 
Fourier Series       continous/periodic      discrete/aperiodic 
Discrete Time FT     discrete/aperiodic      continous/periodic
DFT or DFS           discrete/periodic       discrete/periodic

A better naming would have been

         Name                  Time                  Frequency
Fourier Transform    continous/aperiodic     continous/aperiodic 
Fourier Series       continous/periodic      discrete/aperiodic 
Discrete FT          discrete/aperiodic      continous/periodic
Discrete FS          discrete/periodic       discrete/periodic

so that discrete refers to "discrete in time and periodic in frequency" and "series" refers "discrete in frequency and periodic in time". In other words "series" means sums and "transform" means integrals. Discrete mean sums and continuous means integrals.

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  • $\begingroup$ maybe to add a coda to Hilmar's answer: discrete in either the Time or Frequency domains is equivalent to uniform sampling of a continuous-time/frequency function. and uniform sampling of one continuous domain corresponds to periodicity in the other. so discrete-time must correspond to periodic frequency and discrete-frequency must correspond to periodic time. discrete time and frequency on one side of the transform must correspond to discrete frequency and time on the other side. $\endgroup$ May 8, 2014 at 18:48
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Basically, DFS is used for periodic and infinite sequence. Whereas, DFT is used for non-periodic and finite sequence. Although, They are same Mathematically. But they differ in properties. Practically, we do not have infinite signal. We can say that DFT is extraction of one period from DFS. In other words, DFS is sampling of DFT equally spaced at integer multiple of $\frac{2 \pi}{N}$. DFT is fast and efficient algorithms exits for the computation of the DFT. DFS is adequate for most cases. But FT(Fourier Transform) leads to simpler expression.

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  • $\begingroup$ //"But they differ in properties."// -------------- Abdul, would you care to identify the properties of the DFT and DFS that are operationally different? $\endgroup$ Nov 15, 2021 at 3:30
  • $\begingroup$ //"DFS is sampling of DFT equally spaced at integer multiple of $\frac{2\pi}{N}$ ."// No, both the DFS and DFT (being the same thing) are sampling the DTFT at $N$ equally-spaces points on the unit circle in the z plane. $\endgroup$ Nov 15, 2021 at 4:49

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