the difference between dft and dfs

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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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.

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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. –  Jazzmaniac May 8 '14 at 17:12
"Mathematically, the spaces that DFT and DFS map from are different, specifically in terms of their topology." no, that's not the case at all. –  robert bristow-johnson May 8 '14 at 18:36
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. –  Jazzmaniac May 8 '14 at 20:22
the space that both the DFT and the DFS (since they're just different labels for the same thing) map from and to is $\mathbb{C}^N$. –  robert bristow-johnson May 8 '14 at 21:55
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. –  Jazzmaniac May 9 '14 at 8:41

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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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. –  robert bristow-johnson May 8 '14 at 18:48