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?
Answer
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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