Discrete Fourier series
Introduction In digital signal processing, the term '''Discrete Fourier series''' (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. ''Fourier'') discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. A specific example is the inverse discrete Fourier transform (inverse DFT).
Definition The general form of a DFS is''':'''which are harmonics of a fundamental frequency  for some positive integer   The practical range of  is  because periodicity causes larger values to be redundant. When the  coefficients are derived from an -length DFT, and a factor of  is inserted, this becomes an inverse DFT. And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.=== Example ===A common practice is to create a sequence of length  from a longer  sequence by partitioning it into -length segments and adding them together, pointwise. (see )  That produces one cycle of the periodic summation''':''':Because of periodicity, can be represented as a DFS with  unique coefficients that can be extracted by an -length DFT. :The coefficients are useful because they are also samples of the discrete-time Fourier transform (DTFT) of the  sequence''':''':Here,  represents a sample of a continuous function  with a sampling interval of  and  is the Fourier transform of   The equality is a result of the Poisson summation formula. With definitions   and ''':''':Due to the -periodicity of the  kernel, the summation can be "folded" as follows''':''':
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