Discrete Fourier series

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). 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. 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 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.