Discrete Fourier series

In digital signal processing, a Discrete Fourier series (DFS) a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).

Relation to Fourier series
The exponential form of Fourier series is given by:


 * $$s(t) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i2\pi \frac{k}{P} t},$$

which is periodic with an arbitrary period denoted by $$P.$$ When continuous time $$t$$ is replaced by discrete time $$nT,$$ for integer values of $$n$$ and time interval $$T,$$ the series becomes:


 * $$s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P}nT},\quad n \in \mathbb{Z}.$$

With $$n$$ constrained to integer values, we normally constrain the ratio $$P/T=N$$ to an integer value, resulting in an $$N$$-periodic function:

which are harmonics of a fundamental digital frequency $$1/N.$$ The $$N$$ subscript reminds us of its periodicity. And we note that some authors will refer to just the $$S[k]$$ coefficients themselves as a discrete Fourier series.

Due to the $$N$$-periodicity of the $$e^{i 2\pi \tfrac{k}{N} n}$$ kernel, the infinite summation can be "folded" as follows:

\begin{align} s_{_N}[n] &= \sum_{m=-\infty}^{\infty}\left(\sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k-mN}{N}n}\ S[k-mN]\right)\\ &= \sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k}{N}n} \underbrace{\left(\sum_{m=-\infty}^{\infty}S[k-mN]\right)}_{\triangleq S_N[k]}, \end{align} $$ which is proportional (by a factor of $$N$$) to the inverse DFT of one cycle of the periodic summation, $$S_N.$$