Discrete Fourier series

In digital signal processing, a discrete Fourier series (DFS) is a Fourier series whose sinusoidal components are functions of a discrete variable instead of a continuous variable. The result of the series is also a function of the discrete variable, i.e. a discrete sequence. A Fourier series, by nature, has a discrete set of components with a discrete set of coefficients, also a discrete sequence. So a DFS is a representation of one sequence in terms of another sequence. Well known examples are the Discrete Fourier transform and its inverse transform.

Introduction

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:

$s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}nT},\quad n\in \mathbb {Z} .$

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

Discrete Fourier series

$s_{_{N}}[n]\triangleq s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}$

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 $e^{i2\pi {\tfrac {k}{N}}n}$ kernel, the infinite summation can be "folded" as follows:

${\begin{aligned}s_{_{N}}[n]&=\sum _{m=-\infty }^{\infty }\left(\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k-mN}{N}}n}\ S[k-mN]\right)\\&=\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }S[k-mN]\right)} _{\triangleq S_{N}[k]},\end{aligned}}$

which is the inverse DFT of one cycle of the periodic summation, $S_{N}.$