Discrete Fourier series

In digital signal processing, a discrete Fourier series (DFS) is 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).

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 proportional (by a factor of $N$) to the inverse DFT of one cycle of the periodic summation, $S_{N}.$