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.Template:Rp

Introduction

Relation to Fourier series

The exponential form of Fourier series is given by:

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

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:

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

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

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

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.Template:Rp

Due to the $N$-periodicity of the $e^{i2\pi \frac{k}{N}n}$ kernel, the infinite summation can be "folded" as follows:

$\begin{array}{cc}{s}_{{}_N}[n]& =\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}(\sum _{k=0}^{N-1}{e}^{i2\pi \frac{k-mN}{N}n}S[k-mN])\\ & =\sum _{k=0}^{N-1}{e}^{i2\pi \frac{k}{N}n}\underset{\triangleq S_N[k]}{\underbrace{(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}S[k-mN])}},\end{array}$

which is the inverse DFT of one cycle of the periodic summation, $S_N.$Template:Rp Template:Rp