Discrete Fourier series explained

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 ^{inf ty} _{k =-inf ty} S[k] ⋅ e ^{i 2 \pi (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 ^{inf ty} _{k =-inf ty} S[k] ⋅ e ^{i 2 \pi (k)/(P) nT} , n \in 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} {N} n} kernel, the infinite summation can be "folded" as follows:\begin{align} s _{_{N}} [n] & = \sum ^{inf ty} _{m =-inf ty} \left(\sum ^{N -1} _{k =0} e ^{i} {N}n} S[k - mN]\right)\\ & = \sum ^{N -1} _{k =0} e ^{i} {N}n} \underbrace{\left(\sum ^{inf ty} _{m =-inf ty} 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.}