Discrete Fourier series

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^{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 the inverse DFT of one cycle of the periodic summation, S_N.

References