Discrete Fourier series

In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. A specific example is the inverse discrete Fourier transform (inverse DFT).

Definition

The general form of a DFS is:

Discrete Fourier series

[math]\displaystyle{ \begin{align} s[n] = \sum_k S[k]\cdot e^{i 2\pi \frac{k}{N}n},\quad n \in \mathbb{Z}, \end{align} }[/math]

(Eq.1)

which are harmonics of a fundamental frequency [math]\displaystyle{ 1/N, }[/math] for some positive integer [math]\displaystyle{ N. }[/math] The practical range of [math]\displaystyle{ k, }[/math] is [math]\displaystyle{ [0,\ N-1], }[/math] because periodicity causes larger values to be redundant. When the [math]\displaystyle{ S[k] }[/math] coefficients are derived from an [math]\displaystyle{ N }[/math]-length DFT, and a factor of [math]\displaystyle{ 1/N }[/math] is inserted, this becomes an inverse DFT.  And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.

Example

A common practice is to create a sequence of length [math]\displaystyle{ N }[/math] from a longer [math]\displaystyle{ s[n] }[/math] sequence by partitioning it into [math]\displaystyle{ N }[/math]-length segments and adding them together, pointwise.(see DTFT § L=N×I) That produces one cycle of the periodic summation:

[math]\displaystyle{ s_{_N}[n]\ \triangleq\ \sum_{m=-\infty}^{\infty} s[n-mN], \quad n \in \mathbb{Z}. }[/math]

Because of periodicity, [math]\displaystyle{ s_{_N} }[/math]can be represented as a DFS with [math]\displaystyle{ N }[/math] unique coefficients that can be extracted by an [math]\displaystyle{ N }[/math]-length DFT.   

[math]\displaystyle{ \begin{align} &s_{_N}[n] = \frac{1}{N}\sum_{k=0}^{N-1} S_N[k] \cdot e^{i 2\pi \tfrac{k}{N}n};\quad n \in \mathbb{Z}&& \text{DFS}\\ &S_N[k] \triangleq \sum_{n=0}^{N-1} s_{_N}[n] \cdot e^{-i 2\pi \tfrac{k}{N}n};\quad k \in [0,N-1]&& \text{DFT} \end{align} }[/math]

The coefficients are useful because they are also samples of the discrete-time Fourier transform (DTFT) of the [math]\displaystyle{ s[n] }[/math] sequence:

[math]\displaystyle{ S_{1/T}(f) \triangleq \sum_{n=-\infty}^{\infty}e^{-i 2\pi f nT}\ T\ s(nT) = \sum_{m=-\infty}^{\infty} S\left(f - \tfrac{m}{T}\right), \quad f \in \mathbb{R} }[/math]

Here, [math]\displaystyle{ s(nT) }[/math] represents a sample of a continuous function [math]\displaystyle{ s(t), }[/math] with a sampling interval of [math]\displaystyle{ T, }[/math] and [math]\displaystyle{ S(f) }[/math] is the Fourier transform of [math]\displaystyle{ s(t). }[/math] The equality is a result of the Poisson summation formula. With definitions [math]\displaystyle{ s[n] \triangleq T\ s(nT) }[/math] and [math]\displaystyle{ S[k] \triangleq S_{1/T}\left(\tfrac{k}{NT}\right) }[/math]:

[math]\displaystyle{ S[k] = \sum_{n=-\infty}^{\infty}e^{-i 2\pi \tfrac{k}{N} n}\ s[n];\quad k \in \mathbb{Z}. }[/math]

Due to the [math]\displaystyle{ N }[/math]-periodicity of the [math]\displaystyle{ e^{-i 2\pi \tfrac{k}{N} n} }[/math] kernel, the summation can be "folded" as follows:

[math]\displaystyle{ \begin{align} S[k] &= \sum_{m=-\infty}^{\infty}\left(\sum_{n=0}^{N-1}e^{-i 2\pi \tfrac{k}{N}n}\ s[n-mN]\right)\\ &= \sum_{n=0}^{N-1}e^{-i 2\pi \tfrac{k}{N}n} \underbrace{\left(\sum_{m=-\infty}^{\infty}s[n-mN]\right)}_{s_{_N}[n]}\\ &= S_N[k]. \end{align} }[/math]