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).
The general form of a DFS is:
[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] |
|
( ) |
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.[1]:p.542 (eq 8.4) [2]:p.77 (eq 4.24) And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.[3]:p.85 (eq 15a)
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:
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.[1]:p 543 (eq 8.9):pp 557-558 [2]:p 72 (eq 4.11)
The coefficients are useful because they are also samples of the discrete-time Fourier transform (DTFT) of the [math]\displaystyle{ s[n] }[/math] sequence:
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]:
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:
Original source: https://en.wikipedia.org/wiki/Discrete Fourier series.
Read more |