Discrete Fourier series
Series in digital signal processing
From Wikipedia, the free encyclopedia
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.[1]: ch 8.1
Introduction
Relation to Fourier series
The exponential form of Fourier series is given by:
![{\displaystyle s(t)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}t},}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2d8688dec0f260a18f645edddfb739129ae3d384)
which is periodic with an arbitrary period denoted by When continuous time is replaced by discrete time for integer values of and time interval the series becomes:
![{\displaystyle s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}nT},\quad n\in \mathbb {Z} .}](http://wikimedia.org/api/rest_v1/media/math/render/svg/817251796c3e2d2524dd2ac69b600fd979877352)
With constrained to integer values, we normally constrain the ratio to an integer value, resulting in an -periodic function:
![{\displaystyle s_{_{N}}[n]\triangleq s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ca46e2b919f821d1b0b4989ec2883e029dcd0250)
which are harmonics of a fundamental digital frequency The subscript reminds us of its periodicity. And we note that some authors will refer to just the coefficients themselves as a discrete Fourier series.[2]: p.85 (eq 15a)
![{\displaystyle S[k]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5b988b0ce17df34ae3c6aad5b6f9b5ab0f5fb2f5)
Due to the -periodicity of the kernel, the infinite summation can be "folded" as follows:
![{\displaystyle {\begin{aligned}s_{_{N}}[n]&=\sum _{m=-\infty }^{\infty }\left(\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k-mN}{N}}n}\ S[k-mN]\right)\\&=\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }S[k-mN]\right)} _{\triangleq S_{N}[k]},\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ebcdf617478f8785b6271535a396f26c8cf9e442)
which is the inverse DFT of one cycle of the periodic summation, [1]: p.542 (eq 8.4) [3]: p.77 (eq 4.24)
