Английская Википедия: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: Шаблон:Equation box 1
which are harmonics of a fundamental frequency <math>1/N,</math> for some positive integer <math>N.</math> The practical range of <math>k,</math> is <math>[0,\ N-1],</math> because periodicity causes larger values to be redundant. When the <math>S[k]</math> coefficients are derived from an <math>N</math>-length DFT, and a factor of <math>1/N</math> is inserted, this becomes an inverse DFT.[1]Шаблон:Rp [2]Шаблон:Rp And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.[3]Шаблон:Rp
Example
A common practice is to create a sequence of length <math>N</math> from a longer <math>s[n]</math> sequence by partitioning it into <math>N</math>-length segments and adding them together, pointwise.(see Шаблон:Slink) That produces one cycle of the periodic summation:
- <math>s_{_N}[n]\ \triangleq\ \sum_{m=-\infty}^{\infty} s[n-mN], \quad n \in \mathbb{Z}.</math>
Because of periodicity, <math>s_{_N}</math>can be represented as a DFS with <math>N</math> unique coefficients that can be extracted by an <math>N</math>-length DFT.[1]Шаблон:RpШаблон:Rp [2]Шаблон:Rp
- <math>
\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>s[n]</math> sequence:
- <math>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>s(nT)</math> represents a sample of a continuous function <math>s(t),</math> with a sampling interval of <math>T,</math> and <math>S(f)</math> is the Fourier transform of <math>s(t).</math> The equality is a result of the Poisson summation formula. With definitions <math>s[n] \triangleq T\ s(nT)</math> and <math>S[k] \triangleq S_{1/T}\left(\tfrac{k}{NT}\right)</math>:
- <math>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>N</math>-periodicity of the <math>e^{-i 2\pi \tfrac{k}{N} n}</math> kernel, the summation can be "folded" as follows:
- <math>
\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>
References
