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In digital signal processing, a Discrete Fourier series (DFS) a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).

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^{i2\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:

Discrete Fourier series

s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}

(Eq.1)

which are harmonics of a fundamental digital frequency $1/N.$

Due to the $N$ -periodicity of the $e^{i2\pi {\tfrac {k}{N}}n}$ kernel, the summation can be "folded" as follows:

{\begin{aligned}s(nT)&=\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}}

which is proportional to the inverse DFT of one cycle of the function we've denoted by the periodic summation,

S_{N}.

References

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Cite error: A list-defined reference named "Nuttall" is not used in the content (see the help page).

@@ Line 1: / Line 1: @@
-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 (general)|discrete Fourier transform]] (inverse DFT).
+In [[digital signal processing]], a '''Discrete Fourier series''' (DFS) a [[Fourier series]] whose sinusoidal components are functions of discrete time instead of continuous time.  A specific example is the inverse [[Discrete Fourier transform (general)|discrete Fourier transform]] (inverse DFT).
+==Introduction==
+=== Relation to Fourier series ===
+The exponential form of Fourier series is given by:
+:<math>s(t) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i2\pi \frac{k}{P} t},</math>
+which is periodic with an arbitrary period denoted by <math>P.</math>  When continuous time <math>t</math> is replaced by discrete time <math>nT,</math> for integer values of <math>n</math> and time interval <math>T,</math> the series becomes:
+:<math>s(nT) = \sum_{k=-\infty}^\infty  S[k]\cdot e^{i 2\pi \frac{k}{P}nT},\quad n \in \mathbb{Z}.</math>
+With <math>n</math> constrained to integer values, we normally constrain the ratio <math>P/T=N</math> to an integer value, resulting in an <math>N</math>-periodic function:
-==Definition==
-The general form of a DFS is''':'''
 {{Equation box 1
-|indent =:
+|indent =
 |title=Discrete Fourier series
-|equation = {{NumBlk||<math>
+|equation = {{NumBlk|:|<math>
+s(nT) = \sum_{k=-\infty}^\infty  S[k]\cdot e^{i 2\pi \frac{k}{N}n}
-\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>|{{EquationRef|Eq.1}}}}
 |cellpadding= 6
@@ Line 16: / Line 25: @@
 |background colour=#F5FFFA}}
+which are harmonics of a fundamental digital frequency <math>1/N.</math>
-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.<ref name=Oppenheim/>{{rp|p.542 (eq 8.4)}}&nbsp;<ref name=Prandoni/>{{rp|p.77 (eq 4.24)}}  And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.<ref name=Nuttall/>{{rp|p.85 (eq 15a)}}
-=== 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 {{slink|DTFT|L{{=}}N×I}})  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.<ref name="Oppenheim" />{{rp|p 543 (eq 8.9)}}{{rp|pp 557-558}}&nbsp; &nbsp;<ref name="Prandoni" />{{rp|p 72 (eq 4.11)}}
+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_{_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(nT) &= \sum_{m=-\infty}^{\infty}\left(\sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k-mN}{N}n}\ S[k-mN]\right)\\
-&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}
+&= \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}
 </math>
+:which is proportional to the '''inverse DFT''' of one cycle of the function we've denoted by the periodic summation, <math>
+S_N.
-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>