Discrete Fourier series

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

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### Relation to Fourier series

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The exponential form of Fourier series is given by:

s ( t ) = ∑ k = − ∞ ∞ S [ k ] ⋅ e i 2 π k P t , {\displaystyle 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 . {\displaystyle P.} When continuous time  t {\displaystyle t} is replaced by discrete time  n T , {\displaystyle nT,} for integer values of  n {\displaystyle n} and time interval  T , {\displaystyle T,} the series becomes:

s ( n T ) = ∑ k = − ∞ ∞ S [ k ] ⋅ e i 2 π k P n T , n ∈ Z . {\displaystyle s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}nT},\quad n\in \mathbb {Z} .}

With  n {\displaystyle n} constrained to integer values, we normally constrain the ratio  P / T = N {\displaystyle P/T=N} to an integer value, resulting in an  N {\displaystyle N} -periodic function:

Discrete Fourier series

s N [ n ] ≜ s ( n T ) = ∑ k = − ∞ ∞ S [ k ] ⋅ e i 2 π k N n {\displaystyle s_{_{N}}[n]\triangleq s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}}

which are harmonics of a fundamental digital frequency  1 / N . {\displaystyle 1/N.} The  N {\displaystyle N} subscript reminds us of its periodicity. And we note that some authors will refer to just the  S [ k ] {\displaystyle S[k]} coefficients themselves as a discrete Fourier series.[2]: p.85 (eq 15a)

Due to the  N {\displaystyle N} -periodicity of the  e i 2 π k N n {\displaystyle e^{i2\pi {\tfrac {k}{N}}n}} kernel, the infinite summation can be "folded" as follows:

s N [ n ] = ∑ m = − ∞ ∞ ( ∑ k = 0 N − 1 e i 2 π k − m N N n S [ k − m N ] ) = ∑ k = 0 N − 1 e i 2 π k N n ( ∑ m = − ∞ ∞ S [ k − m N ] ) ⏟ ≜ S N [ k ] , {\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}}}

which is the inverse DFT of one cycle of the periodic summation,  S N . {\displaystyle S_{N}.} [1]: p.542 (eq 8.4) [3]: p.77 (eq 4.24)

## References

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1. ^ a b Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0-13-754920-2. "samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n]. ... The Fourier series coefficients can be interpreted as a sequence of finite length for k=0,...,(N-1), and zero otherwise, or as a periodic sequence defined for all k."
2. ^ Nuttall, Albert H. (Feb 1981). "Some Windows with Very Good Sidelobe Behavior". IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91\. doi:10.1109/TASSP.1981.1163506.
3. ^ Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (PDF) (1 ed.). Boca Raton,FL: CRC Press. pp. 72, 76. ISBN 978-1-4200-7046-0. Retrieved 4 October 2020. "the DFS coefficients for the periodized signal are a discrete set of values for its DTFT"