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Discrete Fourier series

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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.

Introduction

Relation to Fourier series

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\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\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}}\triangleq s(nT)=\sum _{k=-\infty }^{\infty }S\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} coefficients themselves as a discrete Fourier series.

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}}&=\sum _{m=-\infty }^{\infty }\left(\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k-mN}{N}}n}\ S\right)\\&=\sum _{k=0}^{N-1}e^{i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }S\right)} _{\triangleq S_{N}},\end{aligned}}}

which is the inverse DFT of one cycle of the periodic summation, S N . {\displaystyle S_{N}.}  

References

  1. ^ 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 can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x. ... 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
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