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

Relation to Fourier series

The exponential form of Fourier series is given by:

${\displaystyle s(t)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}t},}$

which is periodic with an arbitrary period denoted by ${\displaystyle P.}$ When continuous time ${\displaystyle t}$ is replaced by discrete time ${\displaystyle nT,}$ for integer values of ${\displaystyle n}$ and time interval ${\displaystyle T,}$ the series becomes:

${\displaystyle s(nT)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}nT},\quad n\in \mathbb {Z} .}$

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

Discrete Fourier series

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

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

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

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

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

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[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