Discrete Fourier series

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, $S_{N}.$

Further reading

• Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). "4.2, 8.4". 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.

• 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

• 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.