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.{{rp|ch 8.1}}

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:

: $s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P}nT},\quad n \in \mathbb{Z}.$

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

{{Equation box 1

|title=Discrete Fourier series

|equation = : $s_{_N}[n] \triangleq s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n}$

}}

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

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

: $\begin{align}
s_{_N}[n] &= \sum_{m=-\infty}^{\infty}\left(\sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k-mN}{N}n}\ S[k-mN]\right)\\
&= \sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k}{N}n}
\underbrace{\left(\sum_{m=-\infty}^{\infty}S[k-mN]\right)}_{\triangleq S_N[k]},
\end{align}$

which is the inverse DFT of one cycle of the periodic summation, $S_N.$ {{rp|p.542 (eq 8.4)}} {{rp|p.77 (eq 4.24)}}

References

{{reflist|1|refs=

{{Cite book |ref=Oppenheim |last=Oppenheim |first=Alan V. |authorlink=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |url-access=registration |url=https://archive.org/details/discretetimesign00alan |quote=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.}}

{{cite book |last1=Prandoni |first1=Paolo |last2=Vetterli |first2=Martin |title=Signal Processing for Communications |date=2008 |publisher=CRC Press |location=Boca Raton,FL |isbn=978-1-4200-7046-0 |edition=1 |url=https://www.sp4comm.org/docs/sp4comm.pdf |accessdate=4 October 2020 |pages=72,76 |quote=the DFS coefficients for the periodized signal are a discrete set of values for its DTFT

}}

{{cite journal

| doi =10.1109/TASSP.1981.1163506

| last =Nuttall

| first =Albert H.

| title =Some Windows with Very Good Sidelobe Behavior

| journal =IEEE Transactions on Acoustics, Speech, and Signal Processing

| volume =29

| issue =1

| pages =84–91

| date =Feb 1981

| url =https://zenodo.org/record/1280930

}}

Category:Fourier analysis