Discrete Fourier Series
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digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...
, 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 In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...
and its inverse transform.


Introduction


Relation to Fourier series

The exponential form of Fourier series is given by: :s(t) = \sum_^\infty S cdot e^, 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_^\infty S cdot e^,\quad n \in \mathbb. With n constrained to integer values, we normally constrain the ratio P/T=N to an integer value, resulting in an N-periodic function: 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 /math> coefficients themselves as a discrete Fourier series. Due to the N-periodicity of the e^ kernel, the infinite summation can be "folded" as follows: : \begin s_ &= \sum_^\left(\sum_^e^\ S -mNright)\\ &= \sum_^e^ \underbrace_, \end which is the inverse DFT of one cycle of the periodic summation, S_N. 


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 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. {{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 Fourier analysis