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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 discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).

Introduction

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Relation to Fourier series

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The exponential form of Fourier series is given by:

which is periodic with an arbitrary period denoted by When continuous time is replaced by discrete time for integer values of and time interval the series becomes:

With constrained to integer values, we normally constrain the ratio to an integer value, resulting in an -periodic function:

Discrete Fourier series

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

Due to the -periodicity of the kernel, the infinite summation can be "folded" as follows:

which is proportional (by a factor of ) to the inverse DFT of one cycle of the periodic summation, [2]: p.542 (eq 8.4)  [3]: p.77 (eq 4.24) 

References

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  1. ^ 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.
  2. ^ 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.
  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