In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. A specific example is the inverse discrete Fourier transform (inverse DFT).

Definition

The general form of a DFS is:

Discrete Fourier series

 

 

 

 

(Eq.1)

which are harmonics of a fundamental frequency for some positive integer The practical range of is because periodicity causes larger values to be redundant. When the coefficients are derived from an -length DFT, and a factor of is inserted, this becomes an inverse DFT.[1]:p.542 (eq 8.4) [2]:p.77 (eq 4.24) And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.[3]:p.85 (eq 15a)

Example

A common practice is to create a sequence of length from a longer sequence by partitioning it into -length segments and adding them together, pointwise.(see DTFT § L=N×I) That produces one cycle of the periodic summation:

Because of periodicity, can be represented as a DFS with unique coefficients that can be extracted by an -length DFT.[1]:p 543 (eq 8.9):pp 557-558   [2]:p 72 (eq 4.11)

The coefficients are useful because they are also samples of the discrete-time Fourier transform (DTFT) of the sequence:

Here, represents a sample of a continuous function with a sampling interval of and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): S(f) is the Fourier transform of The equality is a result of the Poisson summation formula. With definitions and :

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

References

  1. 1 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.
  2. 1 2 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
  3. 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.
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