Discrete Fourier series

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

[math]\displaystyle{ \begin{align} x[n] = \sum_k X[k]\cdot e^{i 2\pi \frac{k}{N}n},\quad n \in \mathbb{Z}, \end{align} }[/math]

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(Eq.1)

which are harmonics of a fundamental frequency [math]\displaystyle{ 1/N, }[/math] for some positive integer [math]\displaystyle{ N. }[/math] The practical range of [math]\displaystyle{ k, }[/math] is [math]\displaystyle{ [0,\ N-1], }[/math] because periodicity causes larger values to be redundant. When the [math]\displaystyle{ X[k] }[/math] coefficients are derived from an [math]\displaystyle{ N }[/math]-length DFT, and a factor of [math]\displaystyle{ 1/N }[/math] 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)

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

[math]\displaystyle{ x_{_N}[n]\ \triangleq\ \sum_{m=-\infty}^{\infty} x[n-mN], \quad n \in \mathbb{Z}. }[/math]

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

The coefficients are useful because they are samples of the discrete-time Fourier transform (DTFT) of the [math]\displaystyle{ x[n] }[/math] sequence:

[math]\displaystyle{ X_{1/T}(f) \triangleq \sum_{n=-\infty}^{\infty}e^{-i 2\pi f nT}\ T\ x(nT) = \sum_{m=-\infty}^{\infty} X\left(f - \tfrac{m}{T}\right), \quad f \in \mathbb{R} }[/math]

Here, [math]\displaystyle{ x(nT) }[/math] represents a sample of a continuous function [math]\displaystyle{ x(t), }[/math] with a sampling interval of [math]\displaystyle{ T, }[/math] and [math]\displaystyle{ X(f) }[/math] is the Fourier transform of [math]\displaystyle{ x(t). }[/math] The equality is a result of the Poisson summation formula. With definitions [math]\displaystyle{ x[n] \triangleq T\ x(nT) }[/math] and [math]\displaystyle{ X_N[k] \triangleq X_{1/T}\left(\tfrac{k}{NT}\right) }[/math]:

[math]\displaystyle{ X_N[k] = \sum_{n=-\infty}^{\infty}e^{-i 2\pi \tfrac{k}{N} n}\ x[n], \quad k = 0,...,N-1 }[/math]

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

[math]\displaystyle{ \begin{align} X_N[k] &= \sum_{m=-\infty}^{\infty}\left(\sum_{n=0}^{N-1}e^{-i 2\pi \tfrac{k}{N}n}\ x[n-mN]\right)\\\ &= \sum_{n=0}^{N-1}e^{-i 2\pi \tfrac{k}{N}n} \underbrace{\left(\sum_{m=-\infty}^{\infty}x[n-mN]\right)}_{x_{_N}[n]}\\\ &= \operatorname{DFT}\\{x_{_N}\\}. \end{align} }[/math]

## References

1. ↑ 1.0 1.1 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. https://archive.org/details/discretetimesign00alan. "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. ↑ 2.0 2.1 Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (1 ed.). Boca Raton,FL: CRC Press. pp. 72,76. ISBN 978-1-4200-7046-0. https://www.sp4comm.org/docs/sp4comm.pdf. 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. https://zenodo.org/record/1280930.

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