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

s[n]=∑kS[k]⋅ei2πkNn,n∈ℤ,

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(Eq.1)

which are harmonics of a fundamental frequency 1/N, for some positive integer N. The practical range of k, is [0, N−1], because periodicity causes larger values to be redundant. When the S[k] coefficients are derived from an N-length DFT, and a factor of 1/N 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 N from a longer s[n] sequence by partitioning it into N-length segments and adding them together, pointwise.(see DTFT § L=N×I) That produces one cycle of the periodic summation:

sN[n] ≜ ∑m=−∞∞s[n−mN],n∈ℤ.

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

sN[n]=1N∑k=0N−1SN[k]⋅ei2πkNn;n∈ℤDFSSN[k]≜∑n=0N−1sN[n]⋅e−i2πkNn;k∈[0,N−1]DFT

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

S1/T(f)≜∑n=−∞∞e−i2πfnT T s(nT)=∑m=−∞∞S(f−mT),f∈ℝ

Here, s(nT) represents a sample of a continuous function s(t), with a sampling interval of T, and S(f) is the Fourier transform of s(t). The equality is a result of the Poisson summation formula. With definitions s[n]≜T s(nT) and S[k]≜S1/T(kNT):

S[k]=∑n=−∞∞e−i2πkNn s[n];k∈ℤ.

Due to the N-periodicity of the e−i2πkNn kernel, the summation can be "folded" as follows:

S[k]=∑m=−∞∞(∑n=0N−1e−i2πkNn s[n−mN])=∑n=0N−1e−i2πkNn(∑m=−∞∞s[n−mN])⏟sN[n]=SN[k].

## 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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