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:

{\begin{aligned}s[n]=\sum _{k}S[k]\cdot e^{i2\pi {\frac {k}{N}}n},\quad n\in \mathbb {Z} ,\end{aligned}}

(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:

s_{_{N}}[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }s[n-mN],\quad n\in \mathbb {Z} .

Because of periodicity, $s_{_{N}}$ can 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)}

{\begin{aligned}&s_{_{N}}[n]={\frac {1}{N}}\sum _{k=0}^{N-1}S_{N}[k]\cdot e^{i2\pi {\tfrac {k}{N}}n};\quad n\in \mathbb {Z} &&{\text{DFS}}\\&S_{N}[k]\triangleq \sum _{n=0}^{N-1}s_{_{N}}[n]\cdot e^{-i2\pi {\tfrac {k}{N}}n};\quad k\in [0,N-1]&&{\text{DFT}}\end{aligned}}

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

S_{1/T}(f)\triangleq \sum _{n=-\infty }^{\infty }e^{-i2\pi fnT}\ T\ s(nT)=\sum _{m=-\infty }^{\infty }S\left(f-{\tfrac {m}{T}}\right),\quad f\in \mathbb {R}

Here, $s(nT)$ represents a sample of a continuous function $s(t),$ with a sampling interval of $T,$ 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 $s(t).$ The equality is a result of the Poisson summation formula. With definitions $s[n]\triangleq T\ s(nT)$ and $S[k]\triangleq S_{1/T}\left({\tfrac {k}{NT}}\right)$ :

S[k]=\sum _{n=-\infty }^{\infty }e^{-i2\pi {\tfrac {k}{N}}n}\ s[n];\quad k\in \mathbb {Z} .

Due to the $N$ -periodicity of the $e^{-i2\pi {\tfrac {k}{N}}n}$ kernel, the summation can be "folded" as follows:

{\begin{aligned}S[k]&=\sum _{m=-\infty }^{\infty }\left(\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ s[n-mN]\right)\\&=\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }s[n-mN]\right)} _{s_{_{N}}[n]}\\&=S_{N}[k].\end{aligned}}

References

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.
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
↑ 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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[Oppenheim-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.

[Prandoni-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

[Nuttall-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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