Discrete Fourier series

In digital signal processing, the term Discrete Fourier series (DFS) describes a particular form of the inverse discrete Fourier transform (inverse DFT).[1]^{:p 542}

For a function ${\displaystyle x(t)}$ with Fourier transform ${\displaystyle X(f),}$ the discrete-time Fourier transform (DTFT) of the discrete sequence ${\displaystyle \{x(n),n\in \mathbb {Z} \},}$ is given by a Fourier series:

${\displaystyle \sum _{n=-\infty }^{\infty }e^{-i2\pi fn}\ x(n)=\sum _{m=-\infty }^{\infty }X(f-m),}$

where the right-hand side of the equality is a result of the Poisson summation formula. These formulas are periodic in frequency ${\displaystyle (f)}$ with a period of ${\displaystyle 1}$ (the reciprocal of the sample-interval). A common practice is to compute an arbitrary number ${\displaystyle (N)}$ of samples at frequency intervals of ${\displaystyle 1/N,}$ thereby spanning one cycle of the periodic DTFT:

${\displaystyle \sum _{n=-\infty }^{\infty }e^{-i2\pi {\tfrac {k}{N}}n}\ x(n)=\underbrace {\sum _{m=-\infty }^{\infty }X\left({\tfrac {k}{N}}-m\right)} _{{\tilde {X}}[k]},\quad k=0,...,N-1}$

where the discrete-frequency and periodized (N-periodic) version of ${\displaystyle X}$ is denoted by ${\displaystyle {\tilde {X}}.}$  Due to the N-periodicity of the ${\displaystyle e^{-i2\pi {\tfrac {k}{N}}n}}$ kernel, the left hand side can be "folded" as follows:

${\displaystyle \sum _{m=-\infty }^{\infty }\left(\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ x(n-mN)\right)=\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ \underbrace {\left(\sum _{m=-\infty }^{\infty }x(n-mN)\right)} _{{\tilde {x}}[n]}.}$

Consequently:

Sampling the DTFT

${\displaystyle \sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ {\tilde {x}}[n]={\tilde {X}}[k],\quad k=0,...,N-1}$

(Eq.1)

which is the discrete Fourier transform (DFT) of one cycle of ${\displaystyle {\tilde {x}}[n].}$ The inverse transform is:

Discrete Fourier series

${\displaystyle {\tilde {x}}[n]={\frac {1}{N}}\sum _{k=0}^{N-1}{\tilde {X}}[k]\cdot e^{i2\pi {\frac {n}{N}}k},\quad n\in \mathbb {Z} ,}$   [1]^:p 542 (eq 8.4) [2]^:p 72 (eq. 4.12)

(Eq.2)

which is a representation of the ${\displaystyle {\tilde {x}}[n]}$ sequence in terms of a summation of weighted, harmonically-related complex sinusoids, essentially a Fourier series.[upper-alpha 1] But unlike a conventional Fourier series, its result is a discrete sequence, and the number of frequency components is limited to ${\displaystyle N.}$  Thus the distinction discrete Fourier series.

See also

• Fourier series
• Fast Fourier transform
• Laplace transform
• Discrete Fourier transform
  • DFT matrix
• Discrete-time Fourier transform
• Fractional Fourier transform
• Linear canonical transform
• Fourier sine transform
• Short-time Fourier transform
• Analog signal processing
• Transform (mathematics)

Notes

1. We may note that the same description applies to any inverse DFT. The distinction in this case is that the individual terms of the periodic summation ${\displaystyle {\tilde {x}}[n]}$ are not limited to sequences of length ${\displaystyle N.}$[1]^{:pp 557–558} [2]^{:p 76}

References

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. 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].  url=https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf
2. Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (PDF) (1 ed.). Boca Raton,FL: CRC Press. p. 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