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:

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

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.