Indigital signal processing, adiscrete Fourier series (DFS) is aFourier series whose sinusoidal components are functions of a discrete variable instead of a continuous variable. The result of the series is also a function of the discrete variable, i.e. a discrete sequence. A Fourier series, by nature, has a discrete set of components with a discrete set of coefficients, also a discrete sequence. So a DFS is a representation of one sequence in terms of another sequence. Well known examples are theDiscrete Fourier transform and its inverse transform.^{[1]: ch 8.1}

The exponential form of Fourier series is given by:

${\displaystyle s(t)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S[k]\cdot e^{i2\pi \frac{k}{P}t},}$

which is periodic with an arbitrary period denoted by${\displaystyle P.}$ When continuous time${\displaystyle t}$ is replaced by discrete time${\displaystyle nT,}$ for integer values of${\displaystyle n}$ and time interval${\displaystyle T,}$ the series becomes:

${\displaystyle s(nT)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S[k]\cdot e^{i2\pi \frac{k}{P}nT},n\in \mathbb{Z}.}$

With${\displaystyle n}$ constrained to integer values, we normally constrain the ratio${\displaystyle P/T=N}$ to an integer value, resulting in an${\displaystyle N}$-periodic function:

which are harmonics of a fundamental digital frequency${\displaystyle 1/N.}$ The${\displaystyle N}$ subscript reminds us of its periodicity. And we note that some authors will refer to just the${\displaystyle S[k]}$ coefficients themselves as a discrete Fourier series.^{[2]: p.85 (eq 15a)}

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

which is theinverse DFT of one cycle of theperiodic summation,${\displaystyle S_N.}$^{[1]: p.542 (eq 8.4)  [3]: p.77 (eq 4.24)}

• Fourier analysis