takes a DISCRETE signal as input and outputs a DISCRETE frequency spectrum
transforms data in the time or spatial domain into the frequency domain
should technically be called a Discrete Fourier Series
is like a Fourier Series on data instead of analytic functions
converts a finite sequence of equally-spaced samples of a function 𝑓(𝑥) into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency
is a unitary, invertible, linear transformation
$F : C^{N} \to C^{N}$
most often uses the FFT algorithm to compute the DFT
is the inverse of the Inverse Discrete Fourier Transform (IDFT)

DFT - Definition

DFT transforms a sequence of 𝑁 complex numbers {𝑥₀, 𝑥₁, …, 𝑥_𝑁-1} into another sequence of complex numbers, {𝑋₀, 𝑋₁, …, 𝑋_𝑁-1}, which is defined by:

$X_{k} = \sum_{n = 0}^{N - 1} e^{- \frac{i 2 π}{N} k n} \cdot x_{n}$
$X_{0} X_{1} X_{2} ⋮ X_{N - 1} = 111 ⋮ 1 1 w^{1} w^{2} ⋮ w^{N - 1} 1 w^{2} w^{4} ⋮ w^{2 (N - 1)} \dots \dots \dots ⋱ \dots 1 w^{N - 1} w^{2 (N - 1)} ⋮ w^{(N - 1)^{2}} x_{0} x_{1} x_{2} ⋮ x_{N - 1}$

where:

IDFT - Definition

IDFT transforms a sequence of 𝑁 complex numbers {𝑋₀, 𝑋₁, …, 𝑋_𝑁-1} into another sequence of complex numbers, {𝑥₀, 𝑥₁, …, 𝑥_𝑁-1}, which is defined by:

$x_{k} = \frac{1}{N} \sum_{n = 0}^{N - 1} e^{\frac{i 2 π}{N} k n} \cdot X_{n}$
$x_{0} x_{1} x_{2} ⋮ x_{N - 1} = 111 ⋮ 1 1 W^{1} W^{2} ⋮ W^{N - 1} 1 W^{2} W^{4} ⋮ W^{2 (N - 1)} \dots \dots \dots ⋱ \dots 1 W^{N - 1} W^{2 (N - 1)} ⋮ W^{(N - 1)^{2}} X_{0} X_{1} X_{2} ⋮ X_{N - 1}$

where: