Definition:Discrete Fourier Transform
Definition
Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ be a finite sequence of complex numbers for some $n \in \N_{>0}$.
The discrete Fourier transform of $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ is the sequence $\sequence {\hat x_k}_{1 \mathop \le k \mathop \le n}$ where:
- $\hat x_k = \ds \sum_{r \mathop = 1}^n x_r \map \exp {\dfrac {-2 \pi i k r} n}$
Inverse Discrete Fourier Transform
Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ be a finite sequence of complex numbers for some $n \in \N_{>0}$.
Let $\sequence {\hat x_k}_{1 \mathop \le k \mathop \le n}$ be the discrete Fourier transform of $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$.
The inverse discrete Fourier transform of $\sequence {\hat x_k}_{1 \mathop \le k \mathop \le n}$ is:
- $x_k = \ds \dfrac 1 n \sum_{r \mathop = 1}^n \hat x_r \map \exp {\dfrac {2 \pi i k r} n}$
Also known as
A discrete Fourier transform is also known as a finite Fourier transform.
Also see
- Results about discrete Fourier transforms can be found here.
Source of Name
This entry was named for Joseph Fourier.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discrete Fourier transform
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discrete Fourier transform