Definition:Discrete Fourier Transform/Inverse

Definition

Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ be a finite sequence of complex numbers for some $n \in \N_{>0}$.

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 see

• Definition:Discrete Fourier Transform

• Inverse of Discrete Fourier Transform, proving that the inverse is truly what it is

• Results about discrete Fourier transforms can be found here.

Source of Name

This entry was named for Joseph Fourier.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discrete Fourier transform