Category:Definitions/Discrete Fourier Transforms

This category contains definitions related to Discrete Fourier Transforms.
Related results can be found in Category:Discrete Fourier Transforms.

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:

$\hat x_k = \ds \sum_{r \mathop = 1}^n x_r \map \exp {\dfrac {-2 \pi i k r} n}$