📂Fourier Analysis

Discrete Fourier Transform Properties

Properties¹

Let’s denote the Discrete Fourier Transform as $\mathscr{F}_{N}$ or $\hat{\mathbf{a}}$ given $\mathbf{a} \in \mathbb{C}^{N}$.

• Convolution

  $$\mathscr{F}_{N}(\mathbf{a} \ast \mathbf{b}) = \hat{\mathbf{a}} \hat{\mathbf{b}} = (\hat{a}_{0}\hat{b}_{0}, \dots, \hat{a}_{N-1}\hat{b}_{N-1})$$

  In this case, $\ast$ is the discrete convolution.

Explanation

The Discrete Fourier Transform also satisfies the properties that the Fourier Transform does.