Lecture 16 (Apr 1)

Convolution Theorem

$\mathcal{F}(f*g)(s)= \hat{f}(s) \cdot \hat{g}(s)$.

Discrete Fourier Transform

$$
\hat{f}_n = \sum_{m=0}^{N-1} f_m e^{-2\pi i mn/N}
$$

Inverse Discrete Fourier Transform

$$
f_m = \frac{1}{N} \sum_{n=0}^{N-1} \hat{f}_n e^{2\pi i mn/N}
$$

Fourier Matrix: $F_N=[\omega^{mn}]$ where $\omega=e^{-2\pi i/N}$.

The inverse of the Fourier matrix: $F_N^{-1}=\frac{1}{N} \bar{F}_N$.

Reading Materials

Lecture Notes: p.90-92