Filtering in the Fourier domain

• Let a filter with an impulse response $h$: $$ \begin{aligned} y[t] & = (h*x)[t]\\ y[t] & = \sum_{n=-\infty}^{+\infty} h[n]x[t-n] \end{aligned} $$ then (if the Fourier transform exists) $$ \hat y[\nu] = \hat h[\nu]\hat x[\nu] $$

• For finite sequences (digital signals), the underlying convolution is circular, with period $N$ or $2N$. The Fourier transform must be chosen accordingly using zeros padding.

• Filtering a signal acts directly on the spectrum