Frequency-Domain Convolution

In discrete time,

$$y(n) \isdefs (x\ast h)(n) \isdefs \sum_{m=0}^{\infty} x(m)h(n-m), \quad n=0,1,2,\ldots$$

where $x$ and $h$ are assumed causal

Convolution Theorem:

$$(h\ast x) \;\longleftrightarrow\;H \cdot X$$

or

   DTFT$$_{\omega}(h\ast x)=H(\omega)X(\omega)$$

where $H$ and $X$ are the DTFTs of $h$ and $x$ , respectively.