Convolution

Not that convolution...

Discrete time convolution:

$$ (f \ast g)[n] \equiv \sum_{m = -\infty}^{\infty} f[m]g[n-m] $$

Continuous time convolution:

$$ (f \ast g)(t) \equiv \int_{-\infty}^{\infty} f(\tau)g(t - \tau) d\tau $$

• Convolution is commutative:

$$ f \ast g = g \ast f $$

• Convolution is associative:

$$ f \ast (g \ast h) = (f \ast g) \ast h $$

• Convolution is distributive over addition:

$$ (f + g) \ast x = f \ast x + g \ast x $$

• Convolution in the time/spatial domain is equivalent to multiplication in the frequency domain. The inverse is also true.