Convolution

A convolution is a kind of product that results in a blend/weighted overlap/smooth combination between two functions based on how much one function overlaps with a shifted and flipped version of another. General Convolution

$$(f\ast g)(t)={\int }_{-\infty }^{\infty }f(t-s)g(s)ds$$

Causal Convolution (used with Laplace Transforms)

$$(f\ast g)(t)={\int }_0^{t}f(t-s)g(s)ds$$

Commutative Property

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

Associative Property

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

Distributive Over Addition

$$h\ast x_1+h\ast x_2=h\ast (x_1+x_2)$$