Convolution

The main purpose of using the Impulse Response. This is equivalent to multiplication in the LaPlace domain.

$Y\left(s\right)=P\left(s\right)U\left(s\right)\iff y\left(t\right)=\left(p\ast u\right)\left(t\right)={\int }_0^{t}p\left(t-\tau \right)u\left(T\right)dT$ where the same thing applies for Z Transforms.