Impulse Response

Let $P(s)/G[z]$ be a real, rational, and proper transfer function.

Then the impulse response of $P(s)/G[z]$ is

$$p(t)={\mathcal{L}}^{-1}(P(s))$$ $$g[k]={\mathcal{Z}}^{-1}(G[z])$$

Some more nomenclature.

$$Y(s)=P(s)U(s)⟺y(t)=(p\ast u)(t)={\int }_0^{t}p(t-\tau )u(\tau )d\tau$$ $$Y[z]=G[z]U[z]⟺y[k]=(g\ast u)[k]=\sum _{n=0}^{k}g[k-n]u[n]$$

Laplace and Z Transforms are linear. (distributive properly type shit)