# Impulse Response

For a linear DE with constant coefficients, the Zero-State Response is $Y(s)=H(s)F(s)$

where

- $H(s)$ is the Transfer Function of the DE
- $F(s)$ is the Laplace transform of the forcing term

If the forcing term is the delta impulse function (i.e. $f(t)=δ(t)$), then $F(s)=1$ and thus $Y(s)=H(t)⋅1=H(t).$ Thus we also call $h(t)=L_{−1}{H(s)}$ the impulse response of the DE.

The above holds for Linear DEs with constant coefficients but what about more general LTIs. In general these might not be able to be modelled by a DE… so how do we find their responses to general inputs?

See LTI for theorem 1.