--- title: "Impulse response" sort_title: "Impulse response" date: 2021-03-09 categories: - Mathematics - Physics layout: "concept" --- The **impulse response** $$u_p(t)$$ of a system whose behaviour is described by a linear operator $$\hat{L}$$, is defined as the reponse of the system when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(t)$$: $$\begin{aligned} \boxed{ \hat{L} \{ u_p(t) \} = \delta(t) } \end{aligned}$$ This can be used to find the response $$u(t)$$ of $$\hat{L}$$ to *any* forcing function $$f(t)$$, i.e. not only $$\delta(t)$$, by simply taking the convolution with $$u_p(t)$$: $$\begin{aligned} \hat{L} \{ u(t) \} = f(t) \quad \implies \quad \boxed{ u(t) = (f * u_p)(t) } \end{aligned}$$