Categories: Mathematics, Physics.

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 \(\delta(t)\):

\[\begin{aligned} \boxed{ \hat{L} \{ u_p(t) \} = \delta(t) } \end{aligned}\]

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 \(u_p(t)\):

\[\begin{aligned} \boxed{ \hat{L} \{ u(t) \} = f(t) \quad \implies \quad u(t) = (f * u_p)(t) } \end{aligned}\]

**Proof.** Starting from the definition of \(u_p(t)\), we shift the argument by some constant \(\tau\), and multiply both sides by the constant \(f(\tau)\):

\[\begin{aligned} \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau) \\ \hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau) \end{aligned}\]

*Where \(f(\tau)\) can be moved inside using the linearity of \(\hat{L}\). Integrating over \(\tau\) then gives us:*

\[\begin{aligned} \int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau} &= \int_0^\infty f(\tau) \: \delta(t - \tau) \dd{\tau} = f(t) \end{aligned}\]

*The integral and \(\hat{L}\) are operators of different variables, so we reorder them:*

\[\begin{aligned} \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau} &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t) \end{aligned}\]

**Q.E.D.**

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