Categories: Mathematics, Physics.

# Impulse response

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.