---
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 $u_p(t)$:

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

<div class="accordion">
<input type="checkbox" id="proof-main"/>
<label for="proof-main">Proof</label>
<div class="hidden">
<label for="proof-main">Proof.</label>
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:

$$\begin{aligned}
    \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau}
    &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t)
\end{aligned}$$
</div>
</div>

This is useful for solving initial value problems,
because any initial condition can be satisfied
due to the linearity of $\hat{L}$,
by choosing the initial values of the homogeneous solution $\hat{L}\{ u_h(t) \} = 0$
such that the total solution $(f * u_p)(t) + u_h(t)$
has the desired values.

Meanwhile, for boundary value problems,
the related [fundamental solution](/know/concept/fundamental-solution/)
is preferable.



## References
1.  O. Bang,
    *Applied mathematics for physicists: lecture notes*, 2019,
    unpublished.