---
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 $$u_p(t)$$:

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


{% include proof/start.html id="proof-theorem" -%}
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:

$$\begin{aligned}
    \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau}
    &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t)
\end{aligned}$$
{% include proof/end.html id="proof-theorem" %}


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    unpublished.