---
title: "Impulse response"
sort_title: "Impulse response"
date: 2021-03-09
categories:
- Mathematics
- Physics
layout: "concept"
---

Given a system whose behaviour is described by a linear operator $$\hat{L}$$,
its **impulse response** $$u_\delta(t)$$ is defined as the system's response
when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(t)$$:

$$\begin{aligned}
    \boxed{
        \hat{L} \{ u_\delta(t) \}
        = \delta(t)
    }
\end{aligned}$$

This can be used to find the response $$u(t)$$ of $$\hat{L}$$
to *any* forcing function $$f(t)$$,
by simply taking the convolution with $$u_\delta(t)$$:

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


{% include proof/start.html id="proof-theorem" -%}
Starting from the definition of $$u_\delta(t)$$,
we shift the argument by some constant $$\tau$$,
and multiply both sides by $$f(\tau)$$:

$$\begin{aligned}
    \hat{L} \{ u_\delta(t - \tau) \}
    &= \delta(t - \tau)
    \\
    \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \}
    &= f(\tau) \: \delta(t - \tau)
\end{aligned}$$

Where $$f(\tau)$$ was moved inside thanks to the linearity of $$\hat{L}$$.
Integrating over $$\tau$$ gives us:

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

The integral and $$\hat{L}$$ are operators of different variables, so we reorder them,
and recognize that the resulting integral is a convolution:

$$\begin{aligned}
    f(t)
    &= \hat{L} \int_0^\infty f(\tau) \: u_\delta(t - \tau) \dd{\tau}
    = \hat{L} \Big\{ (f * u_\delta)(t) \Big\}
\end{aligned}$$

Because $$\hat{L} \{ u(t) \} = f(t)$$ by definition,
we then see that $$(f * u_\delta)(t) = u(t)$$.
{% include proof/end.html id="proof-theorem" %}


This is useful for solving initial value problems,
because any initial condition can be satisfied thanks to linearity,
by choosing the initial values of the homogeneous solution $$\hat{L}\{ u_0(t) \} = 0$$
such that the total solution $$(f * u_\delta)(t) + u_0(t)$$ has the desired values.

For boundary value problems, there is the related concept of
a [fundamental solution](/know/concept/fundamental-solution/).



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    unpublished.