1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
|
---
title: "Impulse response"
firstLetter: "I"
publishDate: 2021-03-09
categories:
- Mathematics
- Physics
date: 2021-03-09T20:34:38+01:00
draft: false
markup: pandoc
---
# 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](/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.
|
