From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/impulse-response/index.md | 81 +++++++++++++++++++++++++++ 1 file changed, 81 insertions(+) create mode 100644 source/know/concept/impulse-response/index.md (limited to 'source/know/concept/impulse-response') diff --git a/source/know/concept/impulse-response/index.md b/source/know/concept/impulse-response/index.md new file mode 100644 index 0000000..65849aa --- /dev/null +++ b/source/know/concept/impulse-response/index.md @@ -0,0 +1,81 @@ +--- +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 $u_p(t)$: + +$$\begin{aligned} + \hat{L} \{ u(t) \} = f(t) + \quad \implies \quad + \boxed{ + u(t) = (f * u_p)(t) + } +\end{aligned}$$ + +
+ + + +
+ +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 O. Bang, + *Applied mathematics for physicists: lecture notes*, 2019, + unpublished. -- cgit v1.2.3