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+---
+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+    unpublished.
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