From a8d31faecc733fa4d63fde58ab98a5e9d11029c2 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 2 Apr 2023 16:57:12 +0200
Subject: Improve knowledge base

---
 source/know/concept/impulse-response/index.md | 61 +++++++++++++++------------
 1 file changed, 34 insertions(+), 27 deletions(-)

(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
index 661ed3f..8210f5c 100644
--- a/source/know/concept/impulse-response/index.md
+++ b/source/know/concept/impulse-response/index.md
@@ -8,68 +8,75 @@ categories:
 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
+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/) $$\delta(t)$$:
 
 $$\begin{aligned}
     \boxed{
-        \hat{L} \{ u_p(t) \} = \delta(t)
+        \hat{L} \{ u_\delta(t) \}
+        = \delta(t)
     }
 \end{aligned}$$
 
-This can be used to find the response $$u(t)$$ of $$\hat{L}$$ to
-*any* forcing function $$f(t)$$, i.e. not only $$\delta(t)$$,
-by simply taking the convolution with $$u_p(t)$$:
+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 $$u_\delta(t)$$:
 
 $$\begin{aligned}
-    \hat{L} \{ u(t) \} = f(t)
+    \hat{L} \{ u(t) \}
+    = f(t)
     \quad \implies \quad
     \boxed{
-        u(t) = (f * u_p)(t)
+        u(t)
+        = (f * u_\delta)(t)
     }
 \end{aligned}$$
 
 
 {% include proof/start.html id="proof-theorem" -%}
-Starting from the definition of $$u_p(t)$$,
+Starting from the definition of $$u_\delta(t)$$,
 we shift the argument by some constant $$\tau$$,
-and multiply both sides by the constant $$f(\tau)$$:
+and multiply both sides by $$f(\tau)$$:
 
 $$\begin{aligned}
-    \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau)
+    \hat{L} \{ u_\delta(t - \tau) \}
+    &= \delta(t - \tau)
     \\
-    \hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau)
+    \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \}
+    &= f(\tau) \: \delta(t - \tau)
 \end{aligned}$$
 
-Where $$f(\tau)$$ can be moved inside using the
-linearity of $$\hat{L}$$. Integrating over $$\tau$$ then gives us:
+Where $$f(\tau)$$ was moved inside thanks to the linearity of $$\hat{L}$$.
+Integrating over $$\tau$$ gives us:
 
 $$\begin{aligned}
-    \int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau}
+    \int_0^\infty \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \} \dd{\tau}
     &= \int_0^\infty f(\tau) \: \delta(t - \tau) \dd{\tau}
     = f(t)
 \end{aligned}$$
 
-The integral and $$\hat{L}$$ are operators of different variables, so we reorder them:
+The integral and $$\hat{L}$$ are operators of different variables, so we reorder them,
+and recognize that the resulting integral is a convolution:
 
 $$\begin{aligned}
-    \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau}
-    &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t)
+    f(t)
+    &= \hat{L} \int_0^\infty f(\tau) \: u_\delta(t - \tau) \dd{\tau}
+    = \hat{L} \Big\{ (f * u_\delta)(t) \Big\}
 \end{aligned}$$
+
+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
-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.
+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/).
 
 
 
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