From f9cce7d563d0ea2ac591c31ff7d248ad3d02d1ac Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 3 Jun 2021 19:30:38 +0200
Subject: Expand knowledge base
---
content/know/concept/impulse-response/index.pdc | 19 ++++++++++++-------
1 file changed, 12 insertions(+), 7 deletions(-)
(limited to 'content/know/concept/impulse-response/index.pdc')
diff --git a/content/know/concept/impulse-response/index.pdc b/content/know/concept/impulse-response/index.pdc
index b055fe7..fa921fa 100644
--- a/content/know/concept/impulse-response/index.pdc
+++ b/content/know/concept/impulse-response/index.pdc
@@ -35,9 +35,14 @@ $$\begin{aligned}
}
\end{aligned}$$
-*__Proof.__ Starting from the definition of $u_p(t)$,
+<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 $f(\tau)$:*
+and multiply both sides by the constant $f(\tau)$:
$$\begin{aligned}
\hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau)
@@ -45,8 +50,8 @@ $$\begin{aligned}
\hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau)
\end{aligned}$$
-*Where $f(\tau)$ can be moved inside using the
-linearity of $\hat{L}$. Integrating over $\tau$ then gives us:*
+Where $f(\tau)$ can be moved inside using the
+linearity of $\hat{L}$. Integrating over $\tau$ then gives us:
$$\begin{aligned}
\int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau}
@@ -54,14 +59,14 @@ $$\begin{aligned}
= f(t)
\end{aligned}$$
-*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:
$$\begin{aligned}
\hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau}
&= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t)
\end{aligned}$$
-
-*__Q.E.D.__*
+</div>
+</div>
--
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