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-rw-r--r--source/know/concept/convolution-theorem/index.md23
1 files changed, 9 insertions, 14 deletions
diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md
index 742c8ff..510417a 100644
--- a/source/know/concept/convolution-theorem/index.md
+++ b/source/know/concept/convolution-theorem/index.md
@@ -12,6 +12,8 @@ is equal to a product in the frequency domain. This is especially useful
for computation, replacing an $$\mathcal{O}(n^2)$$ convolution with an
$$\mathcal{O}(n \log(n))$$ transform and product.
+
+
## Fourier transform
The convolution theorem is usually expressed as follows, where
@@ -27,11 +29,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-fourier"/>
-<label for="proof-fourier">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-fourier">Proof.</label>
+
+{% include proof/start.html id="proof-fourier" -%}
We expand the right-hand side of the theorem and
rearrange the integrals:
@@ -57,8 +56,8 @@ $$\begin{aligned}
&= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'}
= B \cdot (\tilde{f} * \tilde{g})(k)
\end{aligned}$$
-</div>
-</div>
+{% include proof/end.html id="proof-fourier" %}
+
## Laplace transform
@@ -79,11 +78,8 @@ $$\begin{aligned}
\boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-laplace"/>
-<label for="proof-laplace">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-laplace">Proof.</label>
+
+{% include proof/start.html id="proof-laplace" -%}
We expand the left-hand side.
Note that the lower integration limit is 0 instead of $$-\infty$$,
because we set both $$f(t)$$ and $$g(t)$$ to zero for $$t < 0$$:
@@ -106,8 +102,7 @@ $$\begin{aligned}
&= \int_0^\infty \tilde{f}(s) \: g(t') \exp(- s t') \dd{t'}
= \tilde{f}(s) \: \tilde{g}(s)
\end{aligned}$$
-</div>
-</div>
+{% include proof/end.html id="proof-laplace" %}