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author | Prefetch | 2022-10-27 20:40:09 +0200 |
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committer | Prefetch | 2022-10-27 20:40:09 +0200 |
commit | 6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch) | |
tree | a8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/convolution-theorem | |
parent | bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff) |
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/convolution-theorem')
-rw-r--r-- | source/know/concept/convolution-theorem/index.md | 23 |
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" %} |