From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 27 Oct 2022 20:40:09 +0200 Subject: Optimize last images, add proof template, improve CSS --- source/know/concept/convolution-theorem/index.md | 23 +++++++++-------------- 1 file changed, 9 insertions(+), 14 deletions(-) (limited to 'source/know/concept/convolution-theorem') 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 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-fourier" %} + ## Laplace transform @@ -79,11 +78,8 @@ $$\begin{aligned} \boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)} \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-laplace" %} -- cgit v1.2.3