From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/convolution-theorem/index.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 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 d4655cf..742c8ff 100644 --- a/source/know/concept/convolution-theorem/index.md +++ b/source/know/concept/convolution-theorem/index.md @@ -9,14 +9,14 @@ layout: "concept" The **convolution theorem** states that a convolution in the direct domain 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. +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 -$\hat{\mathcal{F}}$ is the [Fourier transform](/know/concept/fourier-transform/), -and $A$ and $B$ are constants from its definition: +$$\hat{\mathcal{F}}$$ is the [Fourier transform](/know/concept/fourier-transform/), +and $$A$$ and $$B$$ are constants from its definition: $$\begin{aligned} \boxed{ @@ -46,7 +46,7 @@ $$\begin{aligned} \end{aligned}$$ Then we do the same again, -this time starting from a product in the $x$-domain: +this time starting from a product in the $$x$$-domain: $$\begin{aligned} \hat{\mathcal{F}}\{f(x) \: g(x)\} @@ -63,7 +63,7 @@ $$\begin{aligned} ## Laplace transform -For functions $f(t)$ and $g(t)$ which are only defined for $t \ge 0$, +For functions $$f(t)$$ and $$g(t)$$ which are only defined for $$t \ge 0$$, the convolution theorem can also be stated using the [Laplace transform](/know/concept/laplace-transform/): @@ -71,7 +71,7 @@ $$\begin{aligned} \boxed{(f * g)(t) = \hat{\mathcal{L}}{}^{-1}\{\tilde{f}(s) \: \tilde{g}(s)\}} \end{aligned}$$ -Because the inverse Laplace transform $\hat{\mathcal{L}}{}^{-1}$ is +Because the inverse Laplace transform $$\hat{\mathcal{L}}{}^{-1}$$ is unpleasant, the theorem is often stated using the forward transform instead: @@ -85,8 +85,8 @@ $$\begin{aligned}