From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/convolution-theorem/index.md | 116 +++++++++++++++++++++++ 1 file changed, 116 insertions(+) create mode 100644 source/know/concept/convolution-theorem/index.md (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 new file mode 100644 index 0000000..d6c578b --- /dev/null +++ b/source/know/concept/convolution-theorem/index.md @@ -0,0 +1,116 @@ +--- +title: "Convolution theorem" +date: 2021-02-22 +categories: +- Mathematics +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. + +## 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: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + A \cdot (f * g)(x) &= \hat{\mathcal{F}}{}^{-1}\{\tilde{f}(k) \: \tilde{g}(k)\} \\ + B \cdot (\tilde{f} * \tilde{g})(k) &= \hat{\mathcal{F}}\{f(x) \: g(x)\} + \end{aligned} + } +\end{aligned}$$ + +
+ + + +
+ + +## Laplace transform + +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/): + +$$\begin{aligned} + \boxed{(f * g)(t) = \hat{\mathcal{L}}{}^{-1}\{\tilde{f}(s) \: \tilde{g}(s)\}} +\end{aligned}$$ + +Because the inverse Laplace transform $\hat{\mathcal{L}}{}^{-1}$ is +unpleasant, the theorem is often stated using the forward transform +instead: + +$$\begin{aligned} + \boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)} +\end{aligned}$$ + +
+ + + +
+ + + +## References +1. O. Bang, + *Applied mathematics for physicists: lecture notes*, 2019, + unpublished. -- cgit v1.2.3