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Box-Muller Transformation

A transformation which transforms from a 2-D continuous Uniform Distribution to a 2-D Gaussian Bivariate Distribution (or Complex Gaussian Distribution). If $x_1$ and $x_2$ are uniformly and independently distributed between 0 and 1, then $z_1$ and $z_2$ as defined below have a Gaussian Distribution with Mean $\mu = 0$ and Variance $\sigma^2=1$.

$\displaystyle z_1$ $\textstyle =$ $\displaystyle \sqrt{-2\ln x_1} \cos(2\pi x_2)$ (1)
$\displaystyle z_2$ $\textstyle =$ $\displaystyle \sqrt{-2\ln x_1} \sin(2\pi x_2).$ (2)

This can be verified by solving for $x_1$ and $x_2$,
$\displaystyle x_1$ $\textstyle =$ $\displaystyle e^{-({z_1}^2+{z_2}^2)/2}$ (3)
$\displaystyle x_2$ $\textstyle =$ $\displaystyle {1\over 2\pi} \tan^{-1}\left({z_2\over z_1}\right).$ (4)

Taking the Jacobian yields
$\displaystyle {\partial(x_1,x_2)\over\partial(z_1,z_2)}$ $\textstyle =$ $\displaystyle \left\vert\begin{array}{ccc}{\partial x_1\over\partial z_1} & {\p...
...r\partial z_1} & {\partial x_2\over\partial z_2}\end{array}\right\vert\nonumber$  
  $\textstyle =$ $\displaystyle -\left[{{1\over\sqrt{2\pi}}e^{-{z_1}^2/2}}\right]\left[{{1\over\sqrt{2\pi}}e^{-{z_2}^2/2}}\right].$ (5)

© 1996-9 Eric W. Weisstein