The Complex Gaussian Distribution

This is the beginning of my plan. Or this is a manifesto, a motivation for me. Note what I read, good or bad, old or new, Tao or method. Maybe sometimes not care about the formal usage of the language. From the words I write down, may that someday I can comprehend the theory directly without the words.

The normal or Gaussian distribution is often denoted by $\mathcal N(\mu,\sigma^2)$. When a random variable $X$ is distributed normally with mean $\mu$ and variance $\sigma^2$, we write $X \sim \mathcal N(\mu,\sigma^2)$. The formula for the distribution is

$$f(\boldsymbol x) = \frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{\boldsymbol x^2}{2\sigma^2}).$$

The additive complex Gaussian $\boldsymbol n = \boldsymbol n_r + j\boldsymbol n_i$ distribution is $\mathcal{CN}(0,\sigma^2).$

The real part is $\mathcal N(0,{\sigma^2}/2)$

$$f(\boldsymbol n_r) = \frac{1}{\sqrt{\pi}\sigma}\exp(-\frac{\boldsymbol n_r^2}{\sigma^2}).$$

The imaginary part is $\mathcal N(0,{\sigma^2}/2)$

$$f(\boldsymbol n_r) = \frac{1}{\sqrt{\pi}\sigma}\exp(-\frac{\boldsymbol n_i^2}{\sigma^2}).$$

Because $\boldsymbol n_r$ and $\boldsymbol n_i$ are independent random variables, then

$$\begin{equation*} \begin{aligned}{} f(\boldsymbol n) &= f(\boldsymbol n_r + j\boldsymbol n_i)\\ &= f(\boldsymbol n_r)f(\boldsymbol n_i) \\ &= \frac{1}{\pi\sigma^2}\exp(-\frac{{\boldsymbol n_r^2}+{\boldsymbol n_i^2}}{\sigma^2}) \\ &= \frac{1}{\pi\sigma^2}\exp(-\frac{\|\boldsymbol n\|^2}{\sigma^2}). \end{aligned} \end{equation*}$$

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