GeezerVison

The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable $Z$ whose real and imaginary parts are independent normally distributed random variables with mean zero and variance $1/2$.^{[3]: p. 494 [4]: pp. 501} Formally,

where $\perp \perp$ denotes independence and $Z\sim 𝒞𝒩(0,1)$ denotes that $Z$ is a standard complex normal random variable.

Suppose $X$ and $Y$ are real random variables such that $(X,Y{)}^T$ is a 2-dimensional normal random vector. Then the complex random variable $Z=X+iY$ is called complex normal random variable or complex Gaussian random variable.^{[3]: p. 500}

A n-dimensional complex random vector $𝐙=(Z_1,\dots ,Z_n{)}^T$ is a complex standard normal random vector or complex standard Gaussian random vector if its components are independent and all of them are standard complex normal random variables as defined above.^{[3]: p. 502 [4]: pp. 501} That $𝐙$ is a standard complex normal random vector is denoted $𝐙\sim 𝒞𝒩(0,{𝑰}_n)$.

If $𝐗=(X_1,\dots ,X_n{)}^T$ and $𝐘=(Y_1,\dots ,Y_n{)}^T$ are random vectors in ${ℝ}^n$ such that $[𝐗,𝐘]$ is a normal random vector with $2n$ components. Then we say that the complex random vector

$𝐙=𝐗+i𝐘$

is a complex normal random vector or a complex Gaussian random vector.

A complex random vector $𝐙$ is called circularly symmetric if for every deterministic $\phi \in [-\pi ,\pi )$ the distribution of $e^{i\phi }𝐙$ equals the distribution of $𝐙$.^{[4]: pp. 500–501}

Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix $\Gamma$.

The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e. $\mu =0$ and $C=0$.^{[3]: p. 507 [7]} This is usually denoted

$𝐙\sim 𝒞𝒩(0,\Gamma )$

If $𝐙=𝐗+i𝐘$ is circularly-symmetric (central) complex normal, then the vector $[𝐗,𝐘]$ is multivariate normal with covariance structure

$\left(\begin{array}{c}𝐗\\ 𝐘\end{array}\right)\sim 𝒩(\left[\begin{array}{c}0\\ 0\end{array}\right],\frac{1}{2}\left[\begin{array}{cc}\mathrm{Re}\Gamma & -\mathrm{Im}\Gamma \\ \mathrm{Im}\Gamma & \mathrm{Re}\Gamma \end{array}\right])$

where $\Gamma =E\left[𝐙{𝐙}^H\right]$.

For nonsingular covariance matrix $\Gamma$, its distribution can also be simplified as^{[3]: p. 508}

$f_{𝐙}\left(𝐳\right)=\frac{1}{{\pi }^{n}det\left(\Gamma \right)}{e}^{-(𝐳-𝝁{)}^H{\Gamma }^{-1}(𝐳-𝝁)}$.

Therefore, if the non-zero mean $\mu$ and covariance matrix $\Gamma$ are unknown, a suitable log likelihood function for a single observation vector $z$ would be

$\ln \left(L\right(\mu ,\Gamma \left)\right)=-\ln (det(\Gamma \left)\right)-{\overline{(z-\mu )}}^{\prime }{\Gamma }^{-1}(z-\mu )-n\ln \left(\pi \right).$

The standard complex normal (defined in Eq.1) corresponds to the distribution of a scalar random variable with $\mu =0$, $C=0$ and $\Gamma =1$. Thus, the standard complex normal distribution has density

$f_Z\left(z\right)=\frac{1}{\pi }{e}^{-\bar{z}z}=\frac{1}{\pi }{e}^{-|z{|}^2}.$

The above expression demonstrates why the case $C=0$, $\mu =0$ is called “circularly-symmetric”. The density function depends only on the magnitude of $z$ but not on its argument. As such, the magnitude $\left|z\right|$ of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude $|z{|}^2$ will have the exponential distribution, whereas the argument will be distributed uniformly on $[-\pi ,\pi ]$.

If $\{{𝐙}_1,\dots ,{𝐙}_k\}$ are independent and identically distributed n-dimensional circular complex normal random vectors with $\mu =0$, then the random squared norm

$Q=\sum _{j=1}^k{𝐙}_j^H{𝐙}_j=\sum _{j=1}^k\Vert {𝐙}_j{\Vert }^2$

has the generalized chi-squared distribution and the random matrix

$W=\sum _{j=1}^k{𝐙}_j{𝐙}_j^H$

has the complex Wishart distribution with $k$ degrees of freedom. This distribution can be described by density function

$f\left(w\right)=\frac{det({\Gamma }^{-1}{)}^{k}det(w{)}^{k-n}}{{\pi }^{n(n-1)/2}{\prod }_{j=1}^k(k-j)!}{e}^{-\mathrm{tr}\left({\Gamma }^{-1}w\right)}$

where $k\ge n$, and $w$ is a $n\times n$ nonnegative-definite matrix.