# Complex normal distribution

In probability theory, the family of complex normal distributions, denoted ${\mathcal {CN}}$ or ${\mathcal {N}}_{\mathcal {C}}$, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix $\Gamma$, and the relation matrix $C$. The standard complex normal is the univariate distribution with $\mu =0$, $\Gamma =1$, and $C=0$.
An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: $\mu =0$ and $C=0$. This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.

## Definitions
### Complex standard normal random variable

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 $1/2$.$^{:p.494}$$^{:pp.501}$ Formally,
where $Z\sim {\mathcal {CN}}(0,1)$ denotes that $Z$ is a standard complex normal random variable.

### Complex normal random variable

Suppose $X$ and $Y$ are real random variables such that $(X,Y)^{\mathrm {T} }$ 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.$^{:p.500}$

### Complex standard normal random vector

A n-dimensional complex random vector $\mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\mathrm {T} }$ 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.$^{:p.502}$$^{:pp.501}$ That $\mathbf {Z}$ is a standard complex normal random vector is denoted $\mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})$.

### Complex normal random vector

If $\mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }$ and $\mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\mathrm {T} }$ are random vectors in $\mathbb {R} ^{n}$ such that $[\mathbf {X} ,\mathbf {Y} ]$ is a normal random vector with $2n$ components. Then we say that the complex random vector
    $\mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,$
is a complex normal random vector or a complex Gaussian random vector.

## Mean, covariance, and relation

The complex Gaussian distribution can be described with 3 parameters:
    $\mu =\operatorname {E} [\mathbf {Z} ],\quad \Gamma =\operatorname {E} [(\mathbf {Z} -\mu )({\mathbf {Z} }-\mu )^{\mathrm {H} }],\quad C=\operatorname {E} [(\mathbf {Z} -\mu )(\mathbf {Z} -\mu )^{\mathrm {T} }],$
where $\mathbf {Z} ^{\mathrm {T} }$ denotes matrix transpose of $\mathbf {Z}$, and $\mathbf {Z} ^{\mathrm {H} }$ denotes conjugate transpose.$^{:p.504}$$^{:pp.500}$
Here the location parameter $\mu$ is a n-dimensional complex vector; the covariance matrix $\Gamma$ is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix $C$ is symmetric. The complex normal random vector $\mathbf {Z}$ can now be denoted as$\mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu ,\ \Gamma ,\ C).$Moreover, matrices $\Gamma$ and $C$ are such that the matrix
    $P={\overline {\Gamma }}-{C}^{\mathrm {H} }\Gamma ^{-1}C$
is also non-negative definite where ${\overline {\Gamma }}$ denotes the complex conjugate of $\Gamma$.

## Relationships between covariance matrices

As for any complex random vector, the matrices $\Gamma$ and $C$ can be related to the covariance matrices of $\mathbf {X} =\Re (\mathbf {Z} )$ and $\mathbf {Y} =\Im (\mathbf {Z} )$ via expressions
    ${\begin{aligned}&V_{XX}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma +C],\quad V_{XY}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [-\Gamma +C],\\&V_{YX}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [\Gamma +C],\quad \,V_{YY}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma -C],\end{aligned}}$
and conversely
    ${\begin{aligned}&\Gamma =V_{XX}+V_{YY}+i(V_{YX}-V_{XY}),\\&C=V_{XX}-V_{YY}+i(V_{YX}+V_{XY}).\end{aligned}}$

## Density function

The probability density function for complex normal distribution can be computed as
    ${\begin{aligned}f(z)&={\frac {1}{\pi ^{n}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac {1}{2}}{\begin{bmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{bmatrix}}^{\mathrm {H} }{\begin{bmatrix}\Gamma &C\\{\overline {C}}&{\overline {\Gamma }}\end{bmatrix}}^{\!\!-1}\!{\begin{bmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{bmatrix}}\right\}\\[8pt]&={\tfrac {\sqrt {\det \left({\overline {P^{-1}}}-R^{\ast }P^{-1}R\right)\det(P^{-1})}}{\pi ^{n}}}\,e^{-(z-\mu )^{\ast }{\overline {P^{-1}}}(z-\mu )+\operatorname {Re} \left((z-\mu )^{\intercal }R^{\intercal }{\overline {P^{-1}}}(z-\mu )\right)},\end{aligned}}$
where $R=C^{\mathrm {H} }\Gamma ^{-1}$ and $P={\overline {\Gamma }}-RC$.

## Characteristic function

The characteristic function of complex normal distribution is given by
    $\varphi (w)=\exp \!{\big \i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big{\overline {w}}'\Gamma w+\operatorname {Re} ({\overline {w}}'C{\overline {w}}){\big )}{\big \}},$
where the argument $w$ is an n-dimensional complex vector.

## Properties

* If $\mathbf {Z}$ is a complex normal n-vector, ${\boldsymbol {A}}$ an m×n matrix, and $b$ a constant m-vector, then the linear transform ${\boldsymbol {A}}\mathbf {Z} +b$ will be distributed also complex-normally:
    $Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma A^{\mathrm {H} },\,ACA^{\mathrm {T} })$
* If $\mathbf {Z}$ is a complex normal n-vector, then
    $2{\Big(\mathbf {Z} -\mu )^{\mathrm {H} }{\overline {P^{-1}}}(\mathbf {Z} -\mu )-\operatorname {Re} {\big(\mathbf {Z} -\mu )^{\mathrm {T} }R^{\mathrm {T} }{\overline {P^{-1}}}(\mathbf {Z} -\mu ){\big )}{\Big ]}\ \sim \ \chi ^{2}(2n)$
* Central limit theorem. If $Z_{1},\ldots ,Z_{T}$ are independent and identically distributed complex random variables, then
    ${\sqrt {T}}{\Big{\tfrac {1}{T}}\textstyle \sum _{t=1}^{T}Z_{t}-\operatorname {E} [Z_{t}]{\Big )}\ {\xrightarrow {d}}\ {\mathcal {CN}}(0,\,\Gamma ,\,C),$
    where $\Gamma =\operatorname {E} [ZZ^{\mathrm {H} }]$ and $C=\operatorname {E} [ZZ^{\mathrm {T} }]$.
* The modulus of a complex normal random variable follows a Hoyt distribution.

## Circularly-symmetric central case
### Definition

A complex random vector $\mathbf {Z}$ is called circularly symmetric if for every deterministic $\varphi \in [-\pi ,\pi )$ the distribution of $e^{\mathrm {i} \varphi }\mathbf {Z}$ equals the distribution of $\mathbf {Z}$.$^{:pp.500–501}$
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 $C=0$.$^{:p.507}$ This is usually denoted
    $\mathbf {Z} \sim {\mathcal {CN}}(0,\,\Gamma )$

### Distribution of real and imaginary parts

If $\mathbf {Z} =\mathbf {X} +i\mathbf {Y}$ is circularly-symmetric (central) complex normal, then the vector $[\mathbf {X} ,\mathbf {Y} ]$ is multivariate normal with covariance structure
    ${\begin{pmatrix}\mathbf {X} \\\mathbf {Y} \end{pmatrix}}\ \sim \ {\mathcal {N}}{\Big{\begin{bmatrix}0\\0\end{bmatrix}},\ {\tfrac {1}{2}}{\begin{bmatrix}\operatorname {Re} \,\Gamma &-\operatorname {Im} \,\Gamma \\\operatorname {Im} \,\Gamma &\operatorname {Re} \,\Gamma \end{bmatrix}}{\Big )}$
where $\Gamma =\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{\mathrm {H} }]$.

### Probability density function

For nonsingular covariance matrix $\Gamma$, its distribution can also be simplified as$^{:p.508}$
    $f_{\mathbf {Z} }(\mathbf {z} )={\tfrac {1}{\pi ^{n}\det(\Gamma )}}\,e^{-(\mathbf {z} -\mathbf {\mu } )^{\mathrm {H} }\Gamma ^{-1}(\mathbf {z} -\mathbf {\mu } )}$.
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
    $\ln(L(\mu ,\Gamma ))=-\ln(\det(\Gamma ))-{\overline {(z-\mu )}}'\Gamma ^{-1}(z-\mu )-n\ln(\pi ).$
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
    $f_{Z}(z)={\tfrac {1}{\pi }}e^{-{\overline {z}}z}={\tfrac {1}{\pi }}e^{-|z|^{2}}.$

### Properties

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 $|z|$ 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 $\left\{\mathbf {Z} _{1},\ldots ,\mathbf {Z} _{k}\right\}$ are independent and identically distributed n-dimensional circular complex normal random vectors with $\mu =0$, then the random squared norm
    $Q=\sum _{j=1}^{k}\mathbf {Z} _{j}^{\mathrm {H} }\mathbf {Z} _{j}=\sum _{j=1}^{k}\|\mathbf {Z} _{j}\|^{2}$
has the generalized chi-squared distribution and the random matrix
    $W=\sum _{j=1}^{k}\mathbf {Z} _{j}\mathbf {Z} _{j}^{\mathrm {H} }$
has the complex Wishart distribution with $k$ degrees of freedom. This distribution can be described by density function
    $f(w)={\frac {\det(\Gamma ^{-1})^{k}\det(w)^{k-n}}{\pi ^{n(n-1)/2}\prod _{j=1}^{k}(k-j)!}}\ e^{-\operatorname {tr} (\Gamma ^{-1}w)}$
where $k\geq n$, and $w$ is a $n\times n$ nonnegative-definite matrix.