# Complex normal distribution

Parameters μ ∈ ℂ k — location Γ ∈ ℂk×k — covariance (positive semi-definite matrix) C ∈ ℂk×k — relation (positive semi-definite matrix) ℂk complicated, see text μ μ Γ ${\displaystyle \exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline {w}}'\Gamma w+\operatorname {Re} ({\overline {w}}'C{\overline {w}}){\big )}{\big \}}}$

In probability theory, the family of complex normal distributions characterizes complex random variables whose real and imaginary parts are jointly normal.[1] The complex normal family has three parameters: location parameter μ, covariance matrix Γ, and the relation matrix C. The standard complex normal is the univariate distribution with μ = 0, Γ = 1, and C = 0.

An important subclass of complex normal family is called the circularly-symmetric complex normal and corresponds to the case of zero relation matrix and zero mean: ${\displaystyle \mu =0}$ and ${\displaystyle C=0}$.[2] Circular symmetric complex normal random variables are used extensively in signal processing, and are sometimes referred to as just complex normal in signal processing literature.

## Definition

Suppose X and Y are random vectors in Rk such that vec[X Y] is a 2k-dimensional normal random vector. Then we say that the complex random vector

${\displaystyle Z=X+iY\,}$

has the complex normal distribution. This distribution can be described with 3 parameters:[3]

${\displaystyle \mu =\operatorname {E} [Z],\quad \Gamma =\operatorname {E} [(Z-\mu )({Z}-\mu )^{\dagger }],\quad C=\operatorname {E} [(Z-\mu )(Z-\mu )^{\intercal }],}$

where ${\displaystyle Z^{\intercal }}$ denotes matrix transpose, and ${\displaystyle Z^{\dagger }}$ denotes conjugate transpose. Here the location parameter ${\displaystyle \mu }$ is a k-dimensional complex vector; the covariance matrix ${\displaystyle \Gamma }$ is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix ${\displaystyle C}$ is symmetric. Moreover, matrices ${\displaystyle \Gamma }$ and ${\displaystyle C}$ are such that the matrix

${\displaystyle P={\overline {\Gamma }}-{C}^{\ast }\Gamma ^{-1}C}$

is also non-negative definite.[3]

## Relationships between covariance matrices

Matrices ${\displaystyle \Gamma }$ and ${\displaystyle C}$ can be related to the covariance matrices of ${\displaystyle X}$ and ${\displaystyle Y}$ via expressions

{\displaystyle {\begin{aligned}&V_{xx}\equiv \operatorname {E} [(X-\mu _{x})(X-\mu _{x})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma +C],\quad V_{xy}\equiv \operatorname {E} [(X-\mu _{x})(Y-\mu _{y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [-\Gamma +C],\\&V_{yx}\equiv \operatorname {E} [(Y-\mu _{y})(X-\mu _{x})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [\Gamma +C],\quad \,V_{yy}\equiv \operatorname {E} [(Y-\mu _{y})(Y-\mu _{y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma -C],\end{aligned}}}

and conversely

{\displaystyle {\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

{\displaystyle {\begin{aligned}f(z)&={\frac {1}{\pi ^{k}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac {1}{2}}{\begin{pmatrix}({\overline {z}}-{\overline {\mu }})^{\intercal }&(z-\mu )^{\intercal }\end{pmatrix}}{\begin{pmatrix}\Gamma &C\\{\overline {C}}&{\overline {\Gamma }}\end{pmatrix}}^{\!\!-1}\!{\begin{pmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{pmatrix}}\right\}\\[8pt]&={\tfrac {\sqrt {\det \left({\overline {P^{-1}}}-R^{\ast }P^{-1}R\right)\det(P^{-1})}}{\pi ^{k}}}\,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 ${\displaystyle R=C^{\ast }\Gamma ^{-1}}$ and P = Γ − RC.

## Characteristic function

The characteristic function of complex normal distribution is given by [3]

${\displaystyle \varphi (w)=\exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline {w}}'\Gamma w+\operatorname {Re} ({\overline {w}}'C{\overline {w}}){\big )}{\big \}},}$

where the argument ${\displaystyle w}$ is a k-dimensional complex vector.

## Properties

• If Z is a complex normal k-vector, A an ℓ×k matrix, and b a constant -vector, then the linear transform AZ + b will be distributed also complex-normally:
${\displaystyle Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma {\overline {A}}',\,ACA')}$
• If Z is a complex normal k-vector, then
${\displaystyle 2{\Big [}({\overline {Z}}-{\overline {\mu }})'{\overline {P^{-1}}}(Z-\mu )-\operatorname {Re} {\big (}(Z-\mu )'R'{\overline {P^{-1}}}(Z-\mu ){\big )}{\Big ]}\ \sim \ \chi ^{2}(2k)}$
• Central limit theorem. If z1, …, zT are independent and identically distributed complex random variables, then
${\displaystyle {\sqrt {T}}{\Big (}{\tfrac {1}{T}}\textstyle \sum _{t=1}^{T}z_{t}-\operatorname {E} [z_{t}]{\Big )}\ {\xrightarrow {d}}\ {\mathcal {CN}}(0,\,\Gamma ,\,C),}$
where Γ = E[ zz′ ] and C = E[ zz′ ].

## Circularly-symmetric normal distribution

The 'circularly-symmetric normal distribution [5] corresponds to the case of zero mean and zero relation matrix, μ=0, C=0. If Z = X + iY is circularly-symmetric complex normal, then the vector vec[X Y] is multivariate normal with covariance structure

${\displaystyle {\begin{pmatrix}X\\Y\end{pmatrix}}\ \sim \ {\mathcal {N}}{\Big (}{\begin{bmatrix}\operatorname {Re} \,\mu \\\operatorname {Im} \,\mu \end{bmatrix}},\ {\tfrac {1}{2}}{\begin{bmatrix}\operatorname {Re} \,\Gamma &-\operatorname {Im} \,\Gamma \\\operatorname {Im} \,\Gamma &\operatorname {Re} \,\Gamma \end{bmatrix}}{\Big )}}$

where μ = E[ Z ] = 0 and Γ = E[ ZZ′ ]. This is usually denoted

${\displaystyle Z\sim {\mathcal {CN}}(0,\,\Gamma )}$

and its distribution can also be simplified as

${\displaystyle f(z)={\tfrac {1}{\pi ^{k}\det(\Gamma )}}\,e^{-{\overline {z}}'\;\Gamma ^{-1}\;z}.}$

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

${\displaystyle \ln(L(\mu ,\Gamma ))=-\ln(\det(\Gamma ))-{\overline {(z-\mu )}}'\Gamma ^{-1}(z-\mu )-k\ln(\pi ).}$

The standard complex normal corresponds to the distribution of a scalar random variable with μ = 0, C = 0 and Γ = 1. Thus, the standard complex normal distribution has density

${\displaystyle f(z)={\tfrac {1}{\pi }}e^{-{\overline {z}}z}={\tfrac {1}{\pi }}e^{-|z|^{2}}.}$

This expression demonstrates why the case C = 0, μ = 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 [−ππ].

If {z1, …, zn} are independent and identically distributed k-dimensional circular complex normal random variables with μ = 0, then the random squared norm

${\displaystyle Q=\sum _{j=1}^{n}{\overline {z_{j}'}}z_{j}=\sum _{j=1}^{n}\|z_{j}\|^{2}}$

has the generalized chi-squared distribution and the random matrix

${\displaystyle W=\sum _{j=1}^{n}z_{j}{\overline {z_{j}'}}}$

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

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

where n ≥ k, and w is a k×k nonnegative-definite matrix.