# Complex normal distribution

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 ${\displaystyle \Gamma }$, and the relation matrix ${\displaystyle C}$. The standard complex normal is the univariate distribution with ${\displaystyle \mu =0}$, ${\displaystyle \Gamma =1}$, and ${\displaystyle C=0}$.

Parameters ${\displaystyle \mathbf {\mu } \in \mathbb {C} ^{n}}$ — location${\displaystyle \Gamma \in \mathbb {C} ^{n\times n}}$ — covariance matrix (positive semi-definite matrix) ${\displaystyle C\in \mathbb {C} ^{n\times n}}$ — relation matrix (complex symmetric matrix) ${\displaystyle \mathbb {C} ^{n}}$ complicated, see text ${\displaystyle \mathbf {\mu } }$ ${\displaystyle \mathbf {\mu } }$ ${\displaystyle \Gamma }$ ${\displaystyle \exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline {w}}'\Gamma w+\operatorname {Re} ({\overline {w}}'C{\overline {w}}){\big )}{\big \}}}$

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: ${\displaystyle \mu =0}$ and ${\displaystyle C=0}$.[2] 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 ${\displaystyle Z}$  whose real and imaginary parts are independent normally distributed random variables with mean zero and variance ${\displaystyle 1/2}$ .[3]:p. 494[4]:pp. 501 Formally,

${\displaystyle Z\sim {\mathcal {CN}}(0,1)\quad \iff \quad \Re (Z)\perp \!\!\!\perp \Im (Z){\text{ and }}\Re (Z)\sim {\mathcal {N}}(0,1/2){\text{ and }}\Im (Z)\sim {\mathcal {N}}(0,1/2)}$

(Eq.1)

where ${\displaystyle Z\sim {\mathcal {CN}}(0,1)}$  denotes that ${\displaystyle Z}$  is a standard complex normal random variable.

### Complex normal random variable

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

${\displaystyle Z{\text{ complex normal random variable}}\quad \iff \quad (\Re (Z),\Im (Z))^{\mathrm {T} }{\text{ real normal random vector}}}$

(Eq.2)

### Complex standard normal random vector

A n-dimensional complex random vector ${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\mathrm {T} }}$  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.[3]:p. 502[4]:pp. 501 That ${\displaystyle \mathbf {Z} }$  is a standard complex normal random vector is denoted ${\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})}$ .

${\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})\quad \iff (Z_{1},\ldots ,Z_{n}){\text{ independent}}{\text{ and for }}1\leq i\leq n:Z_{i}\sim {\mathcal {CN}}(0,1)}$

(Eq.3)

### Complex normal random vector

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

${\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,}$

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

${\displaystyle \mathbf {Z} {\text{ complex normal random vector}}\quad \iff \quad (\Re (Z_{1}),\ldots ,\Re (Z_{n}),\Im (Z_{1}),\ldots ,\Im (Z_{n}))^{\mathrm {T} }{\text{ real normal random vector}}}$

(Eq.4)

## Notation

The symbol ${\displaystyle {\mathcal {N}}_{\mathcal {C}}}$  is also used for the complex normal distribution.

## Mean and covariance

The complex Gaussian distribution can be described with 3 parameters:[5]

${\displaystyle \mu =\operatorname {E} [\mathbf {Z} ],\quad \Gamma =\operatorname {E} [(\mathbf {Z} -\mu )({\mathbf {Z} }-\mu )^{\mathrm {H} }],\quad C=\operatorname {E} [(\mathbf {Z} -\mu )(\mathbf {Z} -\mu )^{\mathrm {T} }],}$

where ${\displaystyle \mathbf {Z} ^{\mathrm {T} }}$  denotes matrix transpose of ${\displaystyle \mathbf {Z} }$ , and ${\displaystyle \mathbf {Z} ^{\mathrm {H} }}$  denotes conjugate transpose.[3]:p. 504[4]:pp. 500

Here the location parameter ${\displaystyle \mu }$  is a n-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. The complex normal random vector ${\displaystyle \mathbf {Z} }$  can now be denoted as

${\displaystyle \mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu ,\ \Gamma ,\ C).}$

Moreover, matrices ${\displaystyle \Gamma }$  and ${\displaystyle C}$  are such that the matrix
${\displaystyle P={\overline {\Gamma }}-{C}^{\mathrm {H} }\Gamma ^{-1}C}$

is also non-negative definite where ${\displaystyle {\overline {\Gamma }}}$  denotes the complex conjugate of ${\displaystyle \Gamma }$ .[5]

## Relationships between covariance matrices

As for any complex random vector, the matrices ${\displaystyle \Gamma }$  and ${\displaystyle C}$  can be related to the covariance matrices of ${\displaystyle \mathbf {X} =\Re (\mathbf {Z} )}$  and ${\displaystyle \mathbf {Y} =\Im (\mathbf {Z} )}$  via expressions

{\displaystyle {\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

{\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 ^{n}{\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 ^{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 ${\displaystyle R=C^{\mathrm {H} }\Gamma ^{-1}}$  and ${\displaystyle P={\overline {\Gamma }}-RC}$ .

## Characteristic function

The characteristic function of complex normal distribution is given by[5]

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

where the argument ${\displaystyle w}$  is an n-dimensional complex vector.

## Properties

• If ${\displaystyle \mathbf {Z} }$  is a complex normal n-vector, ${\displaystyle {\boldsymbol {A}}}$  an m×n matrix, and ${\displaystyle b}$  a constant m-vector, then the linear transform ${\displaystyle {\boldsymbol {A}}\mathbf {Z} +b}$  will be distributed also complex-normally:
${\displaystyle Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma A^{\mathrm {H} },\,ACA^{\mathrm {T} })}$
• If ${\displaystyle \mathbf {Z} }$  is a complex normal n-vector, then
${\displaystyle 2{\Big [}(\mathbf {Z} -\mu )^{\mathrm {H} }{\overline {P^{-1}}}(\mathbf {Z} -\mu )-\operatorname {Re} {\big (}(\mathbf {Z} -\mu )^{\mathrm {T} }R^{\mathrm {T} }{\overline {P^{-1}}}(\mathbf {Z} -\mu ){\big )}{\Big ]}\ \sim \ \chi ^{2}(2n)}$
• Central limit theorem. If ${\displaystyle Z_{1},\ldots ,Z_{T}}$  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 ${\displaystyle \Gamma =\operatorname {E} [ZZ^{\mathrm {H} }]}$  and ${\displaystyle C=\operatorname {E} [ZZ^{\mathrm {T} }]}$ .

## Circularly-symmetric central case

### Definition

A complex random vector ${\displaystyle \mathbf {Z} }$  is called circularly symmetric if for every deterministic ${\displaystyle \varphi \in [-\pi ,\pi )}$  the distribution of ${\displaystyle e^{\mathrm {i} \varphi }\mathbf {Z} }$  equals the distribution of ${\displaystyle \mathbf {Z} }$ .[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 ${\displaystyle \Gamma }$ .

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

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

### Distribution of real and imaginary parts

If ${\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} }$  is circularly-symmetric (central) complex normal, then the vector ${\displaystyle [\mathbf {X} ,\mathbf {Y} ]}$  is multivariate normal with covariance structure

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

where ${\displaystyle \mu =\operatorname {E} [\mathbf {Z} ]=0}$  and ${\displaystyle \Gamma =\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{\mathrm {H} }]}$ .

### Probability density function

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

${\displaystyle f_{\mathbf {Z} }(\mathbf {z} )={\tfrac {1}{\pi ^{n}\det(\Gamma )}}\,e^{-\mathbf {z} ^{\mathrm {H} }\Gamma ^{-1}\mathbf {z} }}$ .

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

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

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

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

### Properties

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

If ${\displaystyle \left\{\mathbf {Z} _{1},\ldots ,\mathbf {Z} _{k}\right\}}$  are independent and identically distributed n-dimensional circular complex normal random vectors with ${\displaystyle \mu =0}$ , then the random squared norm

${\displaystyle Q=\sum _{j=1}^{k}\mathbf {Z} _{j}^{\mathrm {H} }\mathbf {Z} _{j}=\sum _{j=1}^{k}\|\mathbf {Z} _{j}\|^{2}}$

has the generalized chi-squared distribution and the random matrix

${\displaystyle W=\sum _{j=1}^{k}\mathbf {Z} _{j}\mathbf {Z} _{j}^{\mathrm {H} }}$

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

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

where ${\displaystyle k\geq n}$ , and ${\displaystyle w}$  is a ${\displaystyle n\times n}$  nonnegative-definite matrix.