Complex standard normal random variable edit

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,

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

Complex normal random variable edit

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}

Complex standard normal random vector edit

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 ${\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})}$ .

Complex normal random vector edit

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

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

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

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

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

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

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

${\displaystyle {\begin{aligned}&\Gamma =V_{XX}+V_{YY}+i(V_{YX}-V_{XY}),\\&C=V_{XX}-V_{YY}+i(V_{YX}+V_{XY}).\end{aligned}}}$ 

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 ${\displaystyle P={\overline {\Gamma }}-RC}$ .

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.

• 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 ${\displaystyle C=\operatorname {E} [ZZ^{\mathrm {T} }]}$ .

• The modulus of a complex normal random variable follows a Hoyt distribution.^[6]

Definition edit

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 edit

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

${\displaystyle {\begin{pmatrix}\mathbf {X} \\\mathbf {Y} \end{pmatrix}}\ \sim \ {\mathcal {N}}{\Big (}{\begin{bmatrix}0\\0\end{bmatrix}},\ {\tfrac {1}{2}}{\begin{bmatrix}\operatorname {Re} \,\Gamma &-\operatorname {Im} \,\Gamma \\\operatorname {Im} \,\Gamma &\operatorname {Re} \,\Gamma \end{bmatrix}}{\Big )}}$ 

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

Probability density function edit

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

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

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 edit

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.

• Complex normal ratio distribution
• Directional statistics § Distribution of the mean (polar form)
• Normal distribution
• Multivariate normal distribution (a complex normal distribution is a bivariate normal distribution)
• Generalized chi-squared distribution
• Wishart distribution
• Complex random variable