Complex normal distribution

In probability theory, the family of complex normal distributions, denoted ${\displaystyle {\mathcal {CN}}}$ or ${\displaystyle {\mathcal {N}}_{\mathcal {C}}}$, 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 ${\displaystyle C=0}$.

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.

La distribución gaussiana compleja se puede describir con 3 parámetros: ^{[ 5 ]}

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

dónde${\displaystyle \mathbf {Z} ^{\mathrm {T} }}$denota la transpuesta de la matriz de${\displaystyle \mathbf {Z} }$, y${\displaystyle \mathbf {Z} ^{\mathrm {H} }}$denota transpuesta conjugada . ^{[ 3 ] : pág. 504 [ 4 ] : págs. 500}

Aquí el parámetro de ubicación${\displaystyle \mu }$es un vector complejo n-dimensional; la matriz de covarianza${\displaystyle \Gamma }$es hermitiana y semidefinida positiva ; y, la matriz de relación o matriz de pseudocovarianza${\displaystyle C}$es simétrico . El vector aleatorio normal complejo${\displaystyle \mathbf {Z} }$ahora se puede denotar como$${\displaystyle \mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu,\ \Gamma,\ C).}$$Además, matrices${\displaystyle \Gamma }$y${\displaystyle C}$son tales que la matriz

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

también es definida no negativa donde${\displaystyle {\overline {\Gamma }}}$denota el conjugado complejo de${\displaystyle \Gamma }$. ^{[ 5 ]}

Como ocurre con cualquier vector aleatorio complejo, las matrices${\displaystyle \Gamma }$y${\displaystyle C}$puede estar relacionado con las matrices de covarianza de${\displaystyle \mathbf {X} =\Re (\mathbf {Z} )}$y${\displaystyle \mathbf {Y} =\Soy (\mathbf {Z} )}$a través de expresiones

${\displaystyle {\begin{aligned}&V_{XX}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma +C],\quad V_{XY}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [-\Gamma +C],\\&V_{YX}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [\Gamma +C],\quad \,V_{YY}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma -C],\end{aligned}}}$

y viceversa

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

La función de densidad de probabilidad para la distribución normal compleja se puede calcular como

${\displaystyle {\begin{aligned}f(z)&={\frac {1}{\pi ^{n}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac {1}{2}}{\begin{bmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{bmatrix}}^{\mathrm {H} }{\begin{bmatrix}\Gamma &C\\{\overline {C}}&{\overline {\Gamma }}\end{bmatrix}}^{\!\!-1}\!{\begin{bmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{bmatrix}}\right\}\\[8pt]&={\tfrac {\sqrt {\det \left({\overline {P^{-1}}}-R^{\ast }P^{-1}R\right)\det(P^{-1})}}{\pi ^{n}}}\,e^{-(z-\mu )^{\ast }{\overline {P^{-1}}}(z-\mu )+\operatorname {Re} \left((z-\mu )^{\intercal }R^{\intercal }{\overline {P^{-1}}}(z-\mu )\right)},\end{aligned}}}$

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

La función característica de la distribución normal compleja viene dada por ^{[ 5 ].}

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

donde el argumento${\displaystyle w}$es un vector complejo n -dimensional.

• Si${\displaystyle \mathbf {Z} }$es un vector normal complejo de n dimensiones,${\displaystyle {\boldsymbol {A}}}$una matriz m×n , y${\displaystyle b}$un vector m constante , luego la transformación lineal${\displaystyle {\boldsymbol {A}}\mathbf {Z} +b}$también se distribuirá de forma compleja-normal:

${\displaystyle Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma A^{\mathrm {H} },\,ACA^{\mathrm {T} })}$

• Si${\displaystyle \mathbf {Z} }$es un vector normal complejo de n dimensiones , entonces

${\displaystyle 2{\Big [}(\mathbf {Z} -\mu )^{\mathrm {H} }{\overline {P^{-1}}}(\mathbf {Z} -\mu )-\operatorname {Re} {\big (}(\mathbf {Z} -\mu )^{\mathrm {T} }R^{\mathrm {T} }{\overline {P^{-1}}}(\mathbf {Z} -\mu ){\big )}{\Big ]}\ \sim \ \chi ^{2}(2n)}$

• Teorema del límite central . Si${\displaystyle Z_{1},\ldots ,Z_{T}}$son variables aleatorias complejas independientes e idénticamente distribuidas, entonces

${\displaystyle {\sqrt {T}}{\Big (}{\tfrac {1}{T}}\textstyle \sum _{t=1}^{T}Z_{t}-\operatorname {E} [Z_{t}]{\Big )}\ {\xrightarrow {d}}\ {\mathcal {CN}}(0,\,\Gamma ,\,C),}$

dónde${\displaystyle \Gamma =\operatorname {E} [ZZ^{\mathrm {H} }]}$y${\displaystyle C=\operatorname {E} [ZZ^{\mathrm {T} }]}$.

• El módulo de una variable aleatoria normal compleja sigue una distribución de Hoyt . ^{[ 6 ]}

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