{{Short description|Statistical distribution of complex random variables}} {{Probability distribution | name = Complex normal | type = multivariate | pdf_image = | cdf_image = | notation = | parameters = <math>\mathbf{\mu} \in \mathbb{C}^n</math> — location<br/> <math>\Gamma \in \mathbb{C}^{n \times n}</math> — covariance matrix (positive semi-definite matrix)<br/> <math>C \in \mathbb{C}^{n \times n}</math> — relation matrix (complex symmetric matrix) | support = <math>\mathbb{C}^n</math> | pdf = complicated, see text | mean = <math>\mathbf{\mu}</math> | mode = <math>\mathbf{\mu}</math> | variance = <math>\Gamma</math> | cf = <math> \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\} </math> }}
In probability theory, the family of '''complex normal distributions''', denoted <math>\mathcal{CN}</math> or <math>\mathcal{N}_{\mathcal{C}}</math>, characterizes complex random variables whose real and imaginary parts are jointly normal.<ref>{{cite journal | first = N.R. | last = Goodman | year = 1963 | title = Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction) | journal = The Annals of Mathematical Statistics | volume = 34 | issue = 1 | pages = 152–177 | jstor = 2991290 | doi=10.1214/aoms/1177704250 | doi-access = free }}</ref> The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix <math>\Gamma</math>, and the ''relation'' matrix <math>C</math>. The '''standard complex normal''' is the univariate distribution with <math>\mu = 0</math>, <math>\Gamma=1</math>, and <math>C=0</math>.
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: <math> \mu = 0 </math> and <math> C=0 </math>.<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R''], pg9.</ref> 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 <math>Z</math> whose real and imaginary parts are independent normally distributed random variables with mean zero and variance <math>1/2</math>.<ref name=Lapidoth>{{cite book | author=Lapidoth, A.| title=A Foundation in Digital Communication| publisher=Cambridge University Press | year=2009 | isbn=9780521193955}}</ref>{{rp|p. 494}}<ref name=TseViswanath>{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press|isbn=9781139444668 |url=https://books.google.com/books?id=GdsLAQAAQBAJ&q=%22random+variable%22}}</ref>{{rp|pp. 501}} Formally,
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>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)</math>|{{EquationRef|Eq.1}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
where <math>Z \sim \mathcal{CN}(0,1)</math> denotes that <math>Z</math> is a standard complex normal random variable.
===Complex normal random variable=== Suppose <math>X</math> and <math>Y</math> are real random variables such that <math>(X,Y)^{\mathrm T}</math> is a 2-dimensional normal random vector. Then the complex random variable <math>Z=X+iY</math> is called '''complex normal random variable''' or '''complex Gaussian random variable'''.<ref name=Lapidoth/>{{rp|p. 500}}
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>Z \text{ complex normal random variable} \quad \iff \quad (\Re(Z),\Im(Z))^{\mathrm T} \text{ real normal random vector} </math>|{{EquationRef|Eq.2}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
===Complex standard normal random vector=== A n-dimensional complex random vector <math>\mathbf{Z}=(Z_1,\ldots,Z_n)^{\mathrm T}</math> 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.<ref name=Lapidoth/>{{rp|p. 502}}<ref name=TseViswanath/>{{rp|pp. 501}} That <math>\mathbf{Z}</math> is a standard complex normal random vector is denoted <math>\mathbf{Z} \sim \mathcal{CN}(0,\boldsymbol{I}_n)</math>.
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>\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)</math>|{{EquationRef|Eq.3}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
===Complex normal random vector=== If <math>\mathbf{X}=(X_1,\ldots,X_n)^{\mathrm T}</math> and <math>\mathbf{Y}=(Y_1,\ldots,Y_n)^{\mathrm T}</math> are random vectors in <math>\mathbb{R}^n</math> such that <math>[\mathbf{X},\mathbf{Y}]</math> is a normal random vector with <math>2n</math> components. Then we say that the complex random vector : <math> \mathbf{Z} = \mathbf{X} + i \mathbf{Y} \, </math> is a '''complex normal random vector''' or a '''complex Gaussian random vector'''.
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>\mathbf{Z} \text{ complex normal random vector} \quad \iff \quad (\Re(\mathbf{Z}^{\mathrm T}),\Im(\mathbf{Z}^{\mathrm T}))^{\mathrm T} = (\Re(Z_1),\ldots,\Re(Z_n),\Im(Z_1),\ldots,\Im(Z_n))^{\mathrm T} \text{ real normal random vector} </math>|{{EquationRef|Eq.4}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
==Mean, covariance, and relation{{anchor|Mean and covariance}}== The complex Gaussian distribution can be described with 3 parameters:<ref name="picinbono">{{cite journal | last = Picinbono | first = Bernard | year = 1996 | title = Second-order complex random vectors and normal distributions | journal = IEEE Transactions on Signal Processing | volume = 44 | issue = 10 | pages = 2637–2640 | doi=10.1109/78.539051 | bibcode = 1996ITSP...44.2637P | url = https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051 }}</ref> : <math> \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}], </math> where <math>\mathbf{Z}^{\mathrm T}</math> denotes matrix transpose of <math>\mathbf{Z}</math>, and <math>\mathbf{Z}^{\mathrm H}</math> denotes conjugate transpose.<ref name=Lapidoth/>{{rp|p. 504}}<ref name=TseViswanath/>{{rp|pp. 500}}
Here the location parameter <math>\mu</math> is a n-dimensional complex vector; the covariance matrix <math>\Gamma</math> is Hermitian and non-negative definite; and, the ''relation matrix'' or ''pseudo-covariance matrix'' <math>C</math> is symmetric. The complex normal random vector <math> \mathbf{Z} </math> can now be denoted as<math display="block"> \mathbf{Z}\ \sim\ \mathcal{CN}(\mu,\ \Gamma,\ C). </math>Moreover, matrices <math>\Gamma</math> and <math>C</math> are such that the matrix : <math> P = \overline{\Gamma} - {C}^{\mathrm H}\Gamma^{-1}C </math> is also non-negative definite where <math>\overline{\Gamma}</math> denotes the complex conjugate of <math>\Gamma</math>.<ref name="picinbono"/>
==Relationships between covariance matrices== {{main|Complex random vector#Covariance matrix and pseudo-covariance matrix}}
As for any complex random vector, the matrices <math>\Gamma</math> and <math>C</math> can be related to the covariance matrices of <math>\mathbf{X} = \Re(\mathbf{Z})</math> and <math>\mathbf{Y} = \Im(\mathbf{Z})</math> via expressions : <math>\begin{align} & 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{align}</math> and conversely : <math>\begin{align} & \Gamma = V_{XX} + V_{YY} + i(V_{YX} - V_{XY}), \\ & C = V_{XX} - V_{YY} + i(V_{YX} + V_{XY}). \end{align}</math>
==Density function== The probability density function for complex normal distribution can be computed as
: <math>\begin{align} f(z) &= \frac{1}{\pi^n\sqrt{\det(\Gamma)\det(P)}}\, \exp\!\left\{-\frac12 \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{align}</math>
where <math>R=C^{\mathrm H} \Gamma^{-1}</math> and <math>P=\overline{\Gamma}-RC</math>.
==Characteristic function== The characteristic function of complex normal distribution is given by<ref name="picinbono"/> : <math> \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\}, </math> where the argument <math>w</math> is an ''n''-dimensional complex vector.
==Properties== * If <math>\mathbf{Z}</math> is a complex normal ''n''-vector, <math>\boldsymbol{A}</math> an ''m×n'' matrix, and <math>b</math> a constant ''m''-vector, then the linear transform <math>\boldsymbol{A}\mathbf{Z}+b</math> will be distributed also complex-normally: : <math> Z\ \sim\ \mathcal{CN}(\mu,\, \Gamma,\, C) \quad \Rightarrow \quad AZ+b\ \sim\ \mathcal{CN}(A\mu+b,\, A \Gamma A^{\mathrm H},\, A C A^{\mathrm T}) </math>
* If <math>\mathbf{Z}</math> is a complex normal ''n''-vector, then : <math> 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) </math>
* '''Central limit theorem'''. If <math>Z_1,\ldots,Z_T</math> are independent and identically distributed complex random variables, then : <math> \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), </math> :where <math>\Gamma = \operatorname{E}[Z Z^{\mathrm H}]</math> and <math>C = \operatorname{E}[Z Z^{\mathrm T}]</math>.
* The modulus of a complex normal random variable follows a Hoyt distribution.<ref>{{cite web |title=The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2) |author=Daniel Wollschlaeger |url=http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html }}{{Dead link|date=July 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
==Circularly-symmetric central case==
===Definition=== A complex random vector <math> \mathbf{Z} </math> is called circularly symmetric if for every deterministic <math> \varphi \in [-\pi,\pi) </math> the distribution of <math> e^{\mathrm i \varphi}\mathbf{Z} </math> equals the distribution of <math> \mathbf{Z} </math>.<ref name=TseViswanath/>{{rp|pp. 500–501}} {{main|Complex random vector#Circular symmetry}}
Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix <math>\Gamma</math>.
The ''circularly-symmetric (central) complex normal distribution'' corresponds to the case of zero mean and zero relation matrix, i.e. <math>\mu = 0</math> and <math>C=0</math>.<ref name=Lapidoth/>{{rp|p. 507}}<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R'']</ref> This is usually denoted :<math>\mathbf{Z} \sim \mathcal{CN}(0,\,\Gamma)</math>
===Distribution of real and imaginary parts=== If <math>\mathbf{Z}=\mathbf{X}+i\mathbf{Y}</math> is circularly-symmetric (central) complex normal, then the vector <math>[\mathbf{X}, \mathbf{Y}]</math> is multivariate normal with covariance structure : <math> \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) </math> where <math>\Gamma=\operatorname{E}[\mathbf{Z} \mathbf{Z}^{\mathrm H}]</math>.
===Probability density function=== For nonsingular covariance matrix <math>\Gamma</math>, its distribution can also be simplified as<ref name=Lapidoth/>{{rp|p. 508}} : <math> f_{\mathbf{Z}}(\mathbf{z}) = \tfrac{1}{\pi^n \det(\Gamma)}\, e^{ -(\mathbf{z}-\mathbf{\mu})^{\mathrm H} \Gamma^{-1} (\mathbf{z}-\mathbf{\mu})} </math>.
Therefore, if the non-zero mean <math>\mu</math> and covariance matrix <math>\Gamma</math> are unknown, a suitable log likelihood function for a single observation vector <math>z</math> would be : <math> \ln(L(\mu,\Gamma)) = -\ln (\det(\Gamma)) -\overline{(z - \mu)}' \Gamma^{-1} (z - \mu) -n \ln(\pi). </math>
The '''standard complex normal''' (defined in {{EquationNote|Eq.1}}) corresponds to the distribution of a scalar random variable with <math>\mu = 0</math>, <math>C=0</math> and <math>\Gamma=1</math>. Thus, the standard complex normal distribution has density
: <math> f_Z(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}. </math>
===Properties=== The above expression demonstrates why the case <math>C=0</math>, <math>\mu = 0</math> is called “circularly-symmetric”. The density function depends only on the magnitude of <math>z</math> but not on its argument. As such, the magnitude <math>|z|</math> of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude <math>|z|^2</math> will have the exponential distribution, whereas the argument will be distributed uniformly on <math>[-\pi,\pi]</math>.
If <math>\left\{ \mathbf{Z}_1,\ldots,\mathbf{Z}_k \right\}</math> are independent and identically distributed ''n''-dimensional circular complex normal random vectors with <math>\mu = 0</math>, then the random squared norm : <math> Q = \sum_{j=1}^k \mathbf{Z}_j^{\mathrm H} \mathbf{Z}_j = \sum_{j=1}^k \| \mathbf{Z}_j \|^2 </math> has the generalized chi-squared distribution and the random matrix : <math> W = \sum_{j=1}^k \mathbf{Z}_j \mathbf{Z}_j^{\mathrm H} </math> has the complex Wishart distribution with <math>k</math> degrees of freedom. This distribution can be described by density function : <math> 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)} </math> where <math>k \ge n</math>, and <math>w</math> is a <math>n \times n</math> nonnegative-definite matrix.
==See also== * Complex normal ratio distribution * {{section link|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
==References== {{More footnotes needed|date=July 2011}} {{reflist}}
{{ProbDistributions|continuous-infinite}}
Category:Continuous distributions Category:Multivariate continuous distributions Category:Complex distributions