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 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: $\mu =0\ {\text{and}}\ C=0$ .^[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 R^k such that vec[X Y] is a 2k-dimensional normal random vector. Then we say that the complex random vector

Z=X+iY\,

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

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

where $Z^{\mathrm {T} }$ denotes matrix transpose, and ${\overline {Z}}$ denotes complex conjugate. Here the location parameter $\mu$ can be an arbitrary k-dimensional complex vector; the covariance matrix $\Gamma$ must be Hermitian and non-negative definite; the relation matrix $C$ should be symmetric. Moreover, matrices $\Gamma$ and $C$ are such that the matrix

P={\overline {\Gamma }}-{\overline {C}}^{\mathrm {T} }\Gamma ^{-1}C

is also non-negative definite.^[3]

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

{\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

{\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

{\begin{aligned}f(z)&={\frac {1}{\pi ^{k}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac 12}{\begin{pmatrix}(\overline {z}-\overline \mu )'&(z-\mu )'\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}}}-\overline {R}'P^{{-1}}R\right)\det(P^{{-1}})}}}{\pi ^{k}}}\,e^{{-(\overline {z}-\overline \mu )'\overline {P^{{-1}}}(z-\mu )+\operatorname {Re}\left((z-\mu )'R'\overline {P^{{-1}}}(z-\mu )\right)}},\end{aligned}}

where R = C′ Γ⁻¹ and P = Γ − RC.

Characteristic function

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

\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 $w$ is a k-dimensional complex vector.

Properties

If Z is a complex normal k-vector, A an ℓ×k matrix, and b a constant ℓ-vector, then the linear transform AZ + b will be distributed also complex-normally:

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

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, …, z_T are independent and identically distributed complex random variables, then

{\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′ ].

The modulus of a complex normal random variable follows a Hoyt distribution.^[4]

Circularly-symmetric and zero mean complex normal distribution

The circularly-symmetric and zero mean complex 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

{\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

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

and its distribution can also be simplified as

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

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

\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

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 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 [−π, π].

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

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

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

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

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

References

↑ Goodman (1963)
↑ bookchapter, Gallager.R, pg9.
1 2 3 Picinbono (1996)
↑ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".
↑ bookchapter, Gallager.R

Goodman, N.R. (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". The Annals of Mathematical Statistics. 34 (1): 152–177. doi:10.1214/aoms/1177704250. JSTOR 2991290.
Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. doi:10.1109/78.539051.

Wollschlaeger, Daniel. "ShotGroups." Hoyt. RDocumentation, n.d. Web. https://www.rdocumentation.org/packages/shotGroups/versions/0.7.1/topics/Hoyt.
Gallager, Robert G (2008). "Circularly-Symmetric Gaussian Random Vectors." (n.d.): n. pag. Pre-print. Web. 9 http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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