 Complex normal distribution

In probability theory, the family of complex normal distributions consists of 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 circular symmetric complex normal and corresponds to the case of zero relation matrix: C = 0. Circular symmetric complex normal random variables are used extensively in signal processing, and are sometimes incorrectly referred to as just complex normal in signal processing literature.
Contents
Definition
Suppose X and Y are random vectors in R^{k} such that vec[X Y] is a 2kdimensional normal random vector. Then we say that the complex random vector
has the complex normal distribution. This distribution can be described with 3 parameters:^{[2]}
where Z ′ denotes matrix transpose, and Z denotes complex conjugate. Here the location parameter μ can be an arbitrary kdimensional complex vector; the covariance matrix Γ must be Hermitian and nonnegative definite; the relation matrix C should be symmetric. Moreover, matrices Γ and C are such that the matrix
is also nonnegative definite.^{[2]}
Matrices Γ and C can be related to the covariance matrices of X and Y via expressions
and conversely
Density function
The probability density function for complex normal distribution can be computed as
where R = C′ Γ^{ −1} and P = Γ − RC.
Characteristic function
The characteristic function of complex normal distribution is given by ^{[2]}
where the argument w is a kdimensional complex vector.
Properties
 If Z is a complex normal kvector, A an ℓ×k matrix, and b a constant ℓvector, then the linear transform AZ + b will be distributed also complexnormally:
 If Z is a complex normal kvector, then
 Central limit theorem. If z_{1}, …, z_{T} are independent and identically distributed complex random variables, then
where Γ = E[ zz′ ] and C = E[ zz′ ].
Circular symmetric complex normal distribution
The circular symmetric complex normal distribution corresponds to the case of zero relation matrix, C=0. If Z = X + iY is circular complex normal, then the vector vec[X Y] is multivariate normal with covariance structure
where μ = E[ Z ] and Γ = E[ ZZ′ ]. This is usually denoted
and its distribution can also be simplified as
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
This expression demonstrates why the case C = 0 is called “circularsymmetric”. 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 {z_{1}, …, z_{n}} are independent and identically distributed kdimensional circular complex normal random variables with μ = 0, then random squared norm
has the Generalized chisquared distribution and the random matrix
has the complex Wishart distribution with n degrees of freedom. This distribution can be described by density function
where n ≥ k, and w is a k×k nonnegativedefinite matrix.
See also
 Normal distribution
 Multivariate normal distribution
 Generalized chisquared distribution
 Wishart distribution
References
 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. JSTOR 2991290.
 Picinbono, Bernard (1996). "Secondorder complex random vectors and normal distributions". IEEE Transactions on Signal Processing 44 (10): 2637–2640.
Categories: Continuous distributions
 Multivariate continuous distributions
 Complex numbers
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