Complex normal distribution

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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: .[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[edit]

Suppose X and Y are random vectors in Rk such that vec[X Y] is a 2k-dimensional normal random vector. Then we say that the complex random vector

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

where denotes matrix transpose, and denotes conjugate transpose. Here the [location parameter] is a k-dimensional complex vector; the [covariance matrix] is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix is symmetric. Moreover, matrices and are such that the matrix

is also non-negative definite.[3]

Relationships between covariance matrices[edit]

Matrices and can be related to the covariance matrices of and via expressions

and conversely

Density function[edit]

The probability density function for complex normal distribution can be computed as

where and P = Γ − RC.

Characteristic function[edit]

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

where the argument is a k-dimensional complex vector.

Properties[edit]

  • If Z is a complex normal k-vector, A an ℓ×k matrix, and b a constant -vector, then the linear transform AZ + b will be distributed also complex-normally:
  • If Z is a complex normal k-vector, then
  • Central limit theorem. If z1, …, zT are independent and identically distributed complex random variables, then
where Γ = E[ zz′ ] and C = E[ zz′ ].

Circularly-symmetric normal distribution[edit]

The 'circularly-symmetric 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

where μ = E[ Z ] = 0 and Γ = E[ ZZ′ ]. This is usually denoted

and its distribution can also be simplified as

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

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, 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 [−ππ].

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

has the Generalized chi-squared 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 nonnegative-definite matrix.

See also[edit]

References[edit]

  • 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.