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: and .[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 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
Matrices and can be related to the covariance matrices of and via expressions
and conversely
Density function
The probability density function for complex normal distribution can be computed as
where and P = Γ − RC.
Characteristic function
The characteristic function of complex normal distribution is given by [3]
where the argument 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:
- 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′ ].
- The modulus of a complex normal random variable follows a Hoyt distribution.[4]
Circularly-symmetric normal distribution
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 a 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, …, zn} are independent and identically distributed k-dimensional circular complex normal random variables with μ = 0, then the 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
- Directional statistics#Distribution of the mean
- Normal distribution
- Multivariate normal distribution (a complex normal distribution is a bivariate normal distribution)
- Generalized chi-squared 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. doi:10.1214/aoms/1177704250.
- 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.