Complex normal distribution
In probability theory, the family of complex normal distributions, denoted formula_1 or formula_2, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: "location" parameter "μ", "covariance" matrix formula_3, and the "relation" matrix formula_4. The standard complex normal is the univariate distribution with formula_5, formula_6, and formula_7. 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: formula_8 and formula_9. 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 formula_10 whose real and imaginary parts are independent normally distributed random variables with mean zero and variance formula_11. Formally, where formula_12 denotes that formula_10 is a standard complex normal random variable. Complex normal random variable. Suppose formula_14 and formula_15 are real random variables such that formula_16 is a 2-dimensional normal random vector. Then the complex random variable formula_17 is called complex normal random variable or complex  Gaussian random variable. Complex standard normal random vector. A n-dimensional complex random vector formula_18 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. That formula_19 is a standard complex normal random vector is denoted formula_20. Complex normal random vector. If formula_21 and formula_22 are random vectors in formula_23 such that formula_24 is a normal random vector with formula_25 components. Then we say that the complex random vector is a complex normal random vector or a complex Gaussian  random vector. Mean, covariance, and relation. The complex Gaussian distribution can be described with 3 parameters: where formula_28 denotes matrix transpose of formula_19, and formula_30 denotes conjugate transpose. Here the location parameter formula_31 is a n-dimensional complex vector; the covariance matrix formula_3 is Hermitian and non-negative definite; and, the "relation matrix" or "pseudo-covariance matrix" formula_4 is symmetric. The complex normal random vector formula_34 can now be denoted asformula_35Moreover, matrices formula_3 and formula_4 are such that the matrix is also non-negative definite where formula_39 denotes the complex conjugate of formula_3. Relationships between covariance matrices. As for any  complex random vector, the matrices formula_3 and formula_4 can be related to the covariance matrices of formula_43 and formula_44 via expressions and conversely Density function. The probability density function for complex normal distribution can be computed as where formula_48 and formula_49. Characteristic function. The characteristic function of complex normal distribution is given by where the argument formula_51 is an "n"-dimensional complex vector. Circularly-symmetric central case. Definition. A complex random vector formula_63 is called circularly symmetric if for every deterministic formula_64 the distribution of formula_65 equals the distribution of formula_63. Central normal complex random vectors that are circularly symmetric are  of particular interest because they are fully specified by the covariance matrix formula_3. The "circularly-symmetric (central) complex normal distribution" corresponds to the case of zero mean and zero relation matrix, i.e. formula_5 and formula_7. This is usually denoted Distribution of real and imaginary parts. If formula_71 is circularly-symmetric (central) complex normal, then the vector formula_72 is multivariate normal with covariance structure where formula_74 and formula_75. Probability density function. For nonsingular covariance matrix formula_3, its distribution can also be simplified as Therefore, if the non-zero mean formula_31 and covariance matrix formula_3 are unknown, a suitable log likelihood function  for a single observation vector formula_80 would be The standard complex normal (defined in )corresponds to the distribution of a scalar random variable with formula_5, formula_7 and formula_6. Thus, the standard complex normal distribution has density Properties. The above expression demonstrates why the case formula_7, formula_5 is called “circularly-symmetric”. The density function depends only on the magnitude of formula_80 but not on its argument. As such, the magnitude formula_89 of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude formula_90 will have the exponential distribution, whereas the argument will be distributed uniformly on  formula_91. If formula_92 are independent and identically distributed "n"-dimensional circular complex normal random vectors with formula_5, then the random squared norm has the generalized chi-squared distribution and the random matrix has the complex Wishart distribution with formula_96 degrees of freedom. This distribution can be described by density function where formula_98, and formula_51 is a formula_100 nonnegative-definite matrix.