— covariance matrix (positive semi-definite matrix) In probability theory, the family of complex normal distributions, denoted

, characterizes complex random variables whose real and imaginary parts are jointly normal.

The standard complex normal is the univariate distribution with

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:

[2] This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.

The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable

whose real and imaginary parts are independent normally distributed random variables with mean zero and variance

is a standard complex normal random variable.

is called complex normal random variable or complex Gaussian random variable.[3]: p.

500 A n-dimensional complex random vector

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.[3]: p.

is a standard complex normal random vector is denoted

The complex Gaussian distribution can be described with 3 parameters:[5] where

is a n-dimensional complex vector; the covariance matrix

The complex normal random vector

[5] As for any complex random vector, the matrices

via expressions and conversely The probability density function for complex normal distribution can be computed as where

The characteristic function of complex normal distribution is given by[5] where the argument

is called circularly symmetric if for every deterministic

500–501 Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix

The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e.

is circularly-symmetric (central) complex normal, then the vector

is multivariate normal with covariance structure where

are unknown, a suitable log likelihood function for a single observation vector

would be The standard complex normal (defined in Eq.1) corresponds to the distribution of a scalar random variable with

Thus, the standard complex normal distribution has density The above expression demonstrates why the case

The density function depends only on the magnitude of

of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude

are independent and identically distributed n-dimensional circular complex normal random vectors with

, then the random squared norm has the generalized chi-squared distribution and the random matrix has the complex Wishart distribution with