↑ arvau

Wikipedia / Complex normal distribution

In probability theory, the family of complex normal distributions, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance, and the relation. The standard complex normal.

An important subclass of complex normal family is called the circularly-symmetric (central) complex normal. 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. Formally,

is a standard complex normal random variable.

Complex normal random variable

is a 2-dimensional normal random vector is called complex normal random variable or complex Gaussian random variable.

Complex standard normal 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..

Complex normal random vector

are random vectors is a normal random vector components. Then we say that the complex random vector

is a complex normal random vector or a complex Gaussian random vector.

Mean, covariance, and relationMean and covariance

The complex Gaussian distribution can be described with 3 parameters:

denotes matrix transpose denotes conjugate transpose.

Here the location parameter is a n-dimensional complex vector; the covariance matrix is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix is symmetric are such that the matrix

.

Relationships between covariance matrices

via expressions

and conversely

Density function

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

.

Characteristic function

The characteristic function of complex normal distribution is given by

is an n-dimensional complex vector.

Properties

  • is a complex normal n an m×n a constant m will be distributed also complex-normally:

    is a complex normal n-vector, then

    Central limit theorem are independent and identically distributed complex random variables, then

  • .

  • The modulus of a complex normal random variable follows a Hoyt distribution.

Circularly-symmetric central case

Definition

is called circularly symmetric.

.

The circularly-symmetric (central) complex normal distribution. This is usually denoted

Distribution of real and imaginary parts

is multivariate normal with covariance structure

.

Probability density function

, its distribution can also be simplified as

  1. .

would be

The standard complex normal (defined in . Thus, the standard complex normal distribution has density

Properties

but not on its argument of a standard complex normal random variable will have the Rayleigh distribution will have the exponential distribution, whereas the argument will be distributed uniformly.

are independent and identically distributed n, then the random squared norm

has the generalized chi-squared distribution and the random matrix

has the complex Wishart distribution degrees of freedom. This distribution can be described by density function

nonnegative-definite matrix.

See also

References