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 , and the relation matrix . The standard complex normal is the univariate distribution with , , and .

Complex normal
Parameters

location
covariance matrix (positive semi-definite matrix)

relation matrix (complex symmetric matrix)
Support
PDF complicated, see text
Mean
Mode
Variance
CF

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

DefinitionsEdit

Complex standard normal random variableEdit

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  .[3]:p. 494[4]:pp. 501 Formally,

 

 

 

 

 

(Eq.1)

where   denotes that   is a standard complex normal random variable.

Complex normal random variableEdit

Suppose   and   are real random variables such that   is a 2-dimensional normal random vector. Then the complex random variable   is called complex normal random variable or complex Gaussian random variable.[3]:p. 500

 

 

 

 

 

(Eq.2)

Complex standard normal random vectorEdit

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. 502[4]:pp. 501 That   is a standard complex normal random vector is denoted  .

 

 

 

 

 

(Eq.3)

Complex normal random vectorEdit

If   and   are random vectors in   such that   is a normal random vector with   components. Then we say that the complex random vector

 

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

 

 

 

 

 

(Eq.4)

NotationEdit

The symbol   is also used for the complex normal distribution.

Mean and covarianceEdit

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

 

where   denotes matrix transpose of  , and   denotes conjugate transpose.[3]:p. 504[4]:pp. 500

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. The complex normal random vector   can now be denoted as

 
Moreover, matrices   and   are such that the matrix
 

is also non-negative definite where   denotes the complex conjugate of  .[5]

Relationships between covariance matricesEdit

As for any complex random vector, the matrices   and   can be related to the covariance matrices of   and   via expressions

 

and conversely

 

Density functionEdit

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

 

where   and  .

Characteristic functionEdit

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

 

where the argument   is an n-dimensional complex vector.

PropertiesEdit

  • If   is a complex normal n-vector,   an m×n matrix, and   a constant m-vector, then the linear transform   will be distributed also complex-normally:
 
  • If   is a complex normal n-vector, then
 
  • Central limit theorem. If   are independent and identically distributed complex random variables, then
 
where   and  .

Circularly-symmetric central caseEdit

DefinitionEdit

A complex random vector   is called circularly symmetric if for every deterministic   the distribution of   equals the distribution of  .[4]:pp. 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.   and  .[3]:p. 507[7] This is usually denoted

 

Distribution of real and imaginary partsEdit

If   is circularly-symmetric (central) complex normal, then the vector   is multivariate normal with covariance structure

 

where   and  .

Probability density functionEdit

For nonsingular covariance matrix  , its distribution can also be simplified as[3]:p. 508

 .

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 (defined in Eq.1)corresponds to the distribution of a scalar random variable with  ,   and  . Thus, the standard complex normal distribution has density

 

PropertiesEdit

The above expression demonstrates why the case  ,   is called “circularly-symmetric”. The density function depends only on the magnitude of   but not on its argument. As such, the magnitude   of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude   will have the exponential distribution, whereas the argument will be distributed uniformly on  .

If   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   degrees of freedom. This distribution can be described by density function

 

where  , and   is a   nonnegative-definite matrix.

See alsoEdit

ReferencesEdit

  1. ^ Goodman (1963)
  2. ^ bookchapter, Gallager.R, pg9.
  3. ^ a b c d e f Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955.
  4. ^ a b c d Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668.
  5. ^ a b c Picinbono (1996)
  6. ^ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".[permanent dead link]
  7. ^ bookchapter, Gallager.R

Further readingEdit