Complex normal distribution

In probability theory, the family of complex normal distributions, denoted or , 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.

Definitions

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Complex standard normal random variable

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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 variable

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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 vector

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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 vector

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

 

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

  (Eq.4)

Mean, covariance, and relation

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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 matrices

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

 

and conversely

 

Density function

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

 

where   and  .

Characteristic function

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

 

where the argument   is an n-dimensional complex vector.

Properties

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  • 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 case

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Definition

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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 parts

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

 

where  .

Probability density function

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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

 

Properties

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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 also

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References

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  1. ^ 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. doi:10.1214/aoms/1177704250. JSTOR 2991290.
  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, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. Bibcode:1996ITSP...44.2637P. doi:10.1109/78.539051.
  6. ^ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".[permanent dead link]
  7. ^ bookchapter, Gallager.R