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Complex normal distribution

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Complex normal
Parameters

location
covariance matrix (positive semi-definite matrix)

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

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 .

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 asMoreover, 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