Complex normal distribution - Alchetron, the free social encyclopedia

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 matrix Γ, and the relation matrix C. The standard complex normal is the univariate distribution with μ = 0, Γ = 1, and C = 0.

An important subclass of complex normal family is called the circularly-symmetric complex normal and corresponds to the case of zero relation matrix and zero mean: μ = 0 and C = 0 . Circular symmetric complex normal random variables are used extensively in signal processing, and are sometimes referred to as just complex normal in signal processing literature.

Definition

Suppose X and Y are random vectors in R^k such that vec[X Y] is a 2k-dimensional normal random vector. Then we say that the complex random vector

Z = X + i Y

has the complex normal distribution. This distribution can be described with 3 parameters:

μ = E ⁡ [ Z ] , Γ = E ⁡ [ ( Z − μ ) ( Z ¯ − μ ¯ ) T ] , C = E ⁡ [ ( Z − μ ) ( Z − μ ) T ] ,

where Z T denotes matrix transpose, and Z ¯ denotes complex conjugate. Here the location parameter μ can be an arbitrary k-dimensional complex vector; the covariance matrix Γ must be Hermitian and non-negative definite; the relation matrix C should be symmetric. Moreover, matrices Γ and C are such that the matrix

P = Γ ¯ − C ¯ T Γ − 1 C

is also non-negative definite.

Matrices Γ and C can be related to the covariance matrices of X and Y via expressions

V x x ≡ E ⁡ [ ( X − μ x ) ( X − μ x ) T ] = 1 2 Re ⁡ [ Γ + C ] , V x y ≡ E ⁡ [ ( X − μ x ) ( Y − μ y ) T ] = 1 2 Im ⁡ [ − Γ + C ] , V y x ≡ E ⁡ [ ( Y − μ y ) ( X − μ x ) T ] = 1 2 Im ⁡ [ Γ + C ] , V y y ≡ E ⁡ [ ( Y − μ y ) ( Y − μ y ) T ] = 1 2 Re ⁡ [ Γ − C ] ,

and conversely

Γ = V x x + V y y + i ( V y x − V x y ) , C = V x x − V y y + i ( V y x + V x y ) .

Density function

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

f ( z ) = 1 π k det ( Γ ) det ( P ) exp { − 1 2 ( ( z ¯ − μ ¯ ) ′ ( z − μ ) ′ ) ( Γ C C ¯ ′ Γ ¯ ) − 1 ( z − μ z ¯ − μ ¯ ) } = det ( P − 1 ¯ − R ¯ ′ P − 1 R ) det ( P − 1 ) π k e − ( z ¯ − μ ¯ ) ′ P − 1 ¯ ( z − μ ) + Re ⁡ ( ( z − μ ) ′ R ′ P − 1 ¯ ( z − μ ) ) ,

where R = C′ Γ⁻¹ and P = Γ − RC.

Characteristic function

The characteristic function of complex normal distribution is given by

φ ( w ) = exp { i Re ⁡ ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ⁡ ( w ¯ ′ C w ¯ ) ) } ,

where the argument w is a k-dimensional complex vector.

Properties

If Z is a complex normal k-vector, A an ℓ×k matrix, and b a constant ℓ-vector, then the linear transform AZ + b will be distributed also complex-normally:

Z ∼ C N ( μ , Γ , C ) ⇒ A Z + b ∼ C N ( A μ + b , A Γ A ¯ ′ , A C A ′ )

If Z is a complex normal k-vector, then

2 [ ( Z ¯ − μ ¯ ) ′ P − 1 ¯ ( Z − μ ) − Re ⁡ ( ( Z − μ ) ′ R ′ P − 1 ¯ ( Z − μ ) ) ] ∼ χ 2 ( 2 k )

Central limit theorem. If z₁, …, z_T are independent and identically distributed complex random variables, then

T ( 1 T ∑ t = 1 T z t − E ⁡ [ z t ] ) → d C N ( 0 , Γ , C ) , where Γ = E[ zz′ ] and C = E[ zz′ ].

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

Circularly-symmetric and zero mean complex normal distribution

The circularly-symmetric and zero mean complex normal distribution corresponds to the case of zero mean and zero relation matrix, μ=0, C=0. If Z = X + iY is circularly-symmetric complex normal, then the vector vec[X Y] is multivariate normal with covariance structure

( X Y ) ∼ N ( [ Re μ Im μ ] , 1 2 [ Re Γ − Im Γ Im Γ Re Γ ] )

where μ = E[ Z ] = 0 and Γ = E[ ZZ′ ]. This is usually denoted

Z ∼ C N ( 0 , Γ )

and its distribution can also be simplified as

f ( z ) = 1 π k det ( Γ ) e − z ¯ ′ Γ − 1 z .

Therefore, if the non-zero mean μ and covariance matrix Γ are unknown, a suitable log likelihood function for a single observation vector z would be

ln ⁡ ( L ( μ , Γ ) ) = − ln ⁡ ( det ( Γ ) ) − ( z − μ ) ¯ ′ Γ − 1 ( z − μ ) − k ln ⁡ ( π ) .

The standard complex normal corresponds to the distribution of a scalar random variable with μ = 0, C = 0 and Γ = 1. Thus, the standard complex normal distribution has density

f ( z ) = 1 π e − z ¯ z = 1 π e − | z | 2 .

This expression demonstrates why the case C = 0, μ = 0 is called “circularly-symmetric”. The density function depends only on the magnitude of z but not on its argument. As such, the magnitude |z| of standard complex normal random variable will have the Rayleigh distribution and the squared magnitude |z|² will have the Exponential distribution, whereas the argument will be distributed uniformly on [−π, π].

If {z₁, …, z_n} are independent and identically distributed k-dimensional circular complex normal random variables with μ = 0, then random squared norm

Q = ∑ j = 1 n z j ′ ¯ z j = ∑ j = 1 n ∥ z j ∥ 2

has the Generalized chi-squared distribution and the random matrix

W = ∑ j = 1 n z j z j ′ ¯

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

f ( w ) = det ( Γ − 1 ) n det ( w ) n − k π k ( k − 1 ) / 2 ∏ j = 1 p ( n − j ) ! e − tr ⁡ ( Γ − 1 w )

where n ≥ k, and w is a k×k nonnegative-definite matrix.

References

Complex normal distribution Wikipedia

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