Complex normal distribution

﻿
Complex normal distribution

In probability theory, the family of complex normal distributions consists of complex random variables whose real and imaginary parts are jointly normal.[1] 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 circular symmetric complex normal and corresponds to the case of zero relation matrix: C = 0. Circular symmetric complex normal random variables are used extensively in signal processing, and are sometimes incorrectly referred to as just complex normal in signal processing literature.

Definition

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

$Z = X + iY \,$

has the complex normal distribution. This distribution can be described with 3 parameters:[2]

$\mu = \operatorname{E}[Z], \quad \Gamma = \operatorname{E}[(Z-\mu)(\overline{Z}-\overline\mu)'], \quad C = \operatorname{E}[(Z-\mu)(Z-\mu)'],$

where Z ′ 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

$P = \overline\Gamma - \overline{C}'\Gamma^{-1}C$

is also non-negative definite.[2]

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

\begin{align} & V_{xx} \equiv \operatorname{E}[(X-\mu_x)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma + C], \quad V_{xy} \equiv \operatorname{E}[(X-\mu_x)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Im}[-\Gamma + C], \\ & V_{yx} \equiv \operatorname{E}[(Y-\mu_y)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Im}[\Gamma + C], \quad\, V_{yy} \equiv \operatorname{E}[(Y-\mu_y)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma - C], \end{align}

and conversely

\begin{align} & \Gamma = V_{xx} + V_{yy} + i(V_{yx} - V_{xy}), \\ & C = V_{xx} - V_{yy} + i(V_{yx} + V_{xy}). \end{align}

Density function

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

\begin{align} f(z) &= \frac{1}{\pi^k\sqrt{\det(\Gamma)\det(P)}}\, \exp\!\left\{-\frac12 \begin{pmatrix}(\overline{z}-\overline\mu)' & (z-\mu)'\end{pmatrix} \begin{pmatrix}\Gamma&C\\\overline{C}'&\overline\Gamma\end{pmatrix}^{\!\!-1}\! \begin{pmatrix}z-\mu \\ \overline{z}-\overline{\mu}\end{pmatrix} \right\} \\[8pt] &= \tfrac{\sqrt{\det\left(\overline{P^{-1}}-\overline{R}'P^{-1}R\right)\det(P^{-1})}}{\pi^k}\, e^{ -(\overline{z}-\overline\mu)'\overline{P^{-1}}(z-\mu) + \operatorname{Re}\left((z-\mu)'R'\overline{P^{-1}}(z-\mu)\right)}, \end{align}

where R = C′ Γ −1 and P = Γ − RC.

Characteristic function

The characteristic function of complex normal distribution is given by [2]

$\varphi(w) = \exp\!\big\{i\operatorname{Re}(\overline{w}'\mu) + \tfrac{1}{4}\big(\overline{w}'\Gamma w + \operatorname{Re}(\overline{w}'C\overline{w})\big)\big\},$

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 -vector, then the linear transform AZ + b will be distributed also complex-normally:
$Z\ \sim\ \mathcal{CN}(\mu,\, \Gamma,\, C) \quad\Rightarrow\quad AZ+b\ \sim\ \mathcal{CN}(A\mu+b,\, A\Gamma\overline{A}',\, ACA')$
• If Z is a complex normal k-vector, then
$2\Big[ (\overline{Z}-\overline\mu)'\overline{P^{-1}}(Z-\mu) - \operatorname{Re}\big((Z-\mu)'R'\overline{P^{-1}}(Z-\mu)\big) \Big]\ \sim\ \chi^2(2k)$
• Central limit theorem. If z1, …, zT are independent and identically distributed complex random variables, then
$\sqrt{T}\Big( \tfrac{1}{T}\textstyle\sum_{t=1}^Tz_t - \operatorname{E}[z_t]\Big) \ \xrightarrow{d}\ \mathcal{CN}(0,\,\Gamma,\,C),$

where Γ = E[ zz′ ] and C = E[ zz′ ].

Circular symmetric complex normal distribution

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

$\begin{pmatrix}X \\ Y\end{pmatrix} \ \sim\ \mathcal{N}\Big( \begin{bmatrix} \operatorname{Re}\,\mu \\ \operatorname{Im}\,\mu \end{bmatrix},\ \tfrac{1}{2}\begin{bmatrix} \operatorname{Re}\,\Gamma & -\operatorname{Im}\,\Gamma \\ \operatorname{Im}\,\Gamma & \operatorname{Re}\,\Gamma \end{bmatrix}\Big)$

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

$Z \sim \mathcal{CN}(\mu,\,\Gamma)$

and its distribution can also be simplified as

$f(z) = \tfrac{1}{\pi^k\det(\Gamma)}\, e^{ -(\overline{z}-\overline\mu)'\Gamma^{-1}(z-\mu) }.$

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

$f(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}.$

This expression demonstrates why the case C = 0 is called “circular-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 [−ππ].

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

$Q = \sum_{j=1}^n \overline{z_j'} z_j = \sum_{j=1}^n \| z_j \|^2$

has the Generalized chi-squared distribution and the random matrix

$W = \sum_{j=1}^n z_j\overline{z_j'}$

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

$f(w) = \frac{\det(\Gamma^{-1})^n\det(w)^{n-k}}{\pi^{k(k-1)/2}\prod_{j=1}^p(n-j)!}\ e^{-\operatorname{tr}(\Gamma^{-1}w)}$

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

References

• 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. JSTOR 2991290.
• Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing 44 (10): 2637–2640.

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

• Multivariate normal distribution — MVN redirects here. For the airport with that IATA code, see Mount Vernon Airport. Probability density function Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the… …   Wikipedia

• normal — I. adjective Etymology: Latin normalis, from norma Date: circa 1696 1. perpendicular; especially perpendicular to a tangent at a point of tangency 2. a. according with, constituting, or not deviating from a norm, rule, or principle b. conforming… …   New Collegiate Dictionary

• Distribution function — This article describes the distribution function as used in physics. You may be looking for the related mathematical concepts of cumulative distribution function or probability density function. In molecular kinetic theory in physics, a particle… …   Wikipedia

• Complex regional pain syndrome — Complex regional pain syndrome/Reflex Sympathetic Dystrophy (CRPS/RSD) Classification and external resources ICD 10 M89.0, G56.4 ICD 9 …   Wikipedia

• Distribution mangagement system — SCADA systems have been a part of utility automation for at least 15 years and contributing to the decision making process of the control rooms. However, majority of the existing solutions are closely related to distribution network data… …   Wikipedia

• Cauchy distribution — Not to be confused with Lorenz curve. Cauchy–Lorentz Probability density function The purple curve is the standard Cauchy distribution Cumulative distribution function …   Wikipedia

• Mixture distribution — See also: Mixture model In probability and statistics, a mixture distribution is the probability distribution of a random variable whose values can be interpreted as being derived in a simple way from an underlying set of other random variables.… …   Wikipedia

• von Mises distribution — von Mises Probability density function The support is chosen to be [ π,π] with μ=0 Cumulative distribution function The support is chosen to be [ π,π] with μ=0 …   Wikipedia

• Rayleigh distribution — Probability distribution name =Rayleigh type =density pdf cdf parameters =sigma>0, support =xin [0;infty) pdf =frac{x expleft(frac{ x^2}{2sigma^2} ight)}{sigma^2} cdf =1 expleft(frac{ x^2}{2sigma^2} ight) mean =sigma sqrt{frac{pi}{2 median… …   Wikipedia