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 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: $\mu = 0 \ \text{and} \ C=0$.^[2] 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

$Z = X + iY \,$

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

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

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.^[3]

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 ^[3]

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

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 ℓ-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, …, z_T 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′ ].

• The modulus of a complex normal random variable follows a Hoyt distribution.^[4]

Circularly-symmetric and zero mean complex normal distribution

The circularly-symmetric and zero mean complex normal distribution ^[5] 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

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

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

$Z \sim \mathcal{CN}(0,\,\Gamma)$

and its distribution can also be simplified as

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

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

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

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, 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

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

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.

See also

• Directional statistics#Distribution of the mean
• Normal distribution
• Multivariate normal distribution (a complex normal distribution is a bivariate normal distribution)
• Generalized chi-squared distribution
• Wishart distribution

References

1. Goodman (1963)
2. bookchapter, Gallager.R, pg9.
3. Picinbono (1996)
4. Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package ‘shotGroups’ version 0.6.2)".
5. bookchapter, Gallager.R

• 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.
• Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing 44 (10): 2637–2640. doi:10.1109/78.539051.