Complex normal distribution

In probability theory, the family of complex normal distributions, denoted $$\mathcal{CN}$$ or $$\mathcal{N}_{\mathcal{C}}$$, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix $$\Gamma$$, and the relation matrix $$C$$. The standard complex normal is the univariate distribution with $$\mu = 0$$, $$\Gamma=1$$, and $$C=0$$.

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: $$ \mu = 0 $$ and $$ C=0 $$. This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.

Complex standard normal random variable
The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable $$Z$$ whose real and imaginary parts are independent normally distributed random variables with mean zero and variance $$1/2$$. Formally,

where $$Z \sim \mathcal{CN}(0,1)$$ denotes that $$Z$$ is a standard complex normal random variable.

Complex normal random variable
Suppose $$X$$ and $$Y$$ are real random variables such that $$(X,Y)^{\mathrm T}$$ is a 2-dimensional normal random vector. Then the complex random variable $$Z=X+iY$$ is called complex normal random variable or complex Gaussian random variable.

Complex standard normal random vector
A n-dimensional complex random vector $$\mathbf{Z}=(Z_1,\ldots,Z_n)^{\mathrm T}$$ 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. That $$\mathbf{Z}$$ is a standard complex normal random vector is denoted $$\mathbf{Z} \sim \mathcal{CN}(0,\boldsymbol{I}_n)$$.

Complex normal random vector
If $$\mathbf{X}=(X_1,\ldots,X_n)^{\mathrm T}$$ and $$\mathbf{Y}=(Y_1,\ldots,Y_n)^{\mathrm T}$$ are random vectors in $$\mathbb{R}^n$$ such that $$[\mathbf{X},\mathbf{Y}]$$ is a normal random vector with $$2n$$ components. Then we say that the complex random vector

\mathbf{Z} = \mathbf{X} + i \mathbf{Y} \, $$ is a complex normal random vector or a complex Gaussian random vector.

Mean, covariance, and relation
The complex Gaussian distribution can be described with 3 parameters:

\mu = \operatorname{E}[\mathbf{Z}], \quad \Gamma = \operatorname{E}[(\mathbf{Z}-\mu)({\mathbf{Z}}-\mu)^{\mathrm H}], \quad C = \operatorname{E}[(\mathbf{Z}-\mu)(\mathbf{Z}-\mu)^{\mathrm T}], $$ where $$\mathbf{Z}^{\mathrm T}$$ denotes matrix transpose of $$\mathbf{Z}$$, and $$\mathbf{Z}^{\mathrm H}$$ denotes conjugate transpose.

Here the location parameter $$\mu$$ is a n-dimensional complex vector; the covariance matrix $$\Gamma$$ is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix $$C$$ is symmetric. The complex normal random vector $$ \mathbf{Z} $$ can now be denoted as$$ \mathbf{Z}\ \sim\ \mathcal{CN}(\mu,\ \Gamma,\ C). $$Moreover, matrices $$\Gamma$$ and $$C$$ are such that the matrix

P = \overline{\Gamma} - {C}^{\mathrm H}\Gamma^{-1}C $$ is also non-negative definite where $$\overline{\Gamma}$$ denotes the complex conjugate of $$\Gamma$$.

Relationships between covariance matrices
As for any complex random vector, the matrices $$\Gamma$$ and $$C$$ can be related to the covariance matrices of $$\mathbf{X} = \Re(\mathbf{Z})$$ and $$\mathbf{Y} = \Im(\mathbf{Z})$$ via expressions
 * $$\begin{align}

& V_{XX} \equiv \operatorname{E}[(\mathbf{X}-\mu_X)(\mathbf{X}-\mu_X)^\mathrm T] = \tfrac{1}{2}\operatorname{Re}[\Gamma + C], \quad V_{XY} \equiv \operatorname{E}[(\mathbf{X}-\mu_X)(\mathbf{Y}-\mu_Y)^\mathrm T] = \tfrac{1}{2}\operatorname{Im}[-\Gamma + C], \\ & V_{YX} \equiv \operatorname{E}[(\mathbf{Y}-\mu_Y)(\mathbf{X}-\mu_X)^\mathrm T] = \tfrac{1}{2}\operatorname{Im}[\Gamma + C], \quad\, V_{YY} \equiv \operatorname{E}[(\mathbf{Y}-\mu_Y)(\mathbf{Y}-\mu_Y)^\mathrm T] = \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^n\sqrt{\det(\Gamma)\det(P)}}\, \exp\!\left\{-\frac12 \begin{pmatrix}(\overline{z}-\overline\mu)^\intercal, & (z-\mu)^\intercal\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}}-R^{\ast} P^{-1}R\right)\det(P^{-1})}}{\pi^n}\, e^{ -(z-\mu)^\ast\overline{P^{-1}}(z-\mu) + \operatorname{Re}\left((z-\mu)^\intercal R^\intercal\overline{P^{-1}}(z-\mu)\right)}, \end{align}$$

where $$R=C^{\mathrm H} \Gamma^{-1}$$ and $$P=\overline{\Gamma}-RC$$.

Characteristic function
The characteristic function of complex normal distribution is given by

\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 an n-dimensional complex vector.

Properties

 * If $$\mathbf{Z}$$ is a complex normal n-vector, $$\boldsymbol{A}$$ an m×n matrix, and $$b$$ a constant m-vector, then the linear transform $$\boldsymbol{A}\mathbf{Z}+b$$ 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 A^{\mathrm H},\, A C A^{\mathrm T}) $$


 * If $$\mathbf{Z}$$ is a complex normal n-vector, then

2\Big[ (\mathbf{Z}-\mu)^{\mathrm H} \overline{P^{-1}}(\mathbf{Z}-\mu) - \operatorname{Re}\big((\mathbf{Z}-\mu)^{\mathrm T} R^{\mathrm T} \overline{P^{-1}}(\mathbf{Z}-\mu)\big) \Big]\ \sim\ \chi^2(2n) $$


 * Central limit theorem. If $$Z_1,\ldots,Z_T$$ are independent and identically distributed complex random variables, then

\sqrt{T}\Big( \tfrac{1}{T}\textstyle\sum_{t=1}^T Z_t - \operatorname{E}[Z_t]\Big) \ \xrightarrow{d}\ \mathcal{CN}(0,\,\Gamma,\,C), $$
 * where $$\Gamma = \operatorname{E}[Z Z^{\mathrm H}]$$ and $$C = \operatorname{E}[Z Z^{\mathrm T}]$$.


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

Definition
A complex random vector $$ \mathbf{Z} $$ is called circularly symmetric if for every deterministic $$ \varphi \in [-\pi,\pi) $$ the distribution of $$ e^{\mathrm i \varphi}\mathbf{Z} $$ equals the distribution of $$ \mathbf{Z} $$.

Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix $$\Gamma$$.

The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e. $$\mu = 0$$ and $$C=0$$. This is usually denoted
 * $$\mathbf{Z} \sim \mathcal{CN}(0,\,\Gamma)$$

Distribution of real and imaginary parts
If $$\mathbf{Z}=\mathbf{X}+i\mathbf{Y}$$ is circularly-symmetric (central) complex normal, then the vector $$[\mathbf{X}, \mathbf{Y}]$$ is multivariate normal with covariance structure

\begin{pmatrix}\mathbf{X} \\ \mathbf{Y}\end{pmatrix} \ \sim\ \mathcal{N}\Big( \begin{bmatrix}                      0 \\                       0                     \end{bmatrix},\                      \tfrac{1}{2}\begin{bmatrix}                       \operatorname{Re}\,\Gamma & -\operatorname{Im}\,\Gamma \\                       \operatorname{Im}\,\Gamma &  \operatorname{Re}\,\Gamma                     \end{bmatrix}\Big) $$ where $$\Gamma=\operatorname{E}[\mathbf{Z} \mathbf{Z}^{\mathrm H}]$$.

Probability density function
For nonsingular covariance matrix $$\Gamma$$, its distribution can also be simplified as

f_{\mathbf{Z}}(\mathbf{z}) = \tfrac{1}{\pi^n \det(\Gamma)}\, e^{ -(\mathbf{z}-\mathbf{\mu})^{\mathrm H} \Gamma^{-1} (\mathbf{z}-\mathbf{\mu})} $$.

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) -n \ln(\pi). $$

The standard complex normal (defined in $$) corresponds to the distribution of a scalar random variable with $$\mu = 0$$, $$C=0$$ and $$\Gamma=1$$. Thus, the standard complex normal distribution has density



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

Properties
The above expression demonstrates why the case $$C=0$$, $$\mu = 0$$ 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 a 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 $$[-\pi,\pi]$$.

If $$\left\{ \mathbf{Z}_1,\ldots,\mathbf{Z}_k \right\}$$ are independent and identically distributed n-dimensional circular complex normal random vectors with $$\mu = 0$$, then the random squared norm

Q = \sum_{j=1}^k \mathbf{Z}_j^{\mathrm H} \mathbf{Z}_j = \sum_{j=1}^k \| \mathbf{Z}_j \|^2 $$ has the generalized chi-squared distribution and the random matrix

W = \sum_{j=1}^k \mathbf{Z}_j \mathbf{Z}_j^{\mathrm H} $$ has the complex Wishart distribution with $$k$$ degrees of freedom. This distribution can be described by density function

f(w) = \frac{\det(\Gamma^{-1})^k\det(w)^{k-n}}{\pi^{n(n-1)/2}\prod_{j=1}^k(k-j)!}\ e^{-\operatorname{tr}(\Gamma^{-1}w)} $$ where $$k \ge n$$, and $$w$$ is a $$n \times n$$ nonnegative-definite matrix.