Complex normal distribution

Definitions

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,

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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.

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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).

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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.

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Mean, covariance, and relation{{anchor|Mean and covariance}}

The complex Gaussian distribution can be described with 3 parameters:{{cite journal : \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

Main article: Complex random vector#Covariance matrix and pseudo-covariance matrix

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{bmatrix} z - \mu \ \overline z -\overline \mu\end{bmatrix}^{\mathrm H} \begin{bmatrix}\Gamma & C \ \overline{C}&\overline\Gamma\end{bmatrix}^{!!-1}! \begin{bmatrix}z-\mu \ \overline{z}-\overline{\mu}\end{bmatrix} \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.

Circularly-symmetric central case

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} . Main article: [Complex random vector#Circular symmetry

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.

References

References

1. [http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R''], pg9.
2. Lapidoth, A.. (2009). "A Foundation in Digital Communication". Cambridge University Press.
3. Tse, David. (2005). "Fundamentals of Wireless Communication". Cambridge University Press.
4. Daniel Wollschlaeger. (July 2019). ["The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)"](http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html }}{{Dead link).
5. {{rp. p. 507[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R'']