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.

Contents

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.

See also

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. 

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