Complex random vector

In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If [math]\displaystyle{ Z_1,\ldots,Z_n }[/math] are complex-valued random variables, then the n-tuple [math]\displaystyle{ \left( Z_1,\ldots,Z_n \right) }[/math] is a complex random vector. Complex random variables can always be considered as pairs of real random vectors: their real and imaginary parts. Some concepts of real random vectors have a straightforward generalization to complex random vectors. For example, the definition of the mean of a complex random vector. Other concepts are unique to complex random vectors.

Applications of complex random vectors are found in digital signal processing.

Definition

A complex random vector [math]\displaystyle{ \mathbf{Z} = (Z_1,\ldots,Z_n)^T }[/math] on the probability space [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] is a function [math]\displaystyle{ \mathbf{Z} \colon \Omega \rightarrow \mathbb{C}^n }[/math] such that the vector [math]\displaystyle{ (\Re{(Z_1)},\Im{(Z_1)},\ldots,\Re{(Z_n)},\Im{(Z_n)})^T }[/math] is a real random vector on [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] where [math]\displaystyle{ \Re{(z)} }[/math] denotes the real part of [math]\displaystyle{ z }[/math] and [math]\displaystyle{ \Im{(z)} }[/math] denotes the imaginary part of [math]\displaystyle{ z }[/math].

Cumulative distribution function

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form [math]\displaystyle{ P(Z \leq 1+3i) }[/math] make no sense. However expressions of the form [math]\displaystyle{ P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) }[/math] make sense. Therefore, the cumulative distribution function [math]\displaystyle{ F_{\mathbf{Z}} : \mathbb{C}^n \mapsto [0,1] }[/math] of a random vector [math]\displaystyle{ \mathbf{Z}=(Z_1,...,Z_n)^T }[/math] is defined as

[math]\displaystyle{ F_{\mathbf{Z}}(\mathbf{z}) = \operatorname{P}(\Re{(Z_1)} \leq \Re{(z_1)} , \Im{(Z_1)} \leq \Im{(z_1)},\ldots,\Re{(Z_n)} \leq \Re{(z_n)} , \Im{(Z_n)} \leq \Im{(z_n)}) }[/math]

(Eq.1)

where [math]\displaystyle{ \mathbf{z} = (z_1,...,z_n)^T }[/math].

Expectation

As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.

[math]\displaystyle{ \operatorname{E}[\mathbf{Z}] = (\operatorname{E}[Z_1],\ldots,\operatorname{E}[Z_n])^T }[/math]

(Eq.2)

Covariance matrix and pseudo-covariance matrix

The covariance matrix (also called second central moment) [math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{Z}} }[/math] contains the covariances between all pairs of components. The covariance matrix of an [math]\displaystyle{ n \times 1 }[/math] random vector is an [math]\displaystyle{ n \times n }[/math] matrix whose [math]\displaystyle{ (i,j) }[/math]^th element is the covariance between the i ^th and the j ^th random variables. Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two. Thus the covariance matrix is a Hermitian matrix.

[math]\displaystyle{ \begin{align} & \operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\mathbf{Z}] = \operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{Z}-\operatorname{E}[\mathbf{Z}])}^H] = \operatorname{E}[\mathbf{Z}\mathbf{Z}^H]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{Z}^H] \\[12pt] \end{align} }[/math]

(Eq.3)

[math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{Z}}= \begin{bmatrix} \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_n - \operatorname{E}[Z_n])}] \\ \\ \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_n - \operatorname{E}[Z_n])}] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_n - \operatorname{E}[Z_n])}] \end{bmatrix} }[/math]

The pseudo-covariance matrix (also called relation matrix) is defined replacing Hermitian transposition by transposition in the definition above.

[math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{Z}}] = \operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{Z}-\operatorname{E}[\mathbf{Z}])}^T] = \operatorname{E}[\mathbf{Z}\mathbf{Z}^T]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{Z}^T] }[/math]

(Eq.4)

[math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{Z}}= \begin{bmatrix} \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_n - \operatorname{E}[Z_n])] \\ \\ \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_n - \operatorname{E}[Z_n])] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_n - \operatorname{E}[Z_n])] \end{bmatrix} }[/math]

Properties

The covariance matrix is a hermitian matrix, i.e.

[math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{Z}}^H = \operatorname{K}_{\mathbf{Z}\mathbf{Z}} }[/math].

The pseudo-covariance matrix is a symmetric matrix, i.e.

[math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{Z}}^T = \operatorname{J}_{\mathbf{Z}\mathbf{Z}} }[/math].

The covariance matrix is a positive semidefinite matrix, i.e.

[math]\displaystyle{ \mathbf{a}^H \operatorname{K}_{\mathbf{Z}\mathbf{Z}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{C}^n }[/math].

Covariance matrices of real and imaginary parts

By decomposing the random vector [math]\displaystyle{ \mathbf{Z} }[/math] into its real part [math]\displaystyle{ \mathbf{X} = \Re{(\mathbf{Z})} }[/math] and imaginary part [math]\displaystyle{ \mathbf{Y} = \Im{(\mathbf{Z})} }[/math] (i.e. [math]\displaystyle{ \mathbf{Z}=\mathbf{X}+i\mathbf{Y} }[/math]), the pair [math]\displaystyle{ (\mathbf{X},\mathbf{Y}) }[/math] has a covariance matrix of the form:

[math]\displaystyle{ \begin{bmatrix} \operatorname{K}_{\mathbf{X}\mathbf{X}} & \operatorname{K}_{\mathbf{Y}\mathbf{X}} \\ \operatorname{K}_{\mathbf{X}\mathbf{Y}} & \operatorname{K}_{\mathbf{Y}\mathbf{Y}} \end{bmatrix} }[/math]

The matrices [math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{Z}} }[/math] and [math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{Z}} }[/math] can be related to the covariance matrices of [math]\displaystyle{ \mathbf{X} }[/math] and [math]\displaystyle{ \mathbf{Y} }[/math] via the following expressions:

[math]\displaystyle{ \begin{align} & \operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^\mathrm T] = \tfrac{1}{2}\operatorname{Re}(\operatorname{K}_{\mathbf{Z}\mathbf{Z}} + \operatorname{J}_{\mathbf{Z}\mathbf{Z}}) \\ & \operatorname{K}_{\mathbf{Y}\mathbf{Y}} = \operatorname{E}[(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^\mathrm T] = \tfrac{1}{2}\operatorname{Re}(\operatorname{K}_{\mathbf{Z}\mathbf{Z}} - \operatorname{J}_{\mathbf{Z}\mathbf{Z}}) \\ & \operatorname{K}_{\mathbf{Y}\mathbf{X}} = \operatorname{E}[(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^\mathrm T] = \tfrac{1}{2}\operatorname{Im}(\operatorname{J}_{\mathbf{Z}\mathbf{Z}} + \operatorname{K}_{\mathbf{Z}\mathbf{Z}}) \\ & \operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^\mathrm T] = \tfrac{1}{2}\operatorname{Im}(\operatorname{J}_{\mathbf{Z}\mathbf{Z}} -\operatorname{K}_{\mathbf{Z}\mathbf{Z}}) \\ \end{align} }[/math]

Conversely:

[math]\displaystyle{ \begin{align} & \operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{K}_{\mathbf{X}\mathbf{X}} + \operatorname{K}_{\mathbf{Y}\mathbf{Y}} + i(\operatorname{K}_{\mathbf{Y}\mathbf{X}} - \operatorname{K}_{\mathbf{X}\mathbf{Y}}) \\ & \operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{K}_{\mathbf{X}\mathbf{X}} - \operatorname{K}_{\mathbf{Y}\mathbf{Y}} + i(\operatorname{K}_{\mathbf{Y}\mathbf{X}} + \operatorname{K}_{\mathbf{X}\mathbf{Y}}) \end{align} }[/math]

Cross-covariance matrix and pseudo-cross-covariance matrix

The cross-covariance matrix between two complex random vectors [math]\displaystyle{ \mathbf{Z},\mathbf{W} }[/math] is defined as:

[math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{cov}[\mathbf{Z},\mathbf{W}] = \operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^H] = \operatorname{E}[\mathbf{Z}\mathbf{W}^H]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}^H] }[/math]

(Eq.5)

[math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}} = \begin{bmatrix} \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_n - \operatorname{E}[W_n])}] \\ \\ \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_n - \operatorname{E}[W_n])}] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_n - \operatorname{E}[W_n])}] \end{bmatrix} }[/math]

And the pseudo-cross-covariance matrix is defined as:

[math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{W}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{W}}] = \operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^T] = \operatorname{E}[\mathbf{Z}\mathbf{W}^T]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}^T] }[/math]

(Eq.6)

[math]\displaystyle{ \operatorname{J}_{\mathbf{Z}\mathbf{W}} = \begin{bmatrix} \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_n - \operatorname{E}[W_n])] \\ \\ \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_n - \operatorname{E}[W_n])] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_n - \operatorname{E}[W_n])] \end{bmatrix} }[/math]

Two complex random vectors [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] are called uncorrelated if

[math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}}=\operatorname{J}_{\mathbf{Z}\mathbf{W}}=0 }[/math].

Independence

Two complex random vectors [math]\displaystyle{ \mathbf{Z}=(Z_1,...,Z_m)^T }[/math] and [math]\displaystyle{ \mathbf{W}=(W_1,...,W_n)^T }[/math] are called independent if

[math]\displaystyle{ F_{\mathbf{Z,W}}(\mathbf{z,w}) = F_{\mathbf{Z}}(\mathbf{z}) \cdot F_{\mathbf{W}}(\mathbf{w}) \quad \text{for all } \mathbf{z},\mathbf{w} }[/math]

(Eq.7)

where [math]\displaystyle{ F_{\mathbf{Z}}(\mathbf{z}) }[/math] and [math]\displaystyle{ F_{\mathbf{W}}(\mathbf{w}) }[/math] denote the cumulative distribution functions of [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] as defined in Eq.1 and [math]\displaystyle{ F_{\mathbf{Z,W}}(\mathbf{z,w}) }[/math] denotes their joint cumulative distribution function. Independence of [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] is often denoted by [math]\displaystyle{ \mathbf{Z} \perp\!\!\!\perp \mathbf{W} }[/math]. Written component-wise, [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] are called independent if

[math]\displaystyle{ F_{Z_1,\ldots,Z_m,W_1,\ldots,W_n}(z_1,\ldots,z_m,w_1,\ldots,w_n) = F_{Z_1,\ldots,Z_m}(z_1,\ldots,z_m) \cdot F_{W_1,\ldots,W_n}(w_1,\ldots,w_n) \quad \text{for all } z_1,\ldots,z_m,w_1,\ldots,w_n }[/math].

Circular symmetry

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

Properties

• The expectation of a circularly symmetric complex random vector is either zero or it is not defined.
• The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.

Proper complex random vectors

A complex random vector [math]\displaystyle{ \mathbf{Z} }[/math] is called proper if the following three conditions are all satisfied:

• [math]\displaystyle{ \operatorname{E}[\mathbf{Z}] = 0 }[/math] (zero mean)
• [math]\displaystyle{ \operatorname{var}[Z_1] \lt \infty , \ldots , \operatorname{var}[Z_n] \lt \infty }[/math] (all components have finite variance)
• [math]\displaystyle{ \operatorname{E}[\mathbf{Z}\mathbf{Z}^T] = 0 }[/math]

Two complex random vectors [math]\displaystyle{ \mathbf{Z},\mathbf{W} }[/math] are called jointly proper is the composite random vector [math]\displaystyle{ (Z_1,Z_2,\ldots,Z_m,W_1,W_2,\ldots,W_n)^T }[/math] is proper.

Properties

• A complex random vector [math]\displaystyle{ \mathbf{Z} }[/math] is proper if, and only if, for all (deterministic) vectors [math]\displaystyle{ \mathbf{c} \in \mathbb{C}^n }[/math] the complex random variable [math]\displaystyle{ \mathbf{c}^T \mathbf{Z} }[/math] is proper.
• Linear transformations of proper complex random vectors are proper, i.e. if [math]\displaystyle{ \mathbf{Z} }[/math] is a proper random vectors with [math]\displaystyle{ n }[/math] components and [math]\displaystyle{ A }[/math] is a deterministic [math]\displaystyle{ m \times n }[/math] matrix, then the complex random vector [math]\displaystyle{ A \mathbf{Z} }[/math] is also proper.
• Every circularly symmetric complex random vector with finite variance of all its components is proper.
• There are proper complex random vectors that are not circularly symmetric.
• A real random vector is proper if and only if it is constant.
• Two jointly proper complex random vectors are uncorrelated if and only if their covariance matrix is zero, i.e. if [math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}} = 0 }[/math].

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is

[math]\displaystyle{ \left| \operatorname{E}[\mathbf{Z}^H \mathbf{W}] \right|^2 \leq \operatorname{E}[\mathbf{Z}^H \mathbf{Z}] \operatorname{E}[|\mathbf{W}^H \mathbf{W}|] }[/math].

Characteristic function

The characteristic function of a complex random vector [math]\displaystyle{ \mathbf{Z} }[/math] with [math]\displaystyle{ n }[/math] components is a function [math]\displaystyle{ \mathbb{C}^n \to \mathbb{C} }[/math] defined by:

[math]\displaystyle{ \varphi_{\mathbf{Z}}(\mathbf{\omega}) = \operatorname{E} \left [ e^{i\Re{(\mathbf{\omega}^H \mathbf{Z})}} \right ] = \operatorname{E} \left [ e^{i( \Re{(\omega_1)}\Re{(Z_1)} + \Im{(\omega_1)}\Im{(Z_1)} + \cdots + \Re{(\omega_n)}\Re{(Z_n)} + \Im{(\omega_n)}\Im{(Z_n)} )} \right ] }[/math]

See also

• Complex normal distribution
• Complex random variable (scalar case)