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 $$Z_1,\ldots,Z_n$$ are complex-valued random variables, then the n-tuple $$\left( Z_1,\ldots,Z_n \right)$$ 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 $$ \mathbf{Z} = (Z_1,\ldots,Z_n)^T $$ on the probability space $$(\Omega,\mathcal{F},P)$$ is a function $$ \mathbf{Z} \colon \Omega \rightarrow \mathbb{C}^n $$ such that the vector $$(\Re{(Z_1)},\Im{(Z_1)},\ldots,\Re{(Z_n)},\Im{(Z_n)})^T $$ is a real random vector on $$(\Omega,\mathcal{F},P)$$ where $$\Re{(z)}$$ denotes the real part of $$z$$ and $$\Im{(z)}$$ denotes the imaginary part of $$z$$.

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

where $$\mathbf{z} = (z_1,...,z_n)^T$$.

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

Covariance matrix and pseudo-covariance matrix
The covariance matrix (also called second central moment) $$ \operatorname{K}_{\mathbf{Z}\mathbf{Z}}$$ contains the covariances between all pairs of components. The covariance matrix of an $$n \times 1$$ random vector is an $$n \times n$$ matrix whose $$(i,j)$$th element is the covariance between the ith and the jth 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.



\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} $$

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



\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} $$

The covariance matrix is a hermitian matrix, i.e.
 * Properties
 * $$\operatorname{K}_{\mathbf{Z}\mathbf{Z}}^H = \operatorname{K}_{\mathbf{Z}\mathbf{Z}}$$.

The pseudo-covariance matrix is a symmetric matrix, i.e.
 * $$\operatorname{J}_{\mathbf{Z}\mathbf{Z}}^T = \operatorname{J}_{\mathbf{Z}\mathbf{Z}}$$.

The covariance matrix is a positive semidefinite matrix, i.e.
 * $$\mathbf{a}^H \operatorname{K}_{\mathbf{Z}\mathbf{Z}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{C}^n$$.

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



\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} $$

The matrices $$\operatorname{K}_{\mathbf{Z}\mathbf{Z}}$$ and $$\operatorname{J}_{\mathbf{Z}\mathbf{Z}}$$ can be related to the covariance matrices of $$\mathbf{X}$$ and $$\mathbf{Y}$$ via the following expressions:
 * $$\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}$$

Conversely:
 * $$\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}$$

Cross-covariance matrix and pseudo-cross-covariance matrix
The cross-covariance matrix between two complex random vectors $$\mathbf{Z},\mathbf{W}$$ is defined as:


 * $$\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} $$

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


 * $$\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} $$

Two complex random vectors $$\mathbf{Z}$$ and $$\mathbf{W}$$ are called uncorrelated if
 * $$\operatorname{K}_{\mathbf{Z}\mathbf{W}}=\operatorname{J}_{\mathbf{Z}\mathbf{W}}=0$$.

Independence
Two complex random vectors $$\mathbf{Z}=(Z_1,...,Z_m)^T$$ and $$\mathbf{W}=(W_1,...,W_n)^T$$ are called independent if

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

Circular symmetry
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 $$ \mathbf{Z} $$.


 * 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 $$\mathbf{Z}$$ is called proper if the following three conditions are all satisfied:
 * $$ \operatorname{E}[\mathbf{Z}] = 0 $$ (zero mean)
 * $$ \operatorname{var}[Z_1] < \infty, \ldots , \operatorname{var}[Z_n] < \infty $$ (all components have finite variance)
 * $$ \operatorname{E}[\mathbf{Z}\mathbf{Z}^T] = 0 $$

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


 * Properties
 * A complex random vector $$\mathbf{Z}$$ is proper if, and only if, for all (deterministic) vectors $$ \mathbf{c} \in \mathbb{C}^n$$ the complex random variable $$\mathbf{c}^T \mathbf{Z}$$ is proper.
 * Linear transformations of proper complex random vectors are proper, i.e. if $$\mathbf{Z}$$ is a proper random vectors with $$n$$ components and $$A$$ is a deterministic $$m \times n$$ matrix, then the complex random vector $$A \mathbf{Z}$$ 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 $$\operatorname{K}_{\mathbf{Z}\mathbf{W}} = 0$$.

Cauchy-Schwarz inequality
The Cauchy-Schwarz inequality for complex random vectors is
 * $$\left| \operatorname{E}[\mathbf{Z}^H \mathbf{W}] \right|^2 \leq \operatorname{E}[\mathbf{Z}^H \mathbf{Z}] \operatorname{E}[|\mathbf{W}^H \mathbf{W}|]$$.

Characteristic function
The characteristic function of a complex random vector $$ \mathbf{Z} $$ with $$ n $$ components is a function $$ \mathbb{C}^n \to \mathbb{C} $$ defined by:


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