Complex random vector

A complex random vector $\backslash mathbf\{Z\}\; =\; (Z\_1,\backslash ldots,Z\_n)^T$ on the probability space $(\backslash Omega,\backslash mathcal\{F\},P)$ is a function $\backslash mathbf\{Z\}\; \backslash colon\; \backslash Omega\; \backslash rightarrow\; \backslash mathbb\{C\}^n$ such that the vector $(\backslash Re\{(Z\_1)\},\backslash Im\{(Z\_1)\},\backslash ldots,\backslash Re\{(Z\_n)\},\backslash Im\{(Z\_n)\})^T$ is a real random vector on $(\backslash Omega,\backslash mathcal\{F\},P)$ where $\backslash Re\{(z)\}$ denotes the real part of $z$ and $\backslash Im\{(z)\}$ denotes the imaginary part of $z$.{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}{{rp|p. 292}}

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form $P(Z\; \backslash leq\; 1+3i)$ make no sense. However expressions of the form $P(\backslash Re\{(Z)\}\; \backslash leq\; 1,\; \backslash Im\{(Z)\}\; \backslash leq\; 3)$ make sense. Therefore, the cumulative distribution function $F\_\{\backslash mathbf\{Z\}\}\; :\; \backslash mathbb\{C\}^n\; \backslash mapsto\; [0,1]$ of a random vector $\backslash mathbf\{Z\}=(Z\_1,...,Z\_n)^T$ is defined as

{{Equation box 1

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|equation = {{NumBlk||$F\_\{\backslash mathbf\{Z\}\}(\backslash mathbf\{z\})\; =\; \backslash operatorname\{P\}(\backslash Re\{(Z\_1)\}\; \backslash leq\; \backslash Re\{(z\_1)\}\; ,\; \backslash Im\{(Z\_1)\}\; \backslash leq\; \backslash Im\{(z\_1)\},\backslash ldots,\backslash Re\{(Z\_n)\}\; \backslash leq\; \backslash Re\{(z\_n)\}\; ,\; \backslash Im\{(Z\_n)\}\; \backslash leq\; \backslash Im\{(z\_n)\})$|{{EquationRef|Eq.1}}}}

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where $\backslash mathbf\{z\}\; =\; (z\_1,...,z\_n)^T$.

As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.{{rp|p. 293}}

{{Equation box 1

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|equation = {{NumBlk||$\backslash operatorname\{E\}[\backslash mathbf\{Z\}]\; =\; (\backslash operatorname\{E\}[Z\_1],\backslash ldots,\backslash operatorname\{E\}[Z\_n])^T$|{{EquationRef|Eq.2}}}}

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{{see also|Covariance matrix#Complex random vector}}

The covariance matrix (also called second central moment) $\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}$ contains the covariances between all pairs of components. The covariance matrix of an $n\; \backslash times\; 1$ random vector is an $n\; \backslash times\; n$ matrix whose $(i,j)$^th element is the covariance between the i ^th and the j ^th random variables.{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}{{rp|p.372}} 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.{{rp|p. 293}}

{{Equation box 1

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\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}

|{{EquationRef|Eq.3}}}}

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:$$

\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.

{{Equation box 1

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\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]

|{{EquationRef|Eq.4}}}}

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:$$

\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}

;Properties

The covariance matrix is a hermitian matrix, i.e.{{rp|p. 293}}

:$\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}^H\; =\; \backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}$.

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

:$\backslash operatorname\{J\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}^T\; =\; \backslash operatorname\{J\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}$.

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

:$\backslash mathbf\{a\}^H\; \backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{Z\}\}\; \backslash mathbf\{a\}\; \backslash ge\; 0\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{a\}\; \backslash in\; \backslash mathbb\{C\}^n$.

{{see also|Complex random variable#Covariance matrix of real and imaginary parts}}

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

:$\backslash begin\{bmatrix\}$

\operatorname{K}_{\mathbf{X}\mathbf{X}} & \operatorname{K}_{\mathbf{X}\mathbf{Y}} \\

\operatorname{K}_{\mathbf{Y}\mathbf{X}} & \operatorname{K}_{\mathbf{Y}\mathbf{Y}}

\end{bmatrix}

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

: $\backslash begin\{align\}$

& \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:

: $\backslash begin\{align\}$

& \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}

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

{{Equation box 1

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|equation = {{NumBlk||$\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; =$

\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] |{{EquationRef|Eq.5}}}}

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:$\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; =$

\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:

{{Equation box 1

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|equation = {{NumBlk||$\backslash operatorname\{J\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; =\; \backslash operatorname\{cov\}[\backslash mathbf\{Z\},\backslash overline\{\backslash mathbf\{W\}\}]\; =\; \backslash operatorname\{E\}[(\backslash mathbf\{Z\}-\backslash operatorname\{E\}[\backslash mathbf\{Z\}])\{(\backslash mathbf\{W\}-\backslash operatorname\{E\}[\backslash mathbf\{W\}])\}^T]\; =\; \backslash operatorname\{E\}[\backslash mathbf\{Z\}\backslash mathbf\{W\}^T]-\backslash operatorname\{E\}[\backslash mathbf\{Z\}]\backslash operatorname\{E\}[\backslash mathbf\{W\}^T]$|{{EquationRef|Eq.6}}}}

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:$\backslash operatorname\{J\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; =$

\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 $\backslash mathbf\{Z\}$ and $\backslash mathbf\{W\}$ are called uncorrelated if

:$\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}=\backslash operatorname\{J\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}=0$.

{{main|Independence (probability theory)}}

Two complex random vectors $\backslash mathbf\{Z\}=(Z\_1,...,Z\_m)^T$ and $\backslash mathbf\{W\}=(W\_1,...,W\_n)^T$ are called independent if

{{Equation box 1

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|equation = {{NumBlk||$F\_\{\backslash mathbf\{Z,W\}\}(\backslash mathbf\{z,w\})\; =\; F\_\{\backslash mathbf\{Z\}\}(\backslash mathbf\{z\})\; \backslash cdot\; F\_\{\backslash mathbf\{W\}\}(\backslash mathbf\{w\})\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{z\},\backslash mathbf\{w\}$|{{EquationRef|Eq.7}}}}

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where $F\_\{\backslash mathbf\{Z\}\}(\backslash mathbf\{z\})$ and $F\_\{\backslash mathbf\{W\}\}(\backslash mathbf\{w\})$ denote the cumulative distribution functions of $\backslash mathbf\{Z\}$ and $\backslash mathbf\{W\}$ as defined in {{EquationNote|Eq.1}} and $F\_\{\backslash mathbf\{Z,W\}\}(\backslash mathbf\{z,w\})$ denotes their joint cumulative distribution function. Independence of $\backslash mathbf\{Z\}$ and $\backslash mathbf\{W\}$ is often denoted by $\backslash mathbf\{Z\}\; \backslash perp\backslash !\backslash !\backslash !\backslash perp\; \backslash mathbf\{W\}$.

Written component-wise, $\backslash mathbf\{Z\}$ and $\backslash mathbf\{W\}$ are called independent if

:$F\_\{Z\_1,\backslash ldots,Z\_m,W\_1,\backslash ldots,W\_n\}(z\_1,\backslash ldots,z\_m,w\_1,\backslash ldots,w\_n)\; =\; F\_\{Z\_1,\backslash ldots,Z\_m\}(z\_1,\backslash ldots,z\_m)\; \backslash cdot\; F\_\{W\_1,\backslash ldots,W\_n\}(w\_1,\backslash ldots,w\_n)\; \backslash quad\; \backslash text\{for\; all\; \}\; z\_1,\backslash ldots,z\_m,w\_1,\backslash ldots,w\_n$.

A complex random vector $\backslash mathbf\{Z\}$ is called circularly symmetric if for every deterministic $\backslash varphi\; \backslash in\; [-\backslash pi,\backslash pi)$ the distribution of $e^\{\backslash mathrm\; i\; \backslash varphi\}\backslash mathbf\{Z\}$ equals the distribution of $\backslash mathbf\{Z\}$.{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press}}{{rp|pp. 500–501}}

;Properties

• The expectation of a circularly symmetric complex random vector is either zero or it is not defined.{{rp|p. 500}}
• The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.{{rp|p. 584}}

A complex random vector $\backslash mathbf\{Z\}$ is called proper if the following three conditions are all satisfied:{{rp|p. 293}}

• $\backslash operatorname\{E\}[\backslash mathbf\{Z\}]\; =\; 0$ (zero mean)
• (all components have finite variance)
• $\backslash operatorname\{E\}[\backslash mathbf\{Z\}\backslash mathbf\{Z\}^T]\; =\; 0$

Two complex random vectors $\backslash mathbf\{Z\},\backslash mathbf\{W\}$ are called jointly proper is the composite random vector $(Z\_1,Z\_2,\backslash ldots,Z\_m,W\_1,W\_2,\backslash ldots,W\_n)^T$ is proper.

;Properties

• A complex random vector $\backslash mathbf\{Z\}$ is proper if, and only if, for all (deterministic) vectors $\backslash mathbf\{c\}\; \backslash in\; \backslash mathbb\{C\}^n$ the complex random variable $\backslash mathbf\{c\}^T\; \backslash mathbf\{Z\}$ is proper.{{rp|p. 293}}
• Linear transformations of proper complex random vectors are proper, i.e. if $\backslash mathbf\{Z\}$ is a proper random vectors with $n$ components and $A$ is a deterministic $m\; \backslash times\; n$ matrix, then the complex random vector $A\; \backslash mathbf\{Z\}$ is also proper.{{rp|p. 295}}
• Every circularly symmetric complex random vector with finite variance of all its components is proper.{{rp|p. 295}}
• There are proper complex random vectors that are not circularly symmetric.{{rp|p. 504}}
• 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 $\backslash operatorname\{K\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; =\; 0$.

The Cauchy-Schwarz inequality for complex random vectors is

:$\backslash left|\; \backslash operatorname\{E\}[\backslash mathbf\{Z\}^H\; \backslash mathbf\{W\}]\; \backslash right|^2\; \backslash leq\; \backslash operatorname\{E\}[\backslash mathbf\{Z\}^H\; \backslash mathbf\{Z\}]\; \backslash operatorname\{E\}[|\backslash mathbf\{W\}^H\; \backslash mathbf\{W\}|]$.

The characteristic function of a complex random vector $\backslash mathbf\{Z\}$ with $n$ components is a function $\backslash mathbb\{C\}^n\; \backslash to\; \backslash mathbb\{C\}$ defined by:{{rp|p. 295}}

: $\backslash varphi\_\{\backslash mathbf\{Z\}\}(\backslash mathbf\{\backslash omega\})\; =\; \backslash operatorname\{E\}\; \backslash left\; [\; e^\{i\backslash Re\{(\backslash mathbf\{\backslash omega\}^H\; \backslash mathbf\{Z\})\}\}\; \backslash right\; ]\; =\; \backslash operatorname\{E\}\; \backslash left\; [\; e^\{i(\; \backslash Re\{(\backslash omega\_1)\}\backslash Re\{(Z\_1)\}\; +\; \backslash Im\{(\backslash omega\_1)\}\backslash Im\{(Z\_1)\}\; +\; \backslash cdots\; +\; \backslash Re\{(\backslash omega\_n)\}\backslash Re\{(Z\_n)\}\; +\; \backslash Im\{(\backslash omega\_n)\}\backslash Im\{(Z\_n)\}\; )\}\; \backslash right\; ]$

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

{{reflist}}

Category:Probability theory

Category:Randomness

Category:Algebra of random variables