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>Z_1,\ldots,Z_n</math> are complex-valued random variables, then the ''n''-tuple <math>\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.
{{Probability fundamentals}}
==Definition== A complex random vector <math> \mathbf{Z} = (Z_1,\ldots,Z_n)^T </math> on the probability space <math>(\Omega,\mathcal{F},P)</math> is a function <math> \mathbf{Z} \colon \Omega \rightarrow \mathbb{C}^n </math> such that the vector <math>(\Re{(Z_1)},\Im{(Z_1)},\ldots,\Re{(Z_n)},\Im{(Z_n)})^T </math> is a real random vector on <math>(\Omega,\mathcal{F},P)</math> where <math>\Re{(z)}</math> denotes the real part of <math>z</math> and <math>\Im{(z)}</math> denotes the imaginary part of <math>z</math>.<ref name=Lapidoth>{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}</ref>{{rp|p. 292}}
==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> P(Z \leq 1+3i) </math> make no sense. However expressions of the form <math> P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) </math> make sense. Therefore, the cumulative distribution function <math>F_{\mathbf{Z}} : \mathbb{C}^n \mapsto [0,1]</math> of a random vector <math>\mathbf{Z}=(Z_1,...,Z_n)^T </math> is defined as
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>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>|{{EquationRef|Eq.1}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
where <math>\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.<ref name=Lapidoth />{{rp|p. 293}}
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math> \operatorname{E}[\mathbf{Z}] = (\operatorname{E}[Z_1],\ldots,\operatorname{E}[Z_n])^T </math>|{{EquationRef|Eq.2}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
==Covariance matrix and pseudo-covariance matrix== {{See also|Covariance matrix#Complex random vector}}
The ''covariance matrix'' (also called ''second central moment'') <math> \operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math> contains the covariances between all pairs of components. The covariance matrix of an <math>n \times 1</math> random vector is an <math>n \times n</math> matrix whose <math>(i,j)</math><sup>th</sup> element is the covariance between the ''i''<sup> th</sup> and the ''j''<sup> th</sup> random variables.<ref name=Gubner>{{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}}</ref>{{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.<ref name=Lapidoth />{{rp|p. 293}}
{{Equation box 1 |indent = |title= |equation = {{NumBlk|| <math> \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> |{{EquationRef|Eq.3}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
:<math> \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.
{{Equation box 1 |indent = |title= |equation = {{NumBlk|| <math> \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> |{{EquationRef|Eq.4}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
:<math> \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.<ref name=Lapidoth/>{{rp|p. 293}} :<math>\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>\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>\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=== {{See also|Complex random variable#Covariance matrix of real and imaginary parts}}
By decomposing the random vector <math>\mathbf{Z}</math> into its real part <math>\mathbf{X} = \Re{(\mathbf{Z})}</math> and imaginary part <math>\mathbf{Y} = \Im{(\mathbf{Z})}</math> (i.e. <math>\mathbf{Z}=\mathbf{X}+i\mathbf{Y}</math>), the pair <math> (\mathbf{X},\mathbf{Y})</math> has a covariance matrix of the form:
:<math>\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}</math>
The matrices <math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math> and <math>\operatorname{J}_{\mathbf{Z}\mathbf{Z}}</math> can be related to the covariance matrices of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> via the following expressions: : <math>\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>\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>\mathbf{Z},\mathbf{W}</math> is defined as:
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math> \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>|{{EquationRef|Eq.5}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
:<math>\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:
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math> \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>|{{EquationRef|Eq.6}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
:<math>\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>\mathbf{Z}</math> and <math>\mathbf{W}</math> are called '''uncorrelated''' if :<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}}=\operatorname{J}_{\mathbf{Z}\mathbf{W}}=0</math>.
==Independence== {{main|Independence (probability theory)}} Two complex random vectors <math>\mathbf{Z}=(Z_1,...,Z_m)^T</math> and <math>\mathbf{W}=(W_1,...,W_n)^T</math> are called '''independent''' if
{{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>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>|{{EquationRef|Eq.7}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
where <math>F_{\mathbf{Z}}(\mathbf{z})</math> and <math>F_{\mathbf{W}}(\mathbf{w})</math> denote the cumulative distribution functions of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> as defined in {{EquationNote|Eq.1}} and <math>F_{\mathbf{Z,W}}(\mathbf{z,w})</math> denotes their joint cumulative distribution function. Independence of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is often denoted by <math>\mathbf{Z} \perp\!\!\!\perp \mathbf{W}</math>. Written component-wise, <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> are called independent if :<math>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> \mathbf{Z} </math> is called circularly symmetric if for every deterministic <math> \varphi \in [-\pi,\pi) </math> the distribution of <math> e^{\mathrm i \varphi}\mathbf{Z} </math> equals the distribution of <math> \mathbf{Z} </math>.<ref name=TseViswanath>{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press}}</ref>{{rp|pp. 500–501}}
;Properties * The expectation of a circularly symmetric complex random vector is either zero or it is not defined.<ref name=TseViswanath />{{rp|p. 500}} * The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.<ref name=TseViswanath />{{rp|p. 584}}
==Proper complex random vectors== A complex random vector <math>\mathbf{Z}</math> is called '''proper''' if the following three conditions are all satisfied:<ref name=Lapidoth />{{rp|p. 293}} * <math> \operatorname{E}[\mathbf{Z}] = 0 </math> (zero mean) * <math> \operatorname{var}[Z_1] < \infty , \ldots , \operatorname{var}[Z_n] < \infty </math> (all components have finite variance) * <math> \operatorname{E}[\mathbf{Z}\mathbf{Z}^T] = 0 </math>
Two complex random vectors <math>\mathbf{Z},\mathbf{W}</math> are called ''' jointly proper''' if the composite random vector <math>(Z_1,Z_2,\ldots,Z_m,W_1,W_2,\ldots,W_n)^T</math> is proper.
;Properties * A complex random vector <math>\mathbf{Z}</math> is proper if, and only if, for all (deterministic) vectors <math> \mathbf{c} \in \mathbb{C}^n</math> the complex random variable <math>\mathbf{c}^T \mathbf{Z}</math> is proper.<ref name=Lapidoth />{{rp|p. 293}} * Linear transformations of proper complex random vectors are proper, i.e. if <math>\mathbf{Z}</math> is a proper random vectors with <math>n</math> components and <math>A</math> is a deterministic <math>m \times n</math> matrix, then the complex random vector <math>A \mathbf{Z}</math> is also proper.<ref name=Lapidoth />{{rp|p. 295}} * Every circularly symmetric complex random vector with finite variance of all its components is proper.<ref name=Lapidoth />{{rp|p. 295}} * There are proper complex random vectors that are not circularly symmetric.<ref name=Lapidoth />{{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 <math>\operatorname{K}_{\mathbf{Z}\mathbf{W}} = 0</math>.
==Cauchy–Schwarz inequality== The Cauchy–Schwarz inequality for complex random vectors is :<math>\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> \mathbf{Z} </math> with <math> n </math> components is a function <math> \mathbb{C}^n \to \mathbb{C} </math> defined by:<ref name=Lapidoth />{{rp|p. 295}}
: <math> \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)
==References== {{reflist}}
Category:Probability theory Category:Randomness Category:Algebra of random variables