Complex random vector explained

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

Z = (Z _{1,\ldots,Z} ^{T} ^{n)} on the probability space (\Omega, l{F},P) is a function Z \colon \Omega → C ^{n} such that the vector (\Re{(Z _{1)},\Im{(Z} _{1)},\ldots,\Re{(Z} _{n)},\Im{(Z} ^{T} ^{n)})} is a real random vector on (\Omega, l{F},P) where \Re{(z)} denotes the real part of z and \Im{(z)} denotes the imaginary part of z .^[1]

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+3 i) 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 _{Z} : C ^{n} \mapsto [ 0 , 1 ] of a random vector Z = (Z _{1,...,Z} ^{T} ^{n)} is defined as

where

z = (z _{1,...,z} ^{T} ^{n)} .

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} _{ZZ} contains the covariances between all pairs of components. The covariance matrix of an n x 1 random vector is an n x n matrix whose (i,j) ^th element is the covariance between the i ^th and the j ^th random variables.^[2] 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} _{ZZ} = \begin{bmatrix} E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(Z} _{1} - \operatorname{E}[Z _{1])}]} & E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(Z} _{2} - \operatorname{E}[Z _{2])}]} & … & E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(Z} _{n} - \operatorname{E}[Z _{n])}]} \ \\ E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(Z} _{1} - \operatorname{E}[Z _{1])}]} & E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(Z} _{2} - \operatorname{E}[Z _{2])}]} & … & E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(Z} _{n} - \operatorname{E}[Z _{n])}]} \ \\ \vdots & \vdots & \ddots & \vdots \ \\ E [(Z _{n} - \operatorname{E}[Z _{n])\overline{(Z} _{1} - \operatorname{E}[Z _{1])}]} & E [(Z _{n} - \operatorname{E}[Z _{n])\overline{(Z} _{2} - \operatorname{E}[Z _{2])}]} & … & 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} _{ZZ} = \begin{bmatrix} E [(Z _{1} - \operatorname{E}[Z _{1])(Z} _{1} - \operatorname{E}[Z _{1])]} & E [(Z _{1} - \operatorname{E}[Z _{1])(Z} _{2} - \operatorname{E}[Z _{2])]} & … & E [(Z _{1} - \operatorname{E}[Z _{1])(Z} _{n} - \operatorname{E}[Z _{n])]} \ \\ E [(Z _{2} - \operatorname{E}[Z _{2])(Z} _{1} - \operatorname{E}[Z _{1])]} & E [(Z _{2} - \operatorname{E}[Z _{2])(Z} _{2} - \operatorname{E}[Z _{2])]} & … & E [(Z _{2} - \operatorname{E}[Z _{2])(Z} _{n} - \operatorname{E}[Z _{n])]} \ \\ \vdots & \vdots & \ddots & \vdots \ \\ E [(Z _{n} - \operatorname{E}[Z _{n])(Z} _{1} - \operatorname{E}[Z _{1])]} & E [(Z _{n} - \operatorname{E}[Z _{n])(Z} _{2} - \operatorname{E}[Z _{2])]} & … & E [(Z _{n} - \operatorname{E}[Z _{n])(Z} _{n} - \operatorname{E}[Z _{n])] \end{bmatrix}}

PropertiesThe covariance matrix is a hermitian matrix, i.e.^[1]

\operatorname{K} ^{H} _{Z Z} = \operatorname{K} _{ZZ} .

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

\operatorname{J} ^{T} _{Z Z} = \operatorname{J} _{ZZ} .

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

a ^{H} \operatorname{K} _{ZZ} a \ge 0 forall a \in C ^{n} .

Covariance matrices of real and imaginary parts

By decomposing the random vector

Z into its real part X = \Re{( Z )} and imaginary part Y = \Im{( Z )} (i.e. Z = X + i Y ), the pair ( X , Y ) has a covariance matrix of the form:\begin{bmatrix} \operatorname{K} _{XX} & \operatorname{K} _{YX} \ \operatorname{K} _{XY} & \operatorname{K} _{YY} \end{bmatrix}

The matrices

\operatorname{K} _{ZZ} and \operatorname{J} _{ZZ} can be related to the covariance matrices of X and Y via the following expressions:\begin{align} & \operatorname{K} _{XX} = \operatorname{E}[( X - \operatorname{E}[ X ])( X - \operatorname{E}[ X ]) ^{T]=} \tfrac{ 1 }{ 2 }\operatorname{Re}(\operatorname{K} _{ZZ} + \operatorname{J} _{ZZ} ) \\ & \operatorname{K} _{YY} = \operatorname{E}[( Y - \operatorname{E}[ Y ])( Y - \operatorname{E}[ Y ]) ^{T]=} \tfrac{ 1 }{ 2 }\operatorname{Re}(\operatorname{K} _{ZZ} - \operatorname{J} _{ZZ} ) \\ & \operatorname{K} _{YX} = \operatorname{E}[( Y - \operatorname{E}[ Y ])( X - \operatorname{E}[ X ]) ^{T]=} \tfrac{ 1 }{ 2 }\operatorname{Im}(\operatorname{J} _{ZZ} + \operatorname{K} _{ZZ} ) \\ & \operatorname{K} _{XY} = \operatorname{E}[( X - \operatorname{E}[ X ])( Y - \operatorname{E}[ Y ]) ^{T]=} \tfrac{ 1 }{ 2 }\operatorname{Im}(\operatorname{J} _{ZZ} - \operatorname{K} _{ZZ} ) \\ \end{align}

Conversely:

\begin{align} & \operatorname{K} _{ZZ} = \operatorname{K} _{XX} + \operatorname{K} _{YY} + i(\operatorname{K} _{YX} - \operatorname{K} _{XY} ) \\ & \operatorname{J} _{ZZ} = \operatorname{K} _{XX} - \operatorname{K} _{YY} + i(\operatorname{K} _{YX} + \operatorname{K} _{XY} ) \end{align}

Cross-covariance matrix and pseudo-cross-covariance matrix

The cross-covariance matrix between two complex random vectors

Z , W is defined as:\operatorname{K} _{ZW} = \begin{bmatrix} E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(W} _{1} - \operatorname{E}[W _{1])}]} & E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(W} _{2} - \operatorname{E}[W _{2])}]} & … & E [(Z _{1} - \operatorname{E}[Z _{1])\overline{(W} _{n} - \operatorname{E}[W _{n])}]} \ \\ E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(W} _{1} - \operatorname{E}[W _{1])}]} & E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(W} _{2} - \operatorname{E}[W _{2])}]} & … & E [(Z _{2} - \operatorname{E}[Z _{2])\overline{(W} _{n} - \operatorname{E}[W _{n])}]} \ \\ \vdots & \vdots & \ddots & \vdots \ \\ E [(Z _{n} - \operatorname{E}[Z _{n])\overline{(W} _{1} - \operatorname{E}[W _{1])}]} & E [(Z _{n} - \operatorname{E}[Z _{n])\overline{(W} _{2} - \operatorname{E}[W _{2])}]} & … & 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} _{ZW} = \begin{bmatrix} E [(Z _{1} - \operatorname{E}[Z _{1])(W} _{1} - \operatorname{E}[W _{1])]} & E [(Z _{1} - \operatorname{E}[Z _{1])(W} _{2} - \operatorname{E}[W _{2])]} & … & E [(Z _{1} - \operatorname{E}[Z _{1])(W} _{n} - \operatorname{E}[W _{n])]} \ \\ E [(Z _{2} - \operatorname{E}[Z _{2])(W} _{1} - \operatorname{E}[W _{1])]} & E [(Z _{2} - \operatorname{E}[Z _{2])(W} _{2} - \operatorname{E}[W _{2])]} & … & E [(Z _{2} - \operatorname{E}[Z _{2])(W} _{n} - \operatorname{E}[W _{n])]} \ \\ \vdots & \vdots & \ddots & \vdots \ \\ E [(Z _{n} - \operatorname{E}[Z _{n])(W} _{1} - \operatorname{E}[W _{1])]} & E [(Z _{n} - \operatorname{E}[Z _{n])(W} _{2} - \operatorname{E}[W _{2])]} & … & E [(Z _{n} - \operatorname{E}[Z _{n])(W} _{n} - \operatorname{E}[W _{n])] \end{bmatrix}}

Two complex random vectors

Z and W are called uncorrelated if\operatorname{K} _{ZW} = \operatorname{J} _{ZW} =0 .

Independence

See main article: Independence (probability theory). Two complex random vectors

Z = (Z _{1,...,Z} ^{T} ^{m)} and W = (W _{1,...,W} ^{T} ^{n)} are called independent if

where

F _{Z} ( z ) and F _{W} ( w ) denote the cumulative distribution functions of Z and W as defined in and F _{Z,W} ( z,w ) denotes their joint cumulative distribution function. Independence of Z and W is often denoted by Z \perp\perp W .Written component-wise, Z and W are called independent ifF _{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)} ⋅ F _{W _{1,\ldots,W} _{n}} (w _{1,\ldots,w} _{n)} forall z _{1,\ldots,z} _{m,w} _{1,\ldots,w} _{n} .

Circular symmetry

A complex random vector

Z is called circularly symmetric if for every deterministic \varphi \in [ - \pi,\pi) the distribution of e ^{i\varphi} Z equals the distribution of Z .^[3]

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

Z is called proper if the following three conditions are all satisfied:\operatorname{E}[ Z ] = 0 (zero mean)\operatorname{var}[Z _{1]} < inf ty , \ldots , \operatorname{var}[Z _{n]} < inf ty (all components have finite variance)\operatorname{E}[ Z Z ^{T]} = 0

Two complex random vectors

Z , W are called jointly proper is the composite random vector (Z _{1,Z} _{2,\ldots,Z} _{m,W} _{1,W} _{2,\ldots,W} ^{T} ^{n)} is proper.

Properties

• A complex random vector

Z is proper if, and only if, for all (deterministic) vectors c \in C ^{n} the complex random variable c ^{T} Z is proper.

• Linear transformations of proper complex random vectors are proper, i.e. if

Z is a proper random vectors with n components and A is a deterministic m x n matrix, then the complex random vector A 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} _{ZW} = 0 .

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is

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

Characteristic function

The characteristic function of a complex random vector

Z with n components is a function C ^{n} \to C defined by:\varphi _{Z} ( \omega ) = \operatorname{E} \left [ e ^{i\Re{( \omega ^{H} Z )} } \right ] = \operatorname \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 ]

See also

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

Notes and References

1. Book: Lapidoth, Amos . 2009 . A Foundation in Digital Communication . Cambridge University Press . 978-0-521-19395-5.
2. Book: Gubner, John A. . 2006 . Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press . 978-0-521-86470-1.
3. Book: Tse, David . 2005 . Fundamentals of Wireless Communication . Cambridge University Press.