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 $\mathbf = (Z_1,\ldots,Z_n)^T$ on the probability space $(\Omega,\mathcal,P)$ is a function $\mathbf \colon \Omega \rightarrow \mathbb^n$ such that the vector $(\Re,\Im,\ldots,\Re,\Im)^T$ is a real random vector on $(\Omega,\mathcal,P)$ where $\Re$ denotes the real part of $z$ and $\Im$ 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 $P(\Re \leq 1, \Im \leq 3)$ make sense. Therefore, the cumulative distribution function F_ : \mathbb^n \mapsto ,1/math> of a random vector $\mathbf=(Z_1,...,Z_n)^T$ is defined as

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

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'') $\operatorname_$ 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 ''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.

:
\operatorname_=
\begin
\mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline& \mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline& \cdots & \mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline\\ \\
\mathrm[(Z_2 - \operatorname _2\overline] & \mathrm[(Z_2 - \operatorname _2\overline] & \cdots & \mathrm[(Z_2 - \operatorname _2\overline] \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\mathrm Z_n - \operatorname[Z_n\overline">_n">Z_n - \operatorname[Z_n\overline& \mathrm Z_n - \operatorname[Z_n\overline">_n">Z_n - \operatorname[Z_n\overline& \cdots & \mathrm Z_n - \operatorname[Z_n\overline]
\end

The ''pseudo-covariance matrix'' (also called ''relation matrix'') is defined replacing #Conjugate transpose, Hermitian transposition by #Transpose, transposition in the definition above.

:
\operatorname_=
\begin
\mathrm Z_1 - \operatorname[Z_1(Z_1 - \operatorname[Z_1">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1(Z_1 - \operatorname[Z_1] & \mathrm Z_1 - \operatorname[Z_1(Z_2 - \operatorname _2] & \cdots & \mathrm Z_1 - \operatorname[Z_1(Z_n - \operatorname[Z_n">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1(Z_n - \operatorname[Z_n] \\ \\
\mathrm[(Z_2 - \operatorname _2(Z_1 - \operatorname[Z_1])] & \mathrm[(Z_2 - \operatorname _2(Z_2 - \operatorname _2] & \cdots & \mathrm[(Z_2 - \operatorname _2(Z_n - \operatorname _n] \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\mathrm Z_n - \operatorname[Z_n(Z_1 - \operatorname[Z_1">_n.html" ;"title="Z_n - \operatorname[Z_n">Z_n - \operatorname[Z_n(Z_1 - \operatorname[Z_1] & \mathrm Z_n - \operatorname[Z_n(Z_2 - \operatorname _2] & \cdots & \mathrm Z_n - \operatorname[Z_n(Z_n - \operatorname _n">_n.html" ;"title="Z_n - \operatorname[Z_n">Z_n - \operatorname[Z_n(Z_n - \operatorname _n\end

;Properties
The covariance matrix is a hermitian matrix, i.e.
:$\operatorname_^H = \operatorname_$.

The pseudo-covariance matrix is a symmetric matrix, i.e.
:$\operatorname_^T = \operatorname_$.

The covariance matrix is a positive semidefinite matrix, i.e.
:$\mathbf^H \operatorname_ \mathbf \ge 0 \quad \text \mathbf \in \mathbb^n$.

Covariance matrices of real and imaginary parts

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

:$\begin \operatorname_ & \operatorname_ \\ \operatorname_ & \operatorname_ \end$

The matrices $\operatorname_$ and $\operatorname_$ can be related to the covariance matrices of $\mathbf$ and $\mathbf$ via the following expressions:
: \begin
& \operatorname_ = \operatorname \mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf">mathbf.html" ;"title="\mathbf-\operatorname[\mathbf">\mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf^\mathrm T] = \tfrac\operatorname(\operatorname_ + \operatorname_) \\
& \operatorname_ = \operatorname \mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf">mathbf.html" ;"title="\mathbf-\operatorname[\mathbf">\mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf^\mathrm T] = \tfrac\operatorname(\operatorname_ - \operatorname_) \\
& \operatorname_ = \operatorname \mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf">mathbf.html" ;"title="\mathbf-\operatorname[\mathbf">\mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf^\mathrm T] = \tfrac\operatorname(\operatorname_ + \operatorname_) \\
& \operatorname_ = \operatorname \mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf">mathbf.html" ;"title="\mathbf-\operatorname[\mathbf">\mathbf-\operatorname[\mathbf(\mathbf-\operatorname[\mathbf^\mathrm T] = \tfrac\operatorname(\operatorname_ -\operatorname_) \\
\end

Conversely:
: $\begin & \operatorname_ = \operatorname_ + \operatorname_ + i(\operatorname_ - \operatorname_) \\ & \operatorname_ = \operatorname_ - \operatorname_ + i(\operatorname_ + \operatorname_) \end$

Cross-covariance matrix and pseudo-cross-covariance matrix

The cross-covariance matrix between two complex random vectors $\mathbf,\mathbf$ is defined as:

:\operatorname_ =
\begin
\mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline& \mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline& \cdots & \mathrm Z_1 - \operatorname[Z_1\overline">_1.html" ;"title="Z_1 - \operatorname[Z_1">Z_1 - \operatorname[Z_1\overline\\ \\
\mathrm[(Z_2 - \operatorname _2\overline] & \mathrm[(Z_2 - \operatorname _2\overline] & \cdots & \mathrm[(Z_2 - \operatorname _2\overline] \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\mathrm Z_n - \operatorname[Z_n\overline">_n">Z_n - \operatorname[Z_n\overline& \mathrm Z_n - \operatorname[Z_n\overline">_n">Z_n - \operatorname[Z_n\overline& \cdots & \mathrm Z_n - \operatorname[Z_n\overline]
\end

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

:\operatorname_ =
\begin
\mathrm Z_1 - \operatorname[Z_1(W_1 - \operatorname[W_1">_1">Z_1 - \operatorname[Z_1(W_1 - \operatorname[W_1">_1<_a>(W_1_-_\operatorname[W_1.html" ;"title="_1">Z_1 - \operatorname[Z_1(W_1 - \operatorname[W_1">_1">Z_1 - \operatorname[Z_1(W_1 - \operatorname[W_1& \mathrm Z_1 - \operatorname[Z_1(W_2 - \operatorname[W_2])] & \cdots & \mathrm Z_1 - \operatorname[Z_1(W_n - \operatorname[W_n])] \\ \\
\mathrm[(Z_2 - \operatorname _2(W_1 - \operatorname _1] & \mathrm _2(W_2 - \operatorname Z_2 - \operatorname _2(W_2 - \operatorname[W_2">_2">Z_2 - \operatorname _2(W_2 - \operatorname[W_2& \cdots & \mathrm[(Z_2 - \operatorname _2(W_n - \operatorname[W_n])] \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\mathrm Z_n - \operatorname[Z_n(W_1 - \operatorname _1] & \mathrm Z_n - \operatorname[Z_n(W_2 - \operatorname[W_2">_n.html" ;"title="Z_n - \operatorname[Z_n">Z_n - \operatorname[Z_n(W_2 - \operatorname[W_2] & \cdots & \mathrm Z_n - \operatorname[Z_n(W_n - \operatorname[W_n])]
\end

Two complex random vectors $\mathbf$ and $\mathbf$ are called uncorrelated if
:$\operatorname_=\operatorname_=0$.

Independence

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

where $F_(\mathbf)$ and $F_(\mathbf)$ denote the cumulative distribution functions of $\mathbf$ and $\mathbf$ as defined in and $F_(\mathbf)$ denotes their joint cumulative distribution function. Independence of $\mathbf$ and $\mathbf$ is often denoted by $\mathbf \perp\!\!\!\perp \mathbf$.
Written component-wise, $\mathbf$ and $\mathbf$ are called independent if
:$F_(z_1,\ldots,z_m,w_1,\ldots,w_n) = F_(z_1,\ldots,z_m) \cdot F_(w_1,\ldots,w_n) \quad \text z_1,\ldots,z_m,w_1,\ldots,w_n$.

Circular symmetry

A complex random vector $\mathbf$ is called circularly symmetric if for every deterministic $\varphi \in [-\pi,\pi)$ the distribution of $e^\mathbf$ equals the distribution of $\mathbf$.

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

Two complex random vectors $\mathbf,\mathbf$ 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 $\mathbf$ is proper if, and only if, for all (deterministic) vectors $\mathbf \in \mathbb^n$ the complex random variable $\mathbf^T \mathbf$ is proper.
* Linear transformations of proper complex random vectors are proper, i.e. if $\mathbf$ is a proper random vectors with $n$ components and $A$ is a deterministic $m \times n$ matrix, then the complex random vector $A \mathbf$ 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 covariace matrix is zero, i.e. if $\operatorname_ = 0$.

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is
:\left, \operatorname mathbf^H \mathbf\^2 \leq \operatorname mathbf^H \mathbf\operatorname Characteristic function (probability theory), characteristic function of a complex random vector $\mathbf$ with $n$ components is a function $\mathbb^n \to \mathbb$ defined by:

: \varphi_(\mathbf) = \operatorname \left e^ \right = \operatorname \left e^ \right /math>

See also

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

References

{{reflist

Probability theory
Randomness
Algebra of random variables>\mathbf^H \mathbf, /math>.

Characteristic function

The Characteristic function (probability theory), characteristic function of a complex random vector $\mathbf$ with $n$ components is a function $\mathbb^n \to \mathbb$ defined by:

: \varphi_(\mathbf) = \operatorname \left e^ \right = \operatorname \left e^ \right /math>

See also

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

References

{{reflist

Probability theory
Randomness
Algebra of random variables