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 formula_1 are complex-valued random variables, then the "n"-tuple formula_2 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 formula_3 on the probability space formula_4 is a function formula_5 such that the vector formula_6 is a real real random vector on formula_4 where formula_8 denotes the real part of formula_9 and formula_10 denotes the imaginary part of formula_9.
Cumulative distribution function.
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form formula_12 make no sense. However expressions of the form formula_13 make sense. Therefore, the cumulative distribution function formula_14 of a random vector formula_15 is defined as
where formula_16.
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.
Definitions.
The covariance matrix (also called second central moment) formula_17 contains the covariances between all pairs of components. The covariance matrix of an formula_18 random vector is an formula_19 matrix whose formula_20th 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.
The pseudo-covariance matrix (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above Hermitian transposition gets replaced by transposition in the definition.
=
\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]
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Properties.
The covariance matrix is a hermitian matrix, i.e.
The pseudo-covariance matrix is a symmetric matrix, i.e.
The covariance matrix is a positive semidefinite matrix, i.e.
Covariance matrices of real and imaginary parts.
By decomposing the random vector formula_26 into its real part formula_27 and imaginary part formula_28 (i.e. formula_29), the matrices formula_30 and formula_31 can be related to the covariance matrices of formula_32 and formula_33 via the following expressions:
and conversely
Cross-covariance matrix and pseudo-cross-covariance matrix.
Definitions.
The cross-covariance matrix between two complex random vectors formula_36 is defined as:
And the pseudo-cross-covariance matrix is defined as:
Uncorrelatedness.
Two complex random vectors formula_26 and formula_40 are called uncorrelated if
Independence.
Two complex random vectors formula_42 and formula_43 are called independent if
where formula_44 and formula_45 denote the cumulative distribution functions of formula_26 and formula_40 as defined in and formula_48 denotes their joint cumulative distribution function. Independence of formula_26 and formula_40 is often denoted by formula_51.
Written component-wise, formula_26 and formula_40 are called independent if
Circular symmetry.
Definition.
A complex random vector formula_55 is called circularly symmetric if for every deterministic formula_56 the distribution of formula_57 equals the distribution of formula_55.
Proper complex random vectors.
Definition.
A complex random vector formula_26 is called proper if the following three conditions are all satisfied:
Two complex random vectors formula_36 are called jointly proper is the composite random vector formula_64 is proper.
Cauchy-Schwarz inequality.
The Cauchy-Schwarz inequality for complex random vectors is
Characteristic function.
The characteristic function of a complex random vector formula_55 with formula_76 components is a function formula_77 defined by: