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 X1, ..., Xn are complex-valued random variables, then the n-tuple (X1, ..., Xn) 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.
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Definition
A complex random vector on the probability space is a function such that the vector is a real real random vector on .[1]:p. 292
Expectation
As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.[1]:p. 293
Covariance matrix and pseudo-covariance matrix
The covariance matrix contains the covariances between all pairs of components. The covariance matrix of an random vector is an matrix whose 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.[1]:p. 293
The pseudo-covariance matrix (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above transposition gets replaced by Hermitian transposition in the definition.
Cross-covariance matrix and pseudo-cross-covariance matrix
The cross-covariance matrix between two complex random vectors is defined as:
And the pseudo-cross-covariance matrix is defined as:
Circular symmetry
A complex random vector is called circularly symmetric if for every deterministic the distribution of equals the distribution of .[2]:pp. 500–501
The expectation of a circularly symmetric complex random vectors is either zero or it is not defined.[2]:p. 500
Proper complex random vectors
Definition
A complex random vector Z is called proper if the following three conditions are all satisfied:[1]:p. 293
- (finite variance)
Properties
- A complex random vector Z is proper if, and only if, for all (deterministic) vectors the complex random variable is proper.[1]:p. 293
- Linear transformations of proper complex random vectors are proper, i.e. if is a proper random vectors with components and is a deterministic matrix, then the complex random vector is also proper.[1]:p. 295
- Every circularly symmetric complex random vector with finite variance of all its components is proper.[1]:p. 295
Characteristic function
The characteristic function of a complex random vector with components is a function defined by:[1]:p. 295