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 are complex-valued random variables, then the n-tuple 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.
A complex random vector on the probability space is a function such that the vector is a real random vector on where denotes the real part of and denotes the imaginary part of .[1]: 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 make no sense. However expressions of the form make sense. Therefore, the cumulative distribution function of a random vector is defined as
(Eq.1)
where .
Expectation
As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.[1]: p. 293
The covariance matrix (also called second central moment) 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.[2]: 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.[1]: p. 293
Two complex random vectors and are called independent if
(Eq.7)
where and denote the cumulative distribution functions of and as defined in Eq.1 and denotes their joint cumulative distribution function. Independence of and is often denoted by . Written component-wise, and are called independent if
.
Circular symmetry
A complex random vector is called circularly symmetric if for every deterministic the distribution of equals the distribution of .[3]: pp. 500–501
Properties
The expectation of a circularly symmetric complex random vector is either zero or it is not defined.[3]: p. 500
The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.[3]: p. 584
Proper complex random vectors
A complex random vector is called proper if the following three conditions are all satisfied:[1]: p. 293
(zero mean)
(all components have finite variance)
Two complex random vectors are called jointly proper is the composite random vector is proper.
Properties
A complex random vector 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
There are proper complex random vectors that are not circularly symmetric.[1]: 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 covariace matrix is zero, i.e. if .
^ abcdefghijLapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN978-0-521-19395-5.
^Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN978-0-521-86470-1.
^ abcTse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press.
February 25, 2023
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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 Z n displaystyle Z 1 ldots Z n are complex valued random variables then the n tuple Z 1 Z n displaystyle 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 Contents 1 Definition 2 Cumulative distribution function 3 Expectation 4 Covariance matrix and pseudo covariance matrix 4 1 Covariance matrices of real and imaginary parts 5 Cross covariance matrix and pseudo cross covariance matrix 6 Independence 7 Circular symmetry 8 Proper complex random vectors 9 Cauchy Schwarz inequality 10 Characteristic function 11 See also 12 ReferencesDefinition EditA complex random vector Z Z 1 Z n T displaystyle mathbf Z Z 1 ldots Z n T on the probability space W F P displaystyle Omega mathcal F P is a function Z W C n displaystyle mathbf Z colon Omega rightarrow mathbb C n such that the vector ℜ Z 1 ℑ Z 1 ℜ Z n ℑ Z n T displaystyle Re Z 1 Im Z 1 ldots Re Z n Im Z n T is a real random vector on W F P displaystyle Omega mathcal F P where ℜ z displaystyle Re z denotes the real part of z displaystyle z and ℑ z displaystyle Im z denotes the imaginary part of z displaystyle z 1 p 292 Cumulative distribution function EditThe generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form P Z 1 3 i displaystyle P Z leq 1 3i make no sense However expressions of the form P ℜ Z 1 ℑ Z 3 displaystyle P Re Z leq 1 Im Z leq 3 make sense Therefore the cumulative distribution function F Z C n 0 1 displaystyle F mathbf Z mathbb C n mapsto 0 1 of a random vector Z Z 1 Z n T displaystyle mathbf Z Z 1 Z n T is defined as F Z z P ℜ Z 1 ℜ z 1 ℑ Z 1 ℑ z 1 ℜ Z n ℜ z n ℑ Z n ℑ z n displaystyle 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 Eq 1 where z z 1 z n T displaystyle mathbf z z 1 z n T Expectation EditAs in the real case the expectation also called expected value of a complex random vector is taken component wise 1 p 293 E Z E Z 1 E Z n T displaystyle operatorname E mathbf Z operatorname E Z 1 ldots operatorname E Z n T Eq 2 Covariance matrix and pseudo covariance matrix EditSee also Covariance matrix Complex random vector The covariance matrix also called second central moment K Z Z displaystyle operatorname K mathbf Z mathbf Z contains the covariances between all pairs of components The covariance matrix of an n 1 displaystyle n times 1 random vector is an n n displaystyle n times n matrix whose i j displaystyle i j th element is the covariance between the ith and the jth random variables 2 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 1 p 293 K Z Z cov Z Z E Z E Z Z E Z H E Z Z H E Z E Z H displaystyle begin aligned amp 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 aligned Eq 3 K Z Z E Z 1 E Z 1 Z 1 E Z 1 E Z 1 E Z 1 Z 2 E Z 2 E Z 1 E Z 1 Z n E Z n E Z 2 E Z 2 Z 1 E Z 1 E Z 2 E Z 2 Z 2 E Z 2 E Z 2 E Z 2 Z n E Z n E Z n E Z n Z 1 E Z 1 E Z n E Z n Z 2 E Z 2 E Z n E Z n Z n E Z n displaystyle operatorname K mathbf Z mathbf Z begin bmatrix mathrm E Z 1 operatorname E Z 1 overline Z 1 operatorname E Z 1 amp mathrm E Z 1 operatorname E Z 1 overline Z 2 operatorname E Z 2 amp cdots amp 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 amp mathrm E Z 2 operatorname E Z 2 overline Z 2 operatorname E Z 2 amp cdots amp mathrm E Z 2 operatorname E Z 2 overline Z n operatorname E Z n vdots amp vdots amp ddots amp vdots mathrm E Z n operatorname E Z n overline Z 1 operatorname E Z 1 amp mathrm E Z n operatorname E Z n overline Z 2 operatorname E Z 2 amp cdots amp mathrm 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 J Z Z cov Z Z E Z E Z Z E Z T E Z Z T E Z E Z T displaystyle 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 Eq 4 J Z Z E Z 1 E Z 1 Z 1 E Z 1 E Z 1 E Z 1 Z 2 E Z 2 E Z 1 E Z 1 Z n E Z n E Z 2 E Z 2 Z 1 E Z 1 E Z 2 E Z 2 Z 2 E Z 2 E Z 2 E Z 2 Z n E Z n E Z n E Z n Z 1 E Z 1 E Z n E Z n Z 2 E Z 2 E Z n E Z n Z n E Z n displaystyle operatorname J mathbf Z mathbf Z begin bmatrix mathrm E Z 1 operatorname E Z 1 Z 1 operatorname E Z 1 amp mathrm E Z 1 operatorname E Z 1 Z 2 operatorname E Z 2 amp cdots amp 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 amp mathrm E Z 2 operatorname E Z 2 Z 2 operatorname E Z 2 amp cdots amp mathrm E Z 2 operatorname E Z 2 Z n operatorname E Z n vdots amp vdots amp ddots amp vdots mathrm E Z n operatorname E Z n Z 1 operatorname E Z 1 amp mathrm E Z n operatorname E Z n Z 2 operatorname E Z 2 amp cdots amp mathrm 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 p 293 K Z Z H K Z Z displaystyle operatorname K mathbf Z mathbf Z H operatorname K mathbf Z mathbf Z The pseudo covariance matrix is a symmetric matrix i e J Z Z T J Z Z displaystyle operatorname J mathbf Z mathbf Z T operatorname J mathbf Z mathbf Z The covariance matrix is a positive semidefinite matrix i e a H K Z Z a 0 for all a C n displaystyle mathbf a H operatorname K mathbf Z mathbf Z mathbf a geq 0 quad text for all mathbf a in mathbb C n Covariance matrices of real and imaginary parts Edit See also Complex random variable Covariance matrix of real and imaginary parts By decomposing the random vector Z displaystyle mathbf Z into its real part X ℜ Z displaystyle mathbf X Re mathbf Z and imaginary part Y ℑ Z displaystyle mathbf Y Im mathbf Z i e Z X i Y displaystyle mathbf Z mathbf X i mathbf Y the pair X Y displaystyle mathbf X mathbf Y has a covariance matrix of the form K X X K Y X K X Y K Y Y displaystyle begin bmatrix operatorname K mathbf X mathbf X amp operatorname K mathbf Y mathbf X operatorname K mathbf X mathbf Y amp operatorname K mathbf Y mathbf Y end bmatrix The matrices K Z Z displaystyle operatorname K mathbf Z mathbf Z and J Z Z displaystyle operatorname J mathbf Z mathbf Z can be related to the covariance matrices of X displaystyle mathbf X and Y displaystyle mathbf Y via the following expressions K X X E X E X X E X T 1 2 Re K Z Z J Z Z K Y Y E Y E Y Y E Y T 1 2 Re K Z Z J Z Z K Y X E Y E Y X E X T 1 2 Im J Z Z K Z Z K X Y E X E X Y E Y T 1 2 Im J Z Z K Z Z displaystyle begin aligned amp 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 amp 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 amp 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 amp 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 aligned Conversely K Z Z K X X K Y Y i K Y X K X Y J Z Z K X X K Y Y i K Y X K X Y displaystyle begin aligned amp 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 amp 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 aligned Cross covariance matrix and pseudo cross covariance matrix EditThe cross covariance matrix between two complex random vectors Z W displaystyle mathbf Z mathbf W is defined as K Z W cov Z W E Z E Z W E W H E Z W H E Z E W H displaystyle 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 Eq 5 K Z W E Z 1 E Z 1 W 1 E W 1 E Z 1 E Z 1 W 2 E W 2 E Z 1 E Z 1 W n E W n E Z 2 E Z 2 W 1 E W 1 E Z 2 E Z 2 W 2 E W 2 E Z 2 E Z 2 W n E W n E Z n E Z n W 1 E W 1 E Z n E Z n W 2 E W 2 E Z n E Z n W n E W n displaystyle operatorname K mathbf Z mathbf W begin bmatrix mathrm E Z 1 operatorname E Z 1 overline W 1 operatorname E W 1 amp mathrm E Z 1 operatorname E Z 1 overline W 2 operatorname E W 2 amp cdots amp 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 amp mathrm E Z 2 operatorname E Z 2 overline W 2 operatorname E W 2 amp cdots amp mathrm E Z 2 operatorname E Z 2 overline W n operatorname E W n vdots amp vdots amp ddots amp vdots mathrm E Z n operatorname E Z n overline W 1 operatorname E W 1 amp mathrm E Z n operatorname E Z n overline W 2 operatorname E W 2 amp cdots amp mathrm 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 J Z W cov Z W E Z E Z W E W T E Z W T E Z E W T displaystyle 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 Eq 6 J Z W E Z 1 E Z 1 W 1 E W 1 E Z 1 E Z 1 W 2 E W 2 E Z 1 E Z 1 W n E W n E Z 2 E Z 2 W 1 E W 1 E Z 2 E Z 2 W 2 E W 2 E Z 2 E Z 2 W n E W n E Z n E Z n W 1 E W 1 E Z n E Z n W 2 E W 2 E Z n E Z n W n E W n displaystyle operatorname J mathbf Z mathbf W begin bmatrix mathrm E Z 1 operatorname E Z 1 W 1 operatorname E W 1 amp mathrm E Z 1 operatorname E Z 1 W 2 operatorname E W 2 amp cdots amp 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 amp mathrm E Z 2 operatorname E Z 2 W 2 operatorname E W 2 amp cdots amp mathrm E Z 2 operatorname E Z 2 W n operatorname E W n vdots amp vdots amp ddots amp vdots mathrm E Z n operatorname E Z n W 1 operatorname E W 1 amp mathrm E Z n operatorname E Z n W 2 operatorname E W 2 amp cdots amp mathrm E Z n operatorname E Z n W n operatorname E W n end bmatrix Two complex random vectors Z displaystyle mathbf Z and W displaystyle mathbf W are called uncorrelated if K Z W J Z W 0 displaystyle operatorname K mathbf Z mathbf W operatorname J mathbf Z mathbf W 0 Independence EditMain article Independence probability theory Two complex random vectors Z Z 1 Z m T displaystyle mathbf Z Z 1 Z m T and W W 1 W n T displaystyle mathbf W W 1 W n T are called independent if F Z W z w F Z z F W w for all z w displaystyle 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 Eq 7 where F Z z displaystyle F mathbf Z mathbf z and F W w displaystyle F mathbf W mathbf w denote the cumulative distribution functions of Z displaystyle mathbf Z and W displaystyle mathbf W as defined in Eq 1 and F Z W z w displaystyle F mathbf Z W mathbf z w denotes their joint cumulative distribution function Independence of Z displaystyle mathbf Z and W displaystyle mathbf W is often denoted by Z W displaystyle mathbf Z perp perp mathbf W Written component wise Z displaystyle mathbf Z and W displaystyle mathbf W are called independent if F Z 1 Z m W 1 W n z 1 z m w 1 w n F Z 1 Z m z 1 z m F W 1 W n w 1 w n for all z 1 z m w 1 w n displaystyle 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 Circular symmetry EditA complex random vector Z displaystyle mathbf Z is called circularly symmetric if for every deterministic f p p displaystyle varphi in pi pi the distribution of e i f Z displaystyle e mathrm i varphi mathbf Z equals the distribution of Z displaystyle mathbf Z 3 pp 500 501 PropertiesThe expectation of a circularly symmetric complex random vector is either zero or it is not defined 3 p 500 The pseudo covariance matrix of a circularly symmetric complex random vector is zero 3 p 584 Proper complex random vectors EditA complex random vector Z displaystyle mathbf Z is called proper if the following three conditions are all satisfied 1 p 293 E Z 0 displaystyle operatorname E mathbf Z 0 zero mean var Z 1 lt var Z n lt displaystyle operatorname var Z 1 lt infty ldots operatorname var Z n lt infty all components have finite variance E Z Z T 0 displaystyle operatorname E mathbf Z mathbf Z T 0 Two complex random vectors Z W displaystyle mathbf Z mathbf W are called jointly proper is the composite random vector Z 1 Z 2 Z m W 1 W 2 W n T displaystyle Z 1 Z 2 ldots Z m W 1 W 2 ldots W n T is proper PropertiesA complex random vector Z displaystyle mathbf Z is proper if and only if for all deterministic vectors c C n displaystyle mathbf c in mathbb C n the complex random variable c T Z displaystyle mathbf c T mathbf Z is proper 1 p 293 Linear transformations of proper complex random vectors are proper i e if Z displaystyle mathbf Z is a proper random vectors with n displaystyle n components and A displaystyle A is a deterministic m n displaystyle m times n matrix then the complex random vector A Z displaystyle A mathbf Z is also proper 1 p 295 Every circularly symmetric complex random vector with finite variance of all its components is proper 1 p 295 There are proper complex random vectors that are not circularly symmetric 1 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 covariace matrix is zero i e if K Z W 0 displaystyle operatorname K mathbf Z mathbf W 0 Cauchy Schwarz inequality EditThe Cauchy Schwarz inequality for complex random vectors is E Z H W 2 E Z H Z E W H W displaystyle 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 Characteristic function EditThe characteristic function of a complex random vector Z displaystyle mathbf Z with n displaystyle n components is a function C n C displaystyle mathbb C n to mathbb C defined by 1 p 295 f Z w E e i ℜ w H Z E e i ℜ w 1 ℜ Z 1 ℑ w 1 ℑ Z 1 ℜ w n ℜ Z n ℑ w n ℑ Z n displaystyle 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 See also EditComplex normal distribution Complex random variable scalar case References Edit a b c d e f g h i j Lapidoth Amos 2009 A Foundation in Digital Communication Cambridge University Press ISBN 978 0 521 19395 5 Gubner John A 2006 Probability and Random Processes for Electrical and Computer Engineers Cambridge University Press ISBN 978 0 521 86470 1 a b c Tse David 2005 Fundamentals of Wireless Communication Cambridge University Press Retrieved from https en wikipedia org w index php title Complex random vector amp oldid 1050002176, wikipedia, wiki, book, books, library,