Complex random vector

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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 Z1,…,Zn are complex-valued random variables, then the n-tuple (Z1,…,Zn) 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.

Part of a series on statistics
Probability theory

Definition

A complex random vector 𝐙=(Z1,…,Zn)T on the probability space (Ω,ℱ,P) is a function 𝐙:Ω→ℂn such that the vector (ℜ(Z1),ℑ(Z1),…,ℜ(Zn),ℑ(Zn))T is a real random vector on (Ω,ℱ,P) where ℜ(z) denotes the real part of z and ℑ(z) denotes the imaginary part of z.[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 P(Z≤1+3i) make no sense. However expressions of the form P(ℜ(Z)≤1,ℑ(Z)≤3) make sense. Therefore, the cumulative distribution function F𝐙:ℂn↦[0,1] of a random vector 𝐙=(Z1,...,Zn)T is defined as

F𝐙(𝐳)=P⁡(ℜ(Z1)≤ℜ(z1),ℑ(Z1)≤ℑ(z1),…,ℜ(Zn)≤ℜ(zn),ℑ(Zn)≤ℑ(zn))

 

 

 

 

(Eq.1)

where 𝐳=(z1,...,zn)T.

Expectation

As 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⁡[Z1],…,E⁡[Zn])T

 

 

 

 

(Eq.2)

Covariance matrix and pseudo-covariance matrix

The covariance matrix (also called second central moment) K𝐙𝐙 contains the covariances between all pairs of components. The covariance matrix of an n×1 random vector is an n×n matrix whose (i,j)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 

K𝐙𝐙=cov⁡[𝐙,𝐙]=E⁡[(𝐙−E⁡[𝐙])(𝐙−E⁡[𝐙])H]=E⁡[𝐙𝐙H]−E⁡[𝐙]E⁡[𝐙H]

 

 

 

 

(Eq.3)

K𝐙𝐙=[E[(Z1−E⁡[Z1])(Z1−E⁡[Z1])‾]E[(Z1−E⁡[Z1])(Z2−E⁡[Z2])‾]⋯E[(Z1−E⁡[Z1])(Zn−E⁡[Zn])‾]E[(Z2−E⁡[Z2])(Z1−E⁡[Z1])‾]E[(Z2−E⁡[Z2])(Z2−E⁡[Z2])‾]⋯E[(Z2−E⁡[Z2])(Zn−E⁡[Zn])‾]⋮⋮⋱⋮E[(Zn−E⁡[Zn])(Z1−E⁡[Z1])‾]E[(Zn−E⁡[Zn])(Z2−E⁡[Z2])‾]⋯E[(Zn−E⁡[Zn])(Zn−E⁡[Zn])‾]]

The pseudo-covariance matrix (also called relation matrix) is defined replacing Hermitian transposition by transposition in the definition above.

J𝐙𝐙=cov⁡[𝐙,𝐙‾]=E⁡[(𝐙−E⁡[𝐙])(𝐙−E⁡[𝐙])T]=E⁡[𝐙𝐙T]−E⁡[𝐙]E⁡[𝐙T]

 

 

 

 

(Eq.4)

J𝐙𝐙=[E[(Z1−E⁡[Z1])(Z1−E⁡[Z1])]E[(Z1−E⁡[Z1])(Z2−E⁡[Z2])]⋯E[(Z1−E⁡[Z1])(Zn−E⁡[Zn])]E[(Z2−E⁡[Z2])(Z1−E⁡[Z1])]E[(Z2−E⁡[Z2])(Z2−E⁡[Z2])]⋯E[(Z2−E⁡[Z2])(Zn−E⁡[Zn])]⋮⋮⋱⋮E[(Zn−E⁡[Zn])(Z1−E⁡[Z1])]E[(Zn−E⁡[Zn])(Z2−E⁡[Z2])]⋯E[(Zn−E⁡[Zn])(Zn−E⁡[Zn])]]
Properties

The covariance matrix is a hermitian matrix, i.e.[1]: p. 293 

K𝐙𝐙H=K𝐙𝐙.

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

J𝐙𝐙T=J𝐙𝐙.

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

𝐚HK𝐙𝐙𝐚≥0for all 𝐚∈ℂn.

Covariance matrices of real and imaginary parts

By decomposing the random vector 𝐙 into its real part 𝐗=ℜ(𝐙) and imaginary part 𝐘=ℑ(𝐙) (i.e. 𝐙=𝐗+i𝐘), the pair (𝐗,𝐘) has a covariance matrix of the form:

[K𝐗𝐗K𝐘𝐗K𝐗𝐘K𝐘𝐘]

The matrices K𝐙𝐙 and J𝐙𝐙 can be related to the covariance matrices of 𝐗 and 𝐘 via the following expressions:

K𝐗𝐗=E⁡[(𝐗−E⁡[𝐗])(𝐗−E⁡[𝐗])T]=12Re⁡(K𝐙𝐙+J𝐙𝐙)K𝐘𝐘=E⁡[(𝐘−E⁡[𝐘])(𝐘−E⁡[𝐘])T]=12Re⁡(K𝐙𝐙−J𝐙𝐙)K𝐘𝐗=E⁡[(𝐘−E⁡[𝐘])(𝐗−E⁡[𝐗])T]=12Im⁡(J𝐙𝐙+K𝐙𝐙)K𝐗𝐘=E⁡[(𝐗−E⁡[𝐗])(𝐘−E⁡[𝐘])T]=12Im⁡(J𝐙𝐙−K𝐙𝐙)

Conversely:

K𝐙𝐙=K𝐗𝐗+K𝐘𝐘+i(K𝐘𝐗−K𝐗𝐘)J𝐙𝐙=K𝐗𝐗−K𝐘𝐘+i(K𝐘𝐗+K𝐗𝐘)

Cross-covariance matrix and pseudo-cross-covariance matrix

The cross-covariance matrix between two complex random vectors 𝐙,𝐖 is defined as:

K𝐙𝐖=cov⁡[𝐙,𝐖]=E⁡[(𝐙−E⁡[𝐙])(𝐖−E⁡[𝐖])H]=E⁡[𝐙𝐖H]−E⁡[𝐙]E⁡[𝐖H]

 

 

 

 

(Eq.5)

K𝐙𝐖=[E[(Z1−E⁡[Z1])(W1−E⁡[W1])‾]E[(Z1−E⁡[Z1])(W2−E⁡[W2])‾]⋯E[(Z1−E⁡[Z1])(Wn−E⁡[Wn])‾]E[(Z2−E⁡[Z2])(W1−E⁡[W1])‾]E[(Z2−E⁡[Z2])(W2−E⁡[W2])‾]⋯E[(Z2−E⁡[Z2])(Wn−E⁡[Wn])‾]⋮⋮⋱⋮E[(Zn−E⁡[Zn])(W1−E⁡[W1])‾]E[(Zn−E⁡[Zn])(W2−E⁡[W2])‾]⋯E[(Zn−E⁡[Zn])(Wn−E⁡[Wn])‾]]

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

J𝐙𝐖=cov⁡[𝐙,𝐖‾]=E⁡[(𝐙−E⁡[𝐙])(𝐖−E⁡[𝐖])T]=E⁡[𝐙𝐖T]−E⁡[𝐙]E⁡[𝐖T]

 

 

 

 

(Eq.6)

J𝐙𝐖=[E[(Z1−E⁡[Z1])(W1−E⁡[W1])]E[(Z1−E⁡[Z1])(W2−E⁡[W2])]⋯E[(Z1−E⁡[Z1])(Wn−E⁡[Wn])]E[(Z2−E⁡[Z2])(W1−E⁡[W1])]E[(Z2−E⁡[Z2])(W2−E⁡[W2])]⋯E[(Z2−E⁡[Z2])(Wn−E⁡[Wn])]⋮⋮⋱⋮E[(Zn−E⁡[Zn])(W1−E⁡[W1])]E[(Zn−E⁡[Zn])(W2−E⁡[W2])]⋯E[(Zn−E⁡[Zn])(Wn−E⁡[Wn])]]

Two complex random vectors 𝐙 and 𝐖 are called uncorrelated if

K𝐙𝐖=J𝐙𝐖=0.

Independence

Main article: Independence (probability theory)

Two complex random vectors 𝐙=(Z1,...,Zm)T and 𝐖=(W1,...,Wn)T are called independent if

F𝐙,𝐖(𝐳,𝐰)=F𝐙(𝐳)⋅F𝐖(𝐰)for all 𝐳,𝐰

 

 

 

 

(Eq.7)

where F𝐙(𝐳) and F𝐖(𝐰) denote the cumulative distribution functions of 𝐙 and 𝐖 as defined in Eq.1 and F𝐙,𝐖(𝐳,𝐰) denotes their joint cumulative distribution function. Independence of 𝐙 and 𝐖 is often denoted by 𝐙⊥⊥𝐖. Written component-wise, 𝐙 and 𝐖 are called independent if

FZ1,…,Zm,W1,…,Wn(z1,…,zm,w1,…,wn)=FZ1,…,Zm(z1,…,zm)⋅FW1,…,Wn(w1,…,wn)for all z1,…,zm,w1,…,wn.

Circular symmetry

A complex random vector 𝐙 is called circularly symmetric if for every deterministic φ∈[−π,π) the distribution of eiφ𝐙 equals the distribution of 𝐙.[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 𝐙 is called proper if the following three conditions are all satisfied:[1]: p. 293 

  • E⁡[𝐙]=0 (zero mean)
  • var⁡[Z1]<∞,…,var⁡[Zn]<∞ (all components have finite variance)
  • E⁡[𝐙𝐙T]=0

Two complex random vectors 𝐙,𝐖 are called jointly proper is the composite random vector (Z1,Z2,…,Zm,W1,W2,…,Wn)T is proper.

Properties
  • A complex random vector 𝐙 is proper if, and only if, for all (deterministic) vectors 𝐜∈ℂn the complex random variable 𝐜T𝐙 is proper.[1]: p. 293 
  • 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×n matrix, then the complex random vector A𝐙 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 covariance matrix is zero, i.e. if K𝐙𝐖=0.

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is

|E⁡[𝐙H𝐖]|2≤E⁡[𝐙H𝐙]E⁡[|𝐖H𝐖|].

Characteristic function

The characteristic function of a complex random vector 𝐙 with n components is a function ℂn→ℂ defined by:[1]: p. 295 

φ𝐙(ω)=E⁡[eiℜ(ωH𝐙)]=E⁡[ei(ℜ(ω1)ℜ(Z1)+ℑ(ω1)ℑ(Z1)+⋯+ℜ(ωn)ℜ(Zn)+ℑ(ωn)ℑ(Zn))]

See also

References

  1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. 
  2. ↑ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1. 
  3. ↑ 3.0 3.1 3.2 Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. 



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