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Definition

For two random vectors X = ( X 1 , … , X m ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}} and Y = ( Y 1 , … , Y n ) T {\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}} , each containing random elements whose expected value and variance exist, the cross-correlation matrix of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } is defined by¹: p.337 

R X Y ≜   E ⁡ [ X Y T ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]}

and has dimensions m × n {\displaystyle m\times n} . Written component-wise:

R X Y = [ E ⁡ [ X 1 Y 1 ] E ⁡ [ X 1 Y 2 ] ⋯ E ⁡ [ X 1 Y n ] E ⁡ [ X 2 Y 1 ] E ⁡ [ X 2 Y 2 ] ⋯ E ⁡ [ X 2 Y n ] ⋮ ⋮ ⋱ ⋮ E ⁡ [ X m Y 1 ] E ⁡ [ X m Y 2 ] ⋯ E ⁡ [ X m Y n ] ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatorname {E} [X_{1}Y_{1}]&\operatorname {E} [X_{1}Y_{2}]&\cdots &\operatorname {E} [X_{1}Y_{n}]\\\\\operatorname {E} [X_{2}Y_{1}]&\operatorname {E} [X_{2}Y_{2}]&\cdots &\operatorname {E} [X_{2}Y_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{m}Y_{1}]&\operatorname {E} [X_{m}Y_{2}]&\cdots &\operatorname {E} [X_{m}Y_{n}]\\\\\end{bmatrix}}}

The random vectors X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } need not have the same dimension, and either might be a scalar value.

Example

For example, if X = ( X 1 , X 2 , X 3 ) T {\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}} and Y = ( Y 1 , Y 2 ) T {\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}} are random vectors, then R X Y {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }} is a 3 × 2 {\displaystyle 3\times 2} matrix whose ( i , j ) {\displaystyle (i,j)} -th entry is E ⁡ [ X i Y j ] {\displaystyle \operatorname {E} [X_{i}Y_{j}]} .

Complex random vectors

If Z = ( Z 1 , … , Z m ) T {\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T}}} and W = ( W 1 , … , W n ) T {\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})^{\rm {T}}} are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of Z {\displaystyle \mathbf {Z} } and W {\displaystyle \mathbf {W} } is defined by

R Z W ≜   E ⁡ [ Z W H ] {\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]}

where H {\displaystyle {}^{\rm {H}}} denotes Hermitian transposition.

Uncorrelatedness

Two random vectors X = ( X 1 , … , X m ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}} and Y = ( Y 1 , … , Y n ) T {\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}} are called uncorrelated if

E ⁡ [ X Y T ] = E ⁡ [ X ] E ⁡ [ Y ] T . {\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}.}

They are uncorrelated if and only if their cross-covariance matrix K X Y {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }} matrix is zero.

In the case of two complex random vectors Z {\displaystyle \mathbf {Z} } and W {\displaystyle \mathbf {W} } they are called uncorrelated if

E ⁡ [ Z W H ] = E ⁡ [ Z ] E ⁡ [ W ] H {\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}}

and

E ⁡ [ Z W T ] = E ⁡ [ Z ] E ⁡ [ W ] T . {\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {T}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {T}}.}

Properties

Relation to the cross-covariance matrix

The cross-correlation is related to the cross-covariance matrix as follows:

K X Y = E ⁡ [ ( X − E ⁡ [ X ] ) ( Y − E ⁡ [ Y ] ) T ] = R X Y − E ⁡ [ X ] E ⁡ [ Y ] T {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\rm {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {Y} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}} Respectively for complex random vectors: K Z W = E ⁡ [ ( Z − E ⁡ [ Z ] ) ( W − E ⁡ [ W ] ) H ] = R Z W − E ⁡ [ Z ] E ⁡ [ W ] H {\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {W} -\operatorname {E} [\mathbf {W} ])^{\rm {H}}]=\operatorname {R} _{\mathbf {Z} \mathbf {W} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}}

See also

• Autocorrelation
• Correlation does not imply causation
• Covariance function
• Pearson product-moment correlation coefficient
• Correlation function (astronomy)
• Correlation function (statistical mechanics)
• Correlation function (quantum field theory)
• Mutual information
• Rate distortion theory
• Radial distribution function

Further reading

• Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.
• Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.
• M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.

References

1. Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1. 978-0-521-86470-1 ↩