Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Complex random vector
open-in-new
<h2 id="definition">Definition</h2> <p>A complex random vector Z = ( Z 1 , … , Z n ) T {\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{T}} on the <a href="/facts/Probability_space/dEcv2VrU">probability space</a> ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> Z : Ω → 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 <a href="/facts/Multivariate_random_variable/qMfooyVf">real random vector</a> on ( Ω , 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} .<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>: p. 292 </p> <h2 id="cumulative-distribution-function">Cumulative distribution function</h2> <p>The 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 </p> <table><tbody><tr><td> 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})})} </td> <td></td> <td>Eq.1</td></tr></tbody></table> <p>where z = ( z 1 , . . . , z n ) T {\displaystyle \mathbf {z} =(z_{1},...,z_{n})^{T}} . </p> <h2 id="expectation">Expectation</h2> <p>As in the real case the expectation (also called <a href="/facts/Expected_value/1XV0JKL8">expected value</a>) of a complex random vector is taken component-wise.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>: p. 293 </p> <table><tbody><tr><td> E [ Z ] = ( E [ Z 1 ] , … , E [ Z n ] ) T {\displaystyle \operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}} </td> <td></td> <td>Eq.2</td></tr></tbody></table> <h2 id="covariance-matrix-and-pseudo-covariance-matrix">Covariance matrix and pseudo-covariance matrix</h2> <p class="note">See also: <a href="/facts/Covariance_matrix/kLhgjHC6">Covariance matrix § Complex random vector</a></p> <p>The <i><a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a></i> (also called <i>second central moment</i>) 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} <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> whose ( i , j ) {\displaystyle (i,j)} th element is the <a href="/facts/Covariance/hILocGNH">covariance</a> between the <i>i</i> th and the <i>j</i> th random variables.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p.372 Unlike in the case of real random variables, the covariance between two random variables involves the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of one of the two. Thus the covariance matrix is a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: p. 293 </p> <table><tbody><tr><td><p> 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}&\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}}} </p></td> <td></td> <td>Eq.3</td></tr></tbody></table> 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}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\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}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\end{bmatrix}}} <p>The <i><a href="/facts/Pseudo-covariance_matrix/kLhgjHC6">pseudo-covariance matrix</a></i> (also called <i>relation matrix</i>) is defined replacing Hermitian transposition by transposition in the definition above. </p> <table><tbody><tr><td><p> 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}]} </p></td> <td></td> <td>Eq.4</td></tr></tbody></table> 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}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\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}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{n}-\operatorname {E} [Z_{n}])]\end{bmatrix}}} Properties <p>The covariance matrix is a <a href="/facts/Hermitian_matrix/qArPblYo">hermitian matrix</a>, i.e.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>: p. 293 </p> K Z Z H = K Z Z {\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }^{H}=\operatorname {K} _{\mathbf {Z} \mathbf {Z} }} . <p>The pseudo-covariance matrix is a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a>, i.e. </p> J Z Z T = J Z Z {\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }^{T}=\operatorname {J} _{\mathbf {Z} \mathbf {Z} }} . <p>The covariance matrix is a <a href="/facts/Positive_semidefinite_matrix/Hbr6AuPS">positive semidefinite matrix</a>, i.e. </p> 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}} . <h3>Covariance matrices of real and imaginary parts</h3> <p class="note">See also: <a href="/facts/Complex_random_variable/h2v6W3rf">Complex random variable § Covariance matrix of real and imaginary parts</a></p> <p>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 <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> of the form: </p> [ K X X K X Y K Y X K Y Y ] {\displaystyle {\begin{bmatrix}\operatorname {K} _{\mathbf {X} \mathbf {X} }&\operatorname {K} _{\mathbf {X} \mathbf {Y} }\\\operatorname {K} _{\mathbf {Y} \mathbf {X} }&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }\end{bmatrix}}} <p>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: </p> 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}&\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} })\\&\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} })\\&\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} })\\&\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}}} <p>Conversely: </p> 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}&\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} })\\&\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}}} <h2 id="cross-covariance-matrix-and-pseudo-cross-covariance-matrix">Cross-covariance matrix and pseudo-cross-covariance matrix</h2> <p>The cross-covariance matrix between two complex random vectors Z , W {\displaystyle \mathbf {Z} ,\mathbf {W} } is defined as: </p> <table><tbody><tr><td> 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}]} </td> <td></td> <td>Eq.5</td></tr></tbody></table> 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}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\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}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\end{bmatrix}}} <p>And the pseudo-cross-covariance matrix is defined as: </p> <table><tbody><tr><td> 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}]} </td> <td></td> <td>Eq.6</td></tr></tbody></table> 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}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\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}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{n}-\operatorname {E} [W_{n}])]\end{bmatrix}}} <p>Two complex random vectors Z {\displaystyle \mathbf {Z} } and W {\displaystyle \mathbf {W} } are called uncorrelated if </p> K Z W = J Z W = 0 {\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0} . <h2 id="independence">Independence</h2> <p class="note">Main article: <a href="/facts/Independence_(probability_theory)/NUzQtnUL">Independence (probability theory)</a></p> <p>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 </p> <table><tbody><tr><td> 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} } </td> <td></td> <td>Eq.7</td></tr></tbody></table> <p>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 </p> 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}} . <h2 id="circular-symmetry">Circular symmetry</h2> <p>A complex random vector Z {\displaystyle \mathbf {Z} } is called circularly symmetric if for every deterministic φ ∈ [ − π , π ) {\displaystyle \varphi \in [-\pi ,\pi )} the distribution of e i φ Z {\displaystyle e^{\mathrm {i} \varphi }\mathbf {Z} } equals the distribution of Z {\displaystyle \mathbf {Z} } .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: pp. 500–501 </p> Properties <ul><li>The expectation of a circularly symmetric complex random vector is either zero or it is not defined.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: p. 500 </li> <li>The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: p. 584 </li></ul> <h2 id="proper-complex-random-vectors">Proper complex random vectors</h2> <p>A complex random vector Z {\displaystyle \mathbf {Z} } is called proper if the following three conditions are all satisfied:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: p. 293 </p> <ul><li> E [ Z ] = 0 {\displaystyle \operatorname {E} [\mathbf {Z} ]=0} (zero mean)</li> <li> var [ Z 1 ] < ∞ , … , var [ Z n ] < ∞ {\displaystyle \operatorname {var} [Z_{1}]<\infty ,\ldots ,\operatorname {var} [Z_{n}]<\infty } (all components have finite variance)</li> <li> E [ Z Z T ] = 0 {\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]=0} </li></ul> <p>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. </p> Properties <ul><li>A 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.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: p. 293 </li> <li>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.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: p. 295 </li> <li>Every circularly symmetric complex random vector with finite variance of all its components is proper.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>: p. 295 </li> <li>There are proper complex random vectors that are not circularly symmetric.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>: p. 504 </li> <li>A real random vector is proper if and only if it is constant.</li> <li>Two jointly proper complex random vectors are uncorrelated if and only if their covariance matrix is zero, i.e. if K Z W = 0 {\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=0} .</li></ul> <h2 id="cauchy-schwarz-inequality">Cauchy-Schwarz inequality</h2> <p>The <a href="/facts/Cauchy-Schwarz_inequality/poMzQUIY">Cauchy-Schwarz inequality</a> for complex random vectors is </p> | 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} |]} . <h2 id="characteristic-function">Characteristic function</h2> <p>The <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> 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:<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>: p. 295 </p> φ Z ( ω ) = E [ e i ℜ ( ω H Z ) ] = E [ e i ( ℜ ( ω 1 ) ℜ ( Z 1 ) + ℑ ( ω 1 ) ℑ ( Z 1 ) + ⋯ + ℜ ( ω n ) ℜ ( Z n ) + ℑ ( ω 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]} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Complex_normal_distribution/KVUmoU1L">Complex normal distribution</a></li> <li><a href="/facts/Complex_random_variable/h2v6W3rf">Complex random variable</a> (scalar case)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1. <a href="978-0-521-86470-1" target="_blank">978-0-521-86470-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5. <a href="978-0-521-19395-5" target="_blank">978-0-521-19395-5</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>