notes6 - ECE 562 Fall 2011 September 14, 2011 Proper...

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Unformatted text preview: ECE 562 Fall 2011 September 14, 2011 Proper Complex Random Variables and Vectors Proper Complex Random Vectors Let Y = Y I + jY Q be a complex random vector. Define the real covariance matrices I = cov [ Y I , Y I ] = E [( Y I- m Y I )( Y I- m Y I ) > ] Q = cov [ Y Q , Y Q ] = E [( Y Q- m Y Q )( Y Q- m Y Q ) > ] IQ = cov [ Y I , Y Q ] = E [( Y I- m Y I )( Y Q- m Y Q ) > ] QI = cov [ Y Q , Y I ] = E [( Y Q- m Y Q )( Y I- m Y I ) > ] (1) Also define the complex covariance matrices = E h ( Y- m Y )( Y- m Y ) i = E h ( Y- m Y )( Y- m Y ) > i (2) where and are, respectively, the covariance and pseudocovariance matrices of Y . Note that = ( I + Q ) + j ( QI- IQ ) = ( I- Q ) + j ( QI + IQ ) (3) Definition 1. Y is said to be a proper complex random vector if pseudocovariance matrix = 0, i.e. if I = Q and QI =- IQ . (4) Note that for proper complex Y , it follows from (3) that = 2 I + j 2 QI . (5) The scalar case In the special case where Y is a scalar, denoted by Y , it is clear that QI = E [( Y I- m I )( Y Q...
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notes6 - ECE 562 Fall 2011 September 14, 2011 Proper...

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