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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 Σ I = cov [ Y I , Y I ] = E [( Y I m Y I )( Y I m Y I ) > ] Σ Q = cov [ Y Q , Y Q ] = E [( Y Q m Y Q )( Y Q m Y Q ) > ] Σ IQ = cov [ Y I , Y Q ] = E [( Y I m Y I )( Y Q m Y Q ) > ] Σ QI = cov [ Y Q , Y I ] = E [( Y Q m Y Q )( Y I m Y I ) > ] (1) Also define the complex covariance matrices Σ = E h ( Y m Y )( Y m Y ) † i ˇ Σ = E h ( Y m Y )( Y m Y ) > i (2) where Σ and ˇ Σ are, respectively, the covariance and pseudocovariance matrices of Y . Note that Σ = (Σ I + Σ Q ) + j (Σ QI Σ IQ ) ˇ Σ = (Σ I Σ Q ) + j (Σ QI + Σ IQ ) (3) Definition 1. Y is said to be a proper complex random vector if pseudocovariance matrix ˇ Σ = 0, i.e. if Σ I = Σ Q and Σ QI = Σ IQ . (4) Note that for proper complex Y , it follows from (3) that Σ = 2Σ I + j 2Σ QI . (5) The scalar case In the special case where Y is a scalar, denoted by Y , it is clear that Σ QI = E [( Y I m I )( Y Q...
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 Fall '09
 Variance, Probability theory, Covariance matrix, White Gaussian noise, complex random vector

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