midterm_review_Sp07

# midterm_review_Sp07 - EE562a Review for Midterm Exam TAs K...

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EE562a: Review for Midterm Exam TAs: K. Raj Kumar and Marcus Urie February 26, 2007 1 Second moment descriptions We have omitted an explicit reference to the index set and sample space variable in the following. They hold for arbitrary random processes and random vectors in particular. Let x and y be real random processes and z = x + ı y . K x = R x - m x m x m z = m x + ım y R z = R x + R y + ı [ R yx - R xy ] ˜ R z = R x - R y + ı [ R yx + R xy ] Can solve for R x , R y , R xy in terms of R z and ˜ R z . Also note that all covariance and correlation matrices (eg. K z and R z ) must be: Hermitian Symmetric (HS) : K z = K z and R z = R z Non-Negative Definite (NND) : a K z a 0 and a R z a 0 for any constant complex vector a . Review what these two properties tell you about the eigenvectors and eigenvalues of K x (and R x ) and know how to check to see if an arbitrary function has these properties (and could thus be a valid covariance matrix).

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