EE562a_HW_2

EE562a_HW_2 - X is singular d Construct an example in which...

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EE 562a Homework 2 Due Monday June 12, 2006 Work 3 problems . Problem 1. Let X =( X 1 ,X 2 ,X 3 ,X 4 ) be a Gaussian random vector where E [ X i ]=0for i =1 , 2 , 3 , 4. Show that E [ X 1 X 2 X 3 X 4 ]= K 12 K 34 + K 13 K 24 + K 14 K 23 where K ij is the i, j element of the covariance matrix K X . Problem 2. Let X ( u )bean n -dimensional random vector with covariance matrix K X and correlation matrix R X .L e t( λ i , e i ) ,i =1 , 2 ,...,n denote the eigenvalue and eigenvector pairs of the covariance matrix K X ,w iththe eigenvectors chosen to form an orthonormal set. a. If K X is non-singular, can R X be singular? Why or why not? b. Show K X can be written in the form K X = n X i =1 λ i e i e i . c. Construct an example in which R
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Unformatted text preview: X is singular. d. Construct an example in which K X and R X are both singular but μ X 6 = 0. e. Verify that R − 1 X = K − 1 X − K − 1 X μ X μ † X K − 1 X 1 + μ † X K − 1 X μ X . Problem 3. Let X ( u ) be a random vector with correlation matrix R X . Let e 1 and e 2 be eigenvectors corresponding to distinct eigenvalues λ 1 and λ 2 , respectively. Assume that || e 1 || = || e 2 || = 1 . Let Y i ( u ) = e † i X ( u ) , i = 1 , 2 . 1 a. Compute E [ | Y 1 ( u ) | 2 ]. b. Compute E [ Y 1 ( u ) Y 2 ( u ) ∗ ]. 2...
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EE562a_HW_2 - X is singular d Construct an example in which...

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