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Unformatted text preview: EE 562a Midterm Exam B SOLUTIONS, 28 Februrary 2007 WRITE YOUR NAME on the exam paper, and indicate if you are an ON-CAMPUS or DEN student. The exam is open book/open notes. You may use a simple calculator (i.e., not a scientific calculator). Do all problems. Note that these problems have different point values, and differ widely in difficulty. You should look over all the problems before starting your work. Write all answers on the test paper. You may use scratch paper to do the problem, but the answer must be written with the problem on the exam paper. If you do work on scratch paper, and wish that work to be considered for partial credit, hand in the scratch paper along with the exam. Part I. Basic Concepts. For each question in this section, circle the best answer from the choices listed. Each problem has only one best answer. There is no partial credit on this section. (3 points each.) 1. Suppose x ( u ) and y ( u ) are jointly Gaussian real random vectors. Which of the following gives enough information to find their joint probability density function? (a) R x and R y (b) R x , R y , m x , m y , and that x ( u ) and y ( u ) are uncorrelated (c) K x , K y , m x and m y (d) R x , K y , m y , and that x ( u ) is mean-zero 2. Which of the following is an affine transformation? (a) ( x, y ) xy + x (b) x | x | (c) x ( x 2 , x 3 ) T (d) x v for a constant n-vector v 3. Suppose that a circular random d-vector x ( u ) has a d d covariance matrix with rank r < d . Which of the following is possible by an affine transformation? (a) x ( u )- y ( u ) with K yy 6 = (b) x ( u )- y ( u ) with K yy * a d d matrix with rank k < r and K yy = (c) x ( u )- k-dimensional white noise, r < k < d (d) x ( u )- d-dimensional white noise 4. Which of the following could be allowed as a preprocessing step without loss of information (with no restriction on the type of allowed operations)? (a) x ( u )- Gx ( u ), for singular G (b) x ( u )- | x ( u ) | (c) x ( u )- v x ( u ) for some vector v (d) x ( u )- ( x ( u )) 3 EE 562a Midterm Exam SOLUTIONS 2 5. Suppose that a random vector x ( u ) has a correlation matrix R x = 2 1 1 2 . Which of the following could be its mean vector m x ? (a) 2 2 (b) p 1 / 2- p 1 / 2 (c)- 1 1 (d) Any vector m x is possible 6. Under which of the following conditions are the random variables x ( u ) and y ( u ) both orthog- onal and uncorrelated?...
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This note was uploaded on 05/06/2008 for the course EE 562a taught by Professor Toddbrun during the Spring '07 term at USC.
- Spring '07