Homework_9

Homework_9 - E[( Y ( A X + B )) 2 ] and E[(( Y Y ) a ( X X...

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ECE600 Random Variables and Waveforms Mark R. Bell Spring 20 1 1 MSEE 336 Homework Assignment #9 Should be compledted before Session 2 6 . Reading Assignment: Read Sections 7-1 through 7-3 of Papoulis. 1. Papoulis 6-78 (7-15 in 3rd edition). 2. Papoulis 6-72 (7-17 in 3rd edition). 3. A signal X consists of a zero-mean Gaussian random variable with variance σ 2 X . Noise N consisting of an independent zero-mean Gaussian random variable with variance σ 2 N is added to X to produce the observation Y = X + N . Suppose we observe that Y = y . (a) Find the minimum mean square error estimate ˆ x MMS ( y )o f X . (b) Find the MAP estimate ˆ x MAP ( y )o f X . 4. Show that if constants A , B , and
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Unformatted text preview: E[( Y ( A X + B )) 2 ] and E[(( Y Y ) a ( X X )) 2 ] are minimum, then a = A . 5. Let X and Y be two jointly distributed Gaussian random variables having distribution N ( x , y , x , y , r xy ) Let V = a X + b Y and W = c X + d Y . Show that V and W are jointly Gaussian and nd the parameters that characterize their joint density. 6. Papoulis 7-2 (8-2 in 3rd edition). 7. Papoulis 7-3 (8-3 in 3rd edition). 8. Papoulis 7-4 (8-4 in 3rd edition). ( Hint: Use characteristic Functions) 9. Papoulis 7-7 (8-7 in 3rd edition). 1...
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