Homework_10

# Homework_10 - 11 Papoulis 9-37(10-37 in 3rd edition 12...

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ECE600 Random Variables and Waveforms Mark R. Bell Spring 20 1 1 MSEE 336 Homework Assignment #10 Should be completed by Final Exam Reading Assignment: Read Sections 7-4, 9-1 through 9-4, 10-1 and 10-2 of Papoulis (8-4, 10-1 through 10-4 and 11-1 and 11-2 in 3rd edition). 1. Papoulis 7-20 (8-20 in 3rd edition). 2. Papoulis 7-25 (8-25 in 3rd edition). 3. Papoulis 7-27 (8-27 in 3rd edition). 4. Consider a random experiment {S , F ,P } with sample space S = { 1 , 2 , 3 ,... } and associated pmf p ( n )= P ( { ω = n } )= α/n 2 . Consider the sequence { X n } of random variables deﬁned by X n ( ω )= ± n, if ω = n , 0 , if ω ± = n . Prove that { X n } converges almost everywhere to X = 0, but does not converge in the mean square sense; that is E {| X n - 0 | 2 } does not converge to zero as n →∞ . 5. Papoulis 7-13 (8-13 in 3rd edition). 6. Papoulis 9-1 (10-1 in 3rd edition). 7. Papoulis 9-2 (10-2 in 3rd edition). 8. Papoulis 9-3 (10-3 in 3rd edition). 9. Papoulis 9-8 (10-8 in 3rd edition). 10. Papoulis 9-10 (10-10 in 3rd edition).

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Unformatted text preview: 11. Papoulis 9-37 (10-37 in 3rd edition). 12. Papoulis 9-39 (10-39 in 3rd edition). 13. Papoulis 9-42 (10-42 in 3rd edition). 14. Let X ( t ) and Y ( t ) be two real, independent, wide-sense stationary random processes deﬁned on a random experiment. Let Z ( t ) be a new random process deﬁned as Z ( t ) = X ( t ) Y ( t ) . Under what conditions is Z ( t ) WSS? – 1 – 15. Let X ( t ) is given by X ( t ) = cos ( ω o t +Θ) , where Θ is a random variable uniformly distributed on the interval [0 , 2 π ) and Y ( t ) is a wide-sense stationary random process with autocorrelation function R Y ( τ ) = e-α | τ | , where α is a positive constant. Assume X ( t ) and Y ( t ) are statistically independent. Let Z ( t ) = X ( t ) Y ( t ) . Find the power spectral density of Z ( t ). – 2 –...
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## This note was uploaded on 10/12/2011 for the course ECE 600 taught by Professor Staff during the Spring '08 term at Purdue.

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Homework_10 - 11 Papoulis 9-37(10-37 in 3rd edition 12...

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