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ps3 - EE 805 Random Processes and Linear Systems OSU Winter...

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EE 805, Random Processes and Linear Systems Jan. 27, 2012 OSU, Winter 2012 Due: Feb. 6, 2012 Problem Set 3 Problem 1 Consider the random process X ( t ) = p sin 2 πf 0 t + B [ n ] π 2 · , nT 6 t < ( n + 1) T, where p and f 0 are known constants and B [ n ] is an iid Bernoulli random sequence taking on values ± 1, each with probability 1/2, and -∞ < n < . Usually, f 0 · T is an integer, and we will assume so here. (a) Sketch a sample path X ( t ). (b) Find m X ( t ) and C XX ( s, t ). (c) Is the process sss? wss? Explain. Problem 2 Let W ( t ) (for t > 0) be a standard Wiener process with variance σ 2 t . (a) Find two times t 1 and t 2 for which the two r.v.s W ( t i ) , W (2) have a correlation coefficient ρ = 0 . 8 for both i = 1 and 2. (b) For arbitrary t and ρ > 0, can two such times t 1 and t 2 be found for which the correlation coefficient between W ( t ) and W ( t i ) is ρ , for both i = 1 and 2? Explain. What if ρ < 0? Problem 3 Let A and B be two uncorrelated r.v.s, each with zero mean and variance σ 2 . Define (for fixed, deterministic ω ) X ( t ) = A cos( ωt ) + B sin( ωt ) Y ( t ) = - A sin( ωt ) + B cos( ωt
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