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StochF02_MidEx2

StochF02_MidEx2 - ECE 330:541 Stochastic Signals and...

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ECE 330:541, Stochastic Signals and Systems Midterm Exam 2 Fall 2002 Instructions: There are three problems on this exam, each with multiple parts. On the booklet(s) provided, please answer the problems and clearly state on each page of the booklet which problem you are solving. Note: The exam will be graded out of 100 points, and will be scaled later to contribute a total of 25 points towards your final grade. 1. Random Phase Signals: Consider the continuous time process X ( t ) = A cos( ω 0 t + φ ), where ω 0 is a known frequency, φ is uniformly distributed in (0 , 2 π ) and independently of the random variable A . (a) ( 10 points ) Is X ( t ) is a wide sense stationary (WSS) process? Hint: You may want to use the fact that cos( A ) cos( B ) = (cos( A - B ) + cos( A + B )) / 2. (b) ( 10 points ) If we take φ to be a known constant, is X ( t ) WSS? 2. Stationary Autoregressive-1 Process: Let e ( t ), for t = 0 , ± 1 , ± 2 , · · · be a real-valued white noise process (i.e. a sequence of uncorrelated real-valued random variables with zero mean and variance σ 2 e ). We may define a process X

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