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Lecture Notes and Homeworks

Lecture Notes and Homeworks - EE 805 Random Processes and...

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EE 805, Random Processes and Linear Systems Feb. 13, 2012 OSU, Winter 2012 Due: Feb. 24, 2011 Problem Set 4 Problem 1 Let the input X ( t ) to a LTI system be white noise with C X ( τ ) = σ 2 δ ( τ ). The LTI system has impulse response h ( t ) = u ( t ) - u ( t - T ) = ( 1 0 t 6 T 0 otherwise . Find C Y ( τ ) for the system’s output Y ( t ). Problem 2 Consider a discrete-time system, whose real-valued input X [ n ] is white noise with C X [ n ] = σ 2 δ [ n ]. The LTI system has impulse response h [ n ] = a n u [ n ], where a is real and | a | < 1. Find C Y X [ k ], C XY [ k ], and C Y [ k ]. Plot these in Matlab using the stem command. Problem 3 The input X ( t ) to a LTI system with transfer function H ( s ) = s + 2 has mean m X = 4 and autocovariance C X ( τ ) = 2 e -| τ | . Find m Y , R Y X ( τ ), and R Y ( τ ). Also, find S Y ( ω ). Problem 4 Consider a random telegraph signal S ( t ) with amplitude A and Poisson switching times with parameter λ . A measured signal X ( t ) = S ( t ) + N ( t ), where N ( t ) is uncorrelated with X ( t ). Assume N ( t ) is zero mean white noise with C N ( τ ) = σ 2 N δ ( τ ).
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