problem set2 - ELEC 531 STATISTICAL SIGNAL PROCESSING...

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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 2 Due: September 13, 2007 Problem 2.1 Determine the mean and correlation function of each of the processes X t defined below. (a) X t is defined by the following equally likely sample functions. X t ( ω 1 ) = 1 X t ( ω 3 ) = sin πt X t ( ω 2 ) = - 2 X t ( ω 4 ) = cos πt (b) X t is defined by X t = cos( At + θ ), where A and θ are statistically independent random variables. θ is uniformly distributed over [0 , 2 π ) and A has the density function p A ( A ) = 1 π (1 + A 2 ) Problem 2.2 (a) Which of the following are valid correlation functions? Indicate your reasoning. (i) R X ( τ ) = e -| τ | - e - 2 | τ | (ii) R X ( τ ) = 5 sin 1000 τ τ (iii) R X ( τ ) = 1 - | τ | T | τ | ≤ T 0 otherwise (iv) R X ( τ ) = 1 | τ | ≤ T 0 otherwise (v) R X ( τ ) = δ ( τ ) + 25 (vi) R X ( τ ) = δ ( τ + 1) + δ ( τ ) + δ ( τ - 1) (b) Which of the following are valid power density spectra? Indicate your reasoning. (i) S X ( f ) = sin πf πf (ii) S X ( f ) = sin πf πf 2 (iii) S X ( f ) = exp - ( f - f 0 ) 2 4 (iv) S X ( f ) = e -| f | - e - 2 | f | (v) S X ( f ) = 1 + 0 . 25 e - j 2 πf (vi) S X ( f ) = 1 | f
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