solution_set5 - EE 511 Solutions to Problem Set 5 1. (a)...

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Unformatted text preview: EE 511 Solutions to Problem Set 5 1. (a) The sample functions are shown in Figure 1.-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1 1 2 t X t (0)-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-1 1 t X t (1)-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-1 1 t X t (2)-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1-1 1 t X t (3) Figure 1: (b) X = 0. X . 5 (0) = 1, X . 5 (1) = − 1, X . 5 (2) = 1, and X . 5 (3) = − 1. X . 25 (0) = 1, X . 25 (1) = 0, X . 25 (2) = − 1, and X . 25 (3) = 0. The marginal CDF’s of X , X . 25 , and X . 5 are shown in Figure 2. (c) Given that X . 5 = − 1, X . 25 = 0 with probability 1. (d) Given that X . 5 = 1, X . 25 = 1 with probability 0 . 5 and X . 25 = − 1 with proba- bility 0 . 5. 2. a) E [ X t ] = 0. E [ X t + τ X t ] = E [ A 2 cos (2 πf c t + Θ) cos (2 πf c ( t + τ ) + Θ)] = A 2 2 [cos 2 πf c τ + E [cos (2 πf c (2 t + τ ) + 2Θ)]] = A 2 2 cos 2 πf c τ b) We can choose any pdf for Θ as long as E [cos (2 πf c (2 t + τ ) + 2Θ)] = 0 and E [cos (2 πf c t + Θ)] = constant for any t , τ . Θ can be defined as follows: Θ =          with prob. 1 4 π 2 with prob. 1 4 π with prob. 1 4 3 π 2 with prob. 1 4 1-2-1.5-1-0.5 0.5 1 1.5 2 2.5 3 0.5 1 x CDf of X F X (x)-2-1.5-1-0.5 0.5 1 1.5 2 2.5 3 0.5 1 x CDF of X 0.25 F X 0.25 (x)-2-1.5-1-0.5 0.5 1 1.5 2 2.5 3 0.5 1 x CDF of X 0.25 F X 0.5 (x) Figure 2: Another possible choice for Θ is: Θ =                              with prob. 1 12 π 4 with prob. 1 6 π 2 with prob. 1 12 3 π 4 with prob. 1 6 π with prob. 1 12 5 π 4 with prob. 1 6 3 π 2 with prob. 1 12 7 π 4 with prob. 1 6 A more general choice for f Θ ( θ ) can be made as follows: (i) Let us assume that the range of Θ is from 0 to 2 π . (ii) The condition for mean to be constant can be obtained as follows: E [cos (2 πf c t + Θ)] = integraltext 2 π cos (2 πf c t + θ ) f Θ ( θ ) dθ = integraltext π cos (2 πf c t + θ ) f Θ ( θ ) dθ + integraltext 2 π π cos (2 πf c t + θ ) f Θ ( θ ) dθ (using θ ′ = θ − π ) = integraltext π cos (2 πf c t + θ ) f Θ ( θ ) dθ + integraltext π cos (2 πf c t + θ ′ + π ) f Θ ( θ ′ + π ) dθ ′ = integraltext π cos (2 πf c t...
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This note was uploaded on 10/26/2011 for the course ECE 171 taught by Professor Tu during the Spring '11 term at Aarhus Universitet.

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solution_set5 - EE 511 Solutions to Problem Set 5 1. (a)...

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