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hw9-Solutions

# hw9-Solutions - EE 351K Probability Statistics and Random...

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Unformatted text preview: EE 351K Probability, Statistics, and Random Processes SPRING 2003 Instructor: Sanjay Shakkottai Random Processes and Stationarity —————————_—__ Probleml Let {1’36 are integers Le. a' can be 0, positive, and negative) be a sequence of i.i.d. Bernoulli random variables with parameter 1), Le. 12={G w.p. 1—1;: 1 w.p. p Let s{t) be the standard pulse function. In other words, 0 t<0, U 3}]. Let us deﬁne the random process (for all times) in the following way. DO Xe) = Z nee — a) .w'=—oo a) Prove that X (t) is not wide sense stationary. b) Let R be a uniform random variable over the interval [0, 1]. We deﬁne the random process Z (t) (for all times) in the following manner. 20!): ‘Z lass—j—R) J=—oo Prove that Z (t) is a wide-sense stationary process. Solution One realization of the random process X (t) can be seen in Figure l. t1ﬁ1+T [big-i"? Figure l: A Realization of X0) For example, the value in the interval [2, 3] is determined by the random variable Y2 since 50‘. ~ 2) = 1 only if 2 g t g 3. Compare E[X(:1)X(r1 + 1—)] with E[X(t'2_)X(tg + r)]. E[X(t1)X(t1 + 7)] = Bum/1] = E[%]E[Y1] = p2 and E[X(t2)X(t2 + T)] = E[Y31’3] = E[Y32] = ,7). Notice that the autocorrelations of two cases are different even though we choose same time difference (7'). Therefore, X (t) is not wide-sense stationary. On the other hand, Z ( t) is wide-sense stationary. Notice that Z (t) is random-time shifted version of X(t). When ‘7' is greater than 1, E[X(t)X(t + 7-)] = E[}’;}’}] = p2(i 96 j) meaning the autocorrelation is not dependent upon the choice of t. Lets think about the case when 1" is smaller than 1. E[X(t)X(t + 7)] = E[Y_11ﬁ]P{R < T) + E[Y_1Y_1]P(R > 7‘) '2 E[Y_1]E[}’ﬁ]r + E[Y31](1 -— 7') = 1027' +p(1 — 7') Therefore, the result is a function of only T , not the choice of 2?. Problem 2- Let us deﬁne a sequence oi'i.i.d. random variables {X1, X2, . . .}, with X,- ~ Uniform[—1,1], 2': 1,2,3,. .. Deﬁne, 1’;- 2 X3, j=1,2,3,... (1) H 3' 1 . 2,- 72m, 321,2,3,... (2) 3 i=1 For each of the four parts, justify your answer. (a). Is-the random process {X,-,i = 1, 2, 3, . . .} wide-sense stationary? (b). Is the random process {1’},j = 1, 2, 3, . . .} wide-sense stationary? (c). Is the random process {2333‘ = l, 2, 3, . . } wide~sense stationary? Solution a) Yes. It is clear that E[X,] = (LW and E[X,X,-] = E[X,~]E[Xj] = 0 ift' 75 3'. Otherwise, ifi = j, E[X,-Xj] = E[Xf] = 1/2. Therefore, the autocorrelation E[X,-X_,-] does not depend on i. Hence, the random process {X,-,i = l, 2, 3, . . .} is wide~sense stationary. b) Yes. EH3] = E[X3] = U,Vj. Also, EDGE-j = EfXng] = E[X§] = 0.5. That is, the autocorrelation E[}’;}f,] 1s a constant and does not depend on '3'. Hence, the random process {YE-J = 1, 2, 3, . . .} is wide— c) No. This is because, 1 r 1 -J"+k EiZjZki = Ekigxdmﬁgmj H U} Tﬁ' C“ + H E: M L; 35;. |_.____, + ti} to. C? + H 3.: Me; ME 3*: 35 I____.| i: 1=1i=1,£-}‘£t 1 = , EX? +0 1 l (i Hence, the autocorrclation, E[Zij+k] depends on 3‘. Thus, the random process {Zj,j = 1, 2,3, . . .} is wide-sense stationary. ...
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hw9-Solutions - EE 351K Probability Statistics and Random...

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