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Unformatted text preview: Chapter 6 \ Random Processes 6.1 Definition and Specification of a Stochastic Process ‘ 6.1 We find the probabilities of the events {X1 = 2', X2 = j} in terms of the probabilities \ of the equivalent events of E: P[X1=1,X2=1] = P[%<{<1]=i 1 1 1 1 3 1 P[X1—1,X2—0] —— P[-2-<€<Z]_Z P[X1=0,X2=0] = p[o<g<i =i => PIXI = 2',X2 =1]: PIXI = z'IPIXz = j] 3112;,- 6 {0,1} => X1, X2 independent RV’s 6.5 a) Since g(t) is zero outside the interval [0,1]: P[X(t) = 0] = 1 for t ¢ [0, 1] For t E [0, 1], we have P[X(t) =11 = P[X(t) = -1] = g 73 74 ‘ Random Processes b) mm) 2 { 3.12%) = 11+(—1>P[X(t> = —11 = 0 35433: c) For t E [0, 1], t + d 6 [0,1], X(t) must be the same value, thus: me=itxu+w=iu==§ P[X(t) = :l:1,X(t +d) = 2|:1] = 0 For t 6 [0,1], t +d ¢ [0,1]: me=itxa+®=m=§ For t ¢ [0,1], t+d ¢ [0, 1]: me=mxa+®=m=1 C'X(t,t + d) = €[X(t)X(t + d)] — mX(t)mX(t + d) = 8[X(t)X(t+d)] {1 te [0,1] andt+d€ [0,1] d) 0 otherwise _—————————__ 6.7 a) We will use conditional probability: WW) 3 as] = P[g(t — T) s m] 1 ==AI%U—flSwW=MhOMA 1 = /o P[g(t — A) g x]d/\ since fT(A) = 1 t = / P[g(u) S w]du after letting u = t — /\ t—1 g(u) (and hence P [g(u) S m]) is a periodic function of u with period 1, so we can change the limits of the above integral to any full period. Thus mm s w] = [12W 3 deu Note that g(u) is deterministic, so 1 u:g(u)S:1: 0 u:g(u)>x mesfl={ 6.1. Definition and Specification of a Stochastic Process 9(U) So finally P[X(t)gx]=/ 1du=/11_$1du=m. was: b) mx(t) = E[X(t)] = [01 x dx = g. The correlation is again found using conditioning on T: E[X(t)X(t + 7-)] [01 EW ‘ TW + T - T)IT = Amman [,1 9(t - A)g(t + r — A)dA j; g<u>g<u + T)du g(u)g(u + 'r) is a periodic function in u so we can change the limits to (0,1): E[X(t)X(t + 7)] = [01 mm + r)du (u) = (1 —u) 75 76 A Random Processes here we assume 0 < 1' < 1 since E[X (t)X (t + 7-)] is periodic in 7'. E[X(t)X(t + r» = fol—Tu — u><1 — u — r)du + [flu — uxz — u — r)du _l 1+L+T__T_ ‘3 2 6 2 6 _1r+r_2 ‘3 2 2 Thus C(tt+ ) — 1—:+T2—l X’ T ‘3 2 2 4 _1_:+T_2 ‘12 2 2 6.10 a) P[H(t) = 1] = P[X(t) 2 0} = P[§ cos 27rt 2 0] = a; = P[H(t) = —1] 8[H(t)] = 1-P[H(t) =1]+(—1)P[H(t) = —1]=o CH(t,t + 7') = 8[H(t)H(t + 7-)] = 1 . P[H(t)H(t + T) = 1] + (—1)P[H(t)H(t + 'r) = -—1] H(t)&H(t+r) H(t)&H(t+7-) same sign opposite sign H(t)H(t + 7') = 1 4:) cos 27rt and cos 27r(t + 7;) have same sign H (t)H (t + 7') = —1 4:} cos 27rt and cos 27r(t + 7') have different sign . __ 1 for t, 7' such that cos 27rt cos 27r(t + 7') = 1 ' ' CH(t’t + T) — { —1 for t,T such that cos 27rt cos 27r(t + 7') = —1 b) P[H(t) = 1] = P[X(t) 2 0] = P[cos(wt + e) 2 0] = § = P[H(t) = —1] 6.1. Definition and Specification of a. Stochastic Process £[H(t)] = 1 (g) + (—1);- = 0 8[H(t)H(t + 7)] = 1 - P[X(t)X(t + r) > 0] +(—1)P[X(t)X(t + r) < 0] 1—P[X(t)X(t+1-)<0] 1 — 2P[X(t)X(t + 1-) < 0] ll P[X(t)X(t + T) < 0] P[cos(wt + 9) cos(w(t + T) + 9) < 0] [% cos wT + écos(2wt + (.07 + 29) < 0] P[cos(2wt + car + 26)) < cos wT] shaded region in figure 27r 1— cos 20 c) P[H(t) = 1] = P[X(t) 2 0] = 1 — Fx(t,(o-) = 1 — P[H(t) = —1] £[H(t)] 1 - P[H(t) = 1] + (—1)P[H(t) = —1] 1 — FX(t)(0') - FX(t)(0')‘ 1 - 2Fx(t)(0_) II ll II 77 78 . Random Processes . d) 5[H(t)X(t)] = 5[|X(t)|]) +X t X t 2 0 HUM“) = { —X8 X8 < 0. 6.14 The covariance matrix is given by: K: [ Cx(t, t) C'X(t, t+s) ]=[ 8328' e—M] Cx(t+s, t) Cx(t+s,t+3) 02 Thus [KP/2 — [0'2 02 — a 2e"’|c72e""']1/2= a4 — 046‘2M = 02V1 — e‘2M and 2 2 s K—l = 04(1 _16—2|a|) [ —age Isl —Oa: I I ] m m Thus the joint pdf is: —l.’b"K_11' f (331 $2) = L:— x = $1 mews) ’ 27T0'2\/1——e-T"l ‘ :62 ...
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