课程名称︰通讯随机过程 课程性质︰选修 课程教师︰钟嘉德 开课学院：电资学院 开课系所︰电信所 电机所 考试日期（年月日）︰ 考试时限（分钟）：3HR 是否需发放奖励金：是 试题 : 1. Determine whether each of the following statements is true or false. If the statement is true, prove it. If the statement is false, give a counterexapmle or explanation. Corrent choice without any proof, counterexample or explanation is not acceptable. (a) For any real random X(μ), E(X^2(μ)) ≧ E^2(X(μ)). (b) If X1(μ) is Gaussian random variable and if, given X1(μ), X2(μ) is conditionally Gaussian distributed, then X2(μ) is marginally Gaussian dustributed. (c) Let X1(μ),X2(μ),...,Xn(μ) be jointly Gaussian random variables with zero mean. If Xi(μ) and Xj(μ) are orthogonal for any i≠j, then X1(μ), X2(μ),...,Xn(μ) are mutually independent. (d) If X(μ,t) and Y(μ,t) are jointly Gaussian random processes, then aX(μ,t) + bY(μ,t) is a Gaussian process for arbitrary real constants a and b with ab≠0. (e) Consider the discrete-time random process S1(μ),S2(μ),S3(μ)...where Sn(μ) is defined by Sn(μ) = Σ Xi(μ) with X1(μ), X2(μ),X3(μ)... (i=1 to n) being independent and identical distributed Gaussian random variables which have zero mean and unit variance. It is true that S1(μ), S2(μ), S3(μ),...is a stationary random process. (f) Every white noise is a Marcov process (g) Let X(μ,t) be a Gaussian random process. If X(μ,t) has mean zero and autocorrelation function Rx(t1,t2) = sin^2(t1-t2)/(t1-t2)^2, then X(μ,t) is strict sense stationary. (h) Let X(μ,t) be a Gaussian random process with zero mean. If E[X(μ,t1)X(μ,t3)] + E[X(μ,t2)X(μ,t4)] = E[X(μ,t1)X(μ,t4)] + E[X(μ,t2)X(μ,t3)] for t1≦t2≦t3≦t4 then X(μ,t) is Marcov. 2. Consider the real-valued Gaussian random process X(μ,t) which have zero mean and autocorrelation Rx(t1,t2)= E(X(μ,t1)X(μ,t2))= 1+|t1|, t1≧t2 1+|t2|, t2≧t1 Answer the following question (a) Determine whether X(μ,t) is wide-sense stationary (b) Find the probability density function of Y(μ) = Σ(n=1 to N) X(μ,n) (c) Evaluate E[X(μ,-1) + X(μ,+1)]^4 3. Let X1(μ),X2(μ),...,Xn(μ) be independent and identical distributed (i.i.d) random variables with the common probability density function (p.d.f) fx(x)= 1/2δ(x+1) + 1/2δ(x-1), where δ(x) is Dirac function. Also let Y1(μ),Y2(μ),...,Yn(μ) be i.i.d Gaussian random variables with zero mean and unit variance. It is also given that Xi(μ) is independent of Yj(μ) for all i,j. Find the p.d.f and characteristic function of the random variable Σ(k=1 to n ) Xk(μ)Yk(μ) ~ 4. Define complex-valued Gaussian random variable Zn(μ) = Xn(μ)+jY(μ), n=1,2,..,N, where {Xn(μ),Yn(μ)}(n=1 to N) are jointly real-valued i.i.d. Gaussian random variables, which have zero mean and unit variance. ~ k ~ N ~* (a) Define Wk(μ) =[Π Zi(μ)]˙[Π Zm(μ)], k=1,2,...,N-1 (* denote i=1 m =k+1 ~ N ~* ~ N ~* the complex conjurate), W (μ) = Π Zm(μ) ,and W (μ)= Π Zi(μ). 0 m=1 N i=1 ~ If N=2, find E[Wk(μ)] for k=0,1,2,...,N (b) Find the characteristic function of a new random variable N ~ 2 V(μ) = Σ |Zn(μ)| n=1 5. Consider a linear and time-invariant system eith continus real input X(μ,t), continuous real output Y(μ,t), continuous real implues resopose h(t), and system H(ω). Let X(μ,t) and Y(μ,t) are both wide-sense stationary random processes with mean η_x and η_y,respectively, autocorrelation function Rx(τ) and Ry(τ), respectively. and power spetrum density Sx(ω) and Sy(ω),respectively. Also let h(t)=1 if |t|<1, and h(t)=0 otherwise. (a) It is known that η_y = α˙η_x with a constant α, Determine α (b) Express Ry(τ) in terms of Rx(τ). (c) Express Sy(ω) in terms of Sx(ω) 6. Cars arrive at a bridge entrance according to a Poisson process of rate λ=15 cars per minute. (a) Find the probability that in a given 4 minute period there are 3 cars arrivals during the first minute and 2 cars arrivals in the last minute. (b) Find the mean and variance od the time of the tenth cae arrival, given that the time of the fifth car arrival is T minutes (c) Assume that the bridge is long enough so that it can virtually accommodate an infinite number of cars. Further, let the time that each car stay on the bridge be independent ane exponetially destributed with mean μ=1 minute. Under this assumption the bridge can be modeled as an M/M/∞ queuing system. Find the mean number of cars on the bridge 7. Consider a discrete-time two-events hpmogeneous Markov chain Xk(μ), k=0,1,2,...,with the following statistical description: Pr(X0(μ)=1) = Pr(X0(μ)=2) = 1/2 Pr(Xn(μ)=1 | Xn-1(μ)=1) = Pr(Xn(μ)=2 | Xn-1(μ)=1) =1/2 Pr(Xn(μ)=2 | Xn(μ)=2) =1 Find the marginal distribution of the increment random variable Z1(μ) = X1(μ)-X0(μ) and Z2(μ) = X2(μ)-X1(μ), and check whether Z1(μ) and Z2(μ) are independent increment. What do you observe? --

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