課程名稱︰通訊隨機過程
課程性質︰選修
課程教師︰鐘嘉德
開課學院：電資學院
開課系所︰電信所 電機所
考試日期（年月日）︰
考試時限（分鐘）：3HR
是否需發放獎勵金：是
試題 :
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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