課程名稱︰隨機訊號與系統
課程性質︰選修
課程教師︰李枝宏 教授
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）︰2012/01/10
考試時限（分鐘）：14:20 ~ 16:20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
NATIONAL TAIWAN UNIVERSITY
FINAL EXAMINATION
COURSE : STOCHASTIC SIGNALS AND SYSTEMS
NOTE : CLOSE BOOKS AND NOTES,DO ALL FIVE PRMBLEMS.
PROBLEM 1 : (20%)
Consider a moving average(MA) process Y[n] generated by a linear system with
random input X[n]. Let the input and output (I/O) of the linear system be
given by Y[n]=(X[n]＋X[n－1])/2 , where {X[n], for all n} is an independent
Bernoulli random sequence. Assume that X[n] = 0 with probability = 1/2 and
X[n]= 1 with probability = 1/2.
(a)
Find the probability mass function (PMF) of Y[n]. You must justify your
answer.(4%)
(b)
Find the conditional probability of Y[n] given Y[n－1]. You must justify your
answer. (4%)
(c)
Find the conditional probability of Y[n] given Y[n－1] and Y[n－2].
You must justify your answer. (4%)
(d)
Is Y[n] a Markov process ? Why ? (8%)
PROBLEM 2 : (20%)
Consider a Markov chain for an internet system receiving a digital binary data
system. Assume that the probability of the next received binary data bit equal
to 0 (state 1) is 1－α if the nth binary bit is 0 , α> 0.On the other hand,
the probability of the next received binary data bit equal to 1 (state 2) is
1-b if the nth binary bit is 1 , b>0.
(a)
Find the state diagram for the two-state data receiving system.(4%)
(b)
Find the one-step state transition probability matrix for the two-state data
receving system.(4%)
(c)
Find the probability of the state that the received binary bit equal to 0 after
two steps if the initial state distribution vector is given by
Π(0)=[p_1(0) , p_2(0)] with p_1(0) = 1/2 . You must justify your answer.(8%)
(d)
Is this Markov chain stationary ? Why ? (4%)
PROBLEM 3 : (20%)
Consider a linear system with random signal input.Assume that a white noise
process X[n] with mean zero and variance σ_XX ^2 is applied to the linear
system with the output signal given by Y[n] = X[n] ＋ c X[n－1] ,
where c is a constant.
(a)
Find the autocorrelation function of Y[n]. You must justify your answer.(5%)
(b)
Find the power density spectrum (PDS) of Y[n].You must justify your answer.(8%)
(c)
Find the average power of Y[n].You must justify your answer.(7%)
PROBLEM 4 : (20%)
Consider that we want to estimate a real random variable Y using the linear
minimum mean-square error (LMMSE) criterion. Assume that the available data
for the estimation is a received random signal X(t), 0≦ t ≦ T and
autocorrelation function of X(t) is given by R_xx(τ) = exp{-β|τ|},β>0.
Let Y = X(T + λ) , λ>0.
(a)
Find the unit impulse response h(t) of the required Wiener filter for
estimating Y. You must justify your answer.(10%)
(b)
Find the corresponding optimum extimate of Y.You must justify your answer.(10%)
PROBLEM 5 : (20%)
Consider the estimate of a ral random signal based on the linear minimum
mean-square error (LMMSE) criterion. Assume that the observed random signal is
given by X(t) = Y(t) + N(t) , where N(t) is a white noise with mean zero and
variance equal to one . The desired signal Y(t) with PDS given by
S_YY(ω) = 1/(1+ω^2) is orthogonal to N(t).
(a)
Find the unit impulse response h(t) of the required Wiener causal filter for
estimating Y(t). You must justify your answer.(10%)
(b)
Find the corresponding mean squared error (MSE). You must justify your answer.
(10%)
GOOD LUCK AND HAVE A HAPPY LUNAR
NEW YEAR!
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