课程名称︰随机讯号与系统 课程性质︰选修 课程教师︰李枝宏 教授 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）︰2012/01/10 考试时限（分钟）：14:20 ~ 16:20 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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