课程名称︰随机信号与系统 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 考试日期（年月日）︰105/1/12 考试时限（分钟）：120分钟 试题 : Problem 1: (20%) Consider that a discrete-time system with the input X[n] that is a white random process with mean = 0 and average power = σ_X^2. Let the output be Y[n] = αY[n-1]+X[n] with |α| < 1. (a) Find the average power of the output Y[n].You must justify your answer. (6%) (b) Find the power density spectrum (PDS) S_YY(f) of Y[n]. You must justify your answer. (7%) (c) Find the cross PDS S_XY(f) of X[n] and Y[n]. You must justift your answer. (7%) Problem 2: (20%) In this problem, we consider the estimation of a random variable Y based on an observation of a received random signal X(t) with autocorrelation function R_XX(t1,t2) = exp{-α|t1-t2|} using the linear minimum mean-square error (LMMSE) criterion. Let Y = X(T+λ), λ > 0, and the time section of the observed random signal X(t) is 0 ≦ t ≦ T. (a) Find the integral equation for solving the impulse response of the required causal Wiener filter for estimation Y = X(T+λ). You must justify your answer. (10%) (b) Find the optimum estimate of Y = X(T+λ). You must justify your answer. (10%) Problem 3: (20%) Consider that we received a random signal X(t) = Y(t) + W(t), where W(t) is a white noise with mean = 0 and variance = σ_W^2. Moreover, the zero-mean signal Y(t) is uncorrelated with W(t) and has the autocorrelation function R_YY(τ) = Aexp{-α|τ|}, where A is a constant and α > 0. Suppose that we ^ want to obtain the optimum estimate Y(t-λ) of Y(t-λ) based on the observations {X(β), -∞<β<∞}, λ is a real number, according to the LMMSE criterion. (a) Find the corresponding optimum linear filter h(t). You must justify your answer. (12%) (b) Find the corresponding minimum mean square error (MSE). You must justify your answer. (8%) Problem 4: (20%) Consider that we received a random signal X(t) = Y(t) + W(t), where W(t) is a white noise with mean = 0 and variance = σ_W^2. Moreover, the zero-mean signal Y(t) is uncorrelated with W(t) and has autocorrelation function R_YY(τ) = Aexp{-α|τ|}, where A is a constant and α > 0. Suppose that we want to obtain the optimum estimate Y(t-λ) of Y(t-λ) based on the observations {X(β),-∞<β< t}, λ is a real number, according to the LMMSE criterion. (a) Find the corresponding optimum linear filter h(t). You must justify your answer. (12%) (b) Find the corresponding minimum mean square error (MSE). You must justify your answer. (8%) Problem 5: (20%) Consider that we received a random signal X[n] = Y[n] + W[n], n ≧ 0, where W[n] is a white noise with variance σ_W^2. Moreover, both of the signal Y[n] and the white noise W[n] are uncorrelated and have zero mean. Suppose that we want to obtain the the optimum estimate of Y[n] based on the observations {X[0], X[1],..., X[n]} according to the LMMSE criterion. (a) Find the requried coefficient vector a in terms of the covariance function of Y[n] and σ_W^2. You must justify your answer. (12%) (b) Repear Part (a) if Y[n] is also a white noise with variance = σ_Y^2. You must justify your answer. (8%) --

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