課程名稱︰隨機信號與系統 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 考試日期（年月日）︰104/11/10 考試時限（分鐘）：120分鐘 試題 : Problem 1 (20%) Assume that we receive a random signal X(t) = Acos(w_c*t)+Bsin(w_c*t) with a constant w_c, where A and B are two uncorrelated zero mean random variables with the same variance σ^2 and having different density functions. Moreover, we create another random signal Y(t) = Bcos(w_c*t)-Asin(w_c*t). (a) Is X(t) a wide-sense stationary(WSS)? Why? (8%) (b) Is Y(t) a WSS? Why? (5%) (c) Are X(t) and Y(t) jointly WSS? Why? (7%) Problem 2 (20%) Assume that we receive a WSS Gaussian signal X(t) with expectation function m_x(t) = E{X(t)} = 0 and autocorrelation function Rxx(τ) = 2^-|τ|. (a) Find the joint probability density function (pdf) of X(t) and X(t+1). Please justify your answer. (10%) (b) Is X(t) a strict-sense stationary (SSS) Gaussian process? Why? (10%) Problem 3 (20%) Assume that we transmit a signal X(t) from a modem as follows: Let X(t) be a rectangular pulse waveform with amplitude = 1 of duration = T if a binary bit 1 is transmitted. In contrast, let X(t) be a rectangular pulse waveform with amplitude = -1 of duration = T if a binary bit 0 is transmitted. Assume that the modem generates the bits 0 and 1 with equal probability. (a) Find the expectation function m_x(t) of X(t). Please justify your answer. (5%) (b) Find the autocorrelation function Rxx(t1,t2) of X(t), Please justify your answer. (7%) (c) Find the correlation coefficient of X(t1) and X(t2). Please justify your answer. (8%) Problem 4 (20%) Assume that we receive a random signal X[n] with expectation equal to zero and autocovariance function -|m-n|σ^2 Cxx[m,n] = 2 We estimate X[k] according to X[k] = aX[k-1] + bX[k-2]. (a) Find the average power of X[n]. Please justify your answer. (5%) (b) Find the best coefficients a and b according to the minimum mean square error criterion. Please justify your answer. (8%) (c) Find the corresponding mean square error. Please justify your answer. (5%) Problem 5 (20%) Assume that we receive an independent, identically distributed random signal X[n] with expectation equal to zero and variance equal to σ^2 at any time n. We generate a random signal Y[n] from X[n] such that １ n Y[n] = ─ Σ X[m]. ｎ m=1 (a) Find the conditional probability P{Y[n]=α|Y[n-1]=β}. Please justify your answer. (8%) (b) Is Y[n] a Markov process? Why? (7%) (c) Is Y[n] WSS? Why? (5%) --

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