[試題] 104-1 李枝宏 隨機信號與系統 期中考
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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