[試題] 104-1 李枝宏 隨機信號與系統 期末考
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）︰105/1/12
考試時限（分鐘）：120分鐘
試題 :
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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