課程名稱︰偵測與評估
課程性質︰選修
課程教師︰李枝宏
開課學院：電資學院
開課系所︰電機所
考試日期（年月日）︰2009/6/14
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Note:close book and class notes, do all five problems.
Problem 1:(20%)
Consider a detection problem with a binary hypothesis given as follows:
H0:r(t)= sqrt(E0)*so(t)+w(t), H1:r(t)= sqrt(s1(t))+w(t),
for 0≦t≦T, where the s0(t) and s1(t) are two known signals with unit energy
for 0≦t≦T.w(t) is a zero-mean white Gaussian noise with variance equal to
σw^2.
(a) Find the appropriate set of complete orthonormal (CON) function over the
time interval [0,T] for performing the Karhunen-Loeve(K-L) series expansion
of r(t).(10%)
(b) Based on part (a), derive the likelihood ratio test required for making a
decision. You must justify your answer.(10%)
Problem 2:(20%)
Consider a nonlinear estimation problem. Let the observed data r(t)=s(t,A)+
w(t), for 0≦t≦T, where the parameter A is to be estimated. w(t) is a
zero-mean white Gaussian noise with variance equal to σw^2.
(a) Using the concept of K-L series expansion, where the integral equation for
solving the maximum likelihood (ML) estimate for a. You must justify your
answer.(10%)
(b) Repeat (a), find the maximum a posteriori(MAP) estimate for A if A is
Gaussian with mean zero and variance σa^2. You must justify your answer(10%)
Problem 3 :(20%)
Consider a detection prpoblem with nonwhite Gaussian noise. Let the received
signal r(t) be as follows:
H1:r(t)=sqrt(E)*s(t)+n(t), H0:r(t)=n(t),
for 0≦t≦T, where s(t) is the known signal with unit energy for 0≦t≦T.n(t)
=w(t)+nc(t) is a zero-mean nonwhite Gaussian noise with autocovariance
function Kn(t1,t2)=Kw(t1-t2)+Kc(t1,t2), where Kw(t1-t2)=No/2*δ(t1-t2) is the
autocovariance function of w(t) and Kc(t1,t2) is the autocovariance function
of nc(t).
(a) Derive the logarithm of the likelihood ratio function required for
detection by using the whitening filter hw(t,u). You must justify your answer.
(12%)
(b) Based on (a), plot the detector system structure.(8%)
Problem 4:(20%)
Consider an estimation problem. Let the observed data r(t)=s(t,a(t))+w(t), for
0≦t≦T, where the random signal a(t) is to be estimated. w(t) is a zero-mean
white Gaussian noise with variance equal to σw^2.
(a) Derive the integral equation specifying the MAP estimate of a(t). You must
justify your answer.(12%)
(b) Repeat (a), if the noise w(t) is nonwhite. You must justify your answer.
(8%)
Problem 5:(20%)
Consider a continuous-waveform estimate problem. Let the received data r(t)=
s(t,x(t))+w(t), for 0≦t≦T, where x(t) is the output of a linear system h(t,u)
with input equal to a(t). The random signal a(t) is to be estimated and w(t) is
a zero-mean white Gaussian noise with variance equal to σw^2.
(a) Dervie the integral equation specifying the MAP estimate of a(t). You must
justify your answer.(12%)
(b)Repeat (a), if the noise w(t) is nonwhite. You must justify your answer(8%)
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