课程名称︰侦测与评估 课程性质︰选修 课程教师︰李枝宏 开课学院：电资学院 开课系所︰电机所 考试日期（年月日）︰2009/6/14 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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