課程名稱︰偵測與評估 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︰電機所 電信所 考試日期（年月日）︰2012/6/18 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） Problem 1 In this problem, we consider an observation r = m+n，where m and n are two Poisson random variables with pdf given by Pn(N)= exp(-λ)λ^N / (N!)； N = 0,1,2,...and Pm|a(M|A) = exp(-A)(A^M)/(M!)；M=0,1,2,...， a>=0 (a) Find the conditional pdf Pr|a(R|A) with R=0,1,2,...(5%) (b) Find the ML estimate of "a" Is the estimate unbiased? (5%) (c) Find the corresopnding Cramer-Rao lower bound for the conditional mean -square estimate of "a"on the basis of observing r. Is the ML estimate of "a" efficient? (10%) Problem 2 Consider a detection problem with binary hypothesis given as follow: H0: r(t) = sqrt(E0)*s0(t)+w(t) H1: r(t)= sqrt(E1)*s1(t) +w(t) for 0≦t≦T, where s0(t) and s1(t) are two known signals with unit erergy and correlation equal to ρ for 0≦t≦T，w(t) is a zero-mean white Gaussian noise with the variance equal to (σ_w)^2 (a) Find the appropriate set of complete orthonormal (CON) basis function over the time interval [0,T] based on s0(t) and s1(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. (10%) Problem 3 Consider an estimate problem. Let the observed data y(t)= x(t,A) + n(t)， for 0≦t≦T，where the parameter A is said to be estimated. n(t) is a zero-mean white Gaussian noise with variance equal to (σ_n)^2 (a) Using the concept of K-L series expansion, derive the integral equation for solving the manimun likelihood (ML) estimate of the nonlinear parameter "A". (5%) (b) Find the ML estimate of "A" from (a) under the assumption that x(t)=A*x(t) with the erergy of x(t) equal to E in 0≦t≦T. (5%) (c) Find the error variance of the ML estimate of "A" obtained from (b) (5%) (d) Is the ML estimate of "A" obtained from (c) efficient? (5%) Problem 4 Consider an estimate problem. Let the observed data y(t)=s(t,A) + n(t) for 0≦t≦T, where the Gaussian random parameter A with zero-mean and variance (σ_n)^2 (a) Using the concept of K-L series expansion, derive the equation for solving the maximun a posteriori (MAP) estimate for "A". (7%) (b) Find the MAP estimate of "A" from (a) under the assumption that x(t,A) = A*x(t) with the energy of x(t) equal to E in 0≦t≦T. (7%) (c) Find the error variance of the MAP estimate of "A" obtained from (b) (6%) Problem 5 Consider a detection problem with nonwhite Gaussian noise. Let the received signal r(t) be as follow: H0: r(t) = n(t) H1: r(t) = m(t) + n(t) for 0≦t≦T where m(t) is the known signal with energy E in 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) Find the integral equation including Kn(t1,t2) required for finding a whitening filter hw(t,u). (7%) T (b) Let the Qn(t1,t2) = ∫ [hw(u,t1)][hw(u,t2)] du, 0≦ t1,t2 ≦ T 0 show that Qn(t1,t2) is an inverse kernel of Kn(t1,t2). (5%) (c) Find the solution for Qn(t1,t2). this sokution must in terms of the eigenvalues and eigenfunctions of Kc(t1,t2). (8%) --

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