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）︰104/6/23 考試時限（分鐘）：120 試題 : Problem1:(15%) In this problem, we consider the following binary detection problem: H1:r1=K+wi, i=1,2,...,N. H0:r1=wi, i=1,2,...,N. where K is a constant and the noise samples wi are independent zero-mean Gaussian random variables with variance σ^2. Let the data vector r=[r1,r2,...rN]^T be used. (a) Find the moment-generating function ψl|H0(s) of the log likelihood ratio function l(R) (the sufficient statistic) on H0 and the resulting semi-invariant moment generating function μ(s). You must justify your answer.(5%) (b) Find the tightest upper bound for the probability Pf of a false-alarm by using μ(s). You must justify your answer.(5%) (c) Find the tightest upper bound for the probability PM of a miss by using μ(s). You must justify your answer.(5%) Problem 2: (25%) In this priblem, we consider the following binary detection problem: H1:r(t)= √E1 s1(t) + n(t), 0≦t≦T Ho:r(t)= √E0 s0(t) + n(t), 0≦t≦T where n(t) is an additive white Gaussian noise (AWGN) with mean =0 and T T variance = N0/2, ∫sk(t)^2dt=1, k=0,1.∫s0(t)^2dt=ρ. 0 0 The a priori probabilities of the two hypothese are equal. (a) Find an appropriate set {ψi(t)} of completely orthonormal (CON) functions with ψ1(t)=s1(t) for this detection problem. You must justify your answer.(6%) (b) Find the corressponding likelihood ratio test for performing the detection. You must justify your answer.(7%) (c) Plot the resulting decision space and Determine the decision line if E1=E0, ρ=-1 if the criterion is mininum probility of error for this detection problem You must justify your answer.(12%) Problem 3:(40%) Assume that the signal waveform is given by r(t) = s(t,A)+n(t), 0≦t≦T, where n(t) is an AWGN with mean function =0 and variance function =N0/2. T Moreover, s(t,A)= A√E s(t), ∫s(t)^2dt=1. 0 (a) Let the set {ψi(t)} of completely orthonormal (CON) functions be the eigenfunctions obtained from the autocorrelation finction of r(t). Do you think that directtly using {ψi(t)} is an appropriate manner to solve this problem? Why? (6%) (b) If your answer to Part(a) is NO, then Find an approriate CON set for solving this problem. You must justify your answer.(6%) ^ (c) Based on Part(b), Find the maximum likelihood (ML) estimate aml of the real parameter A. You must justify your answer.(6%) ^ (d) Is aml of Part(c) boased? You must justify your answer.(4%) (e) From Part(c), Find the variance of the corresponding estimation error. You must justify your answer.(7%) ^ (f) Find themaximum a posteriori (MAP) estimate amap of the random parameter A which is Gaussian with mean function =0 and variance (σa)^2. You must justify your answer.(6%) (g) From Part(f), Find the variance of the corresponding estimation error. You must justify your answer.(5%) Problem 4:(20%) In this problem, we consider a simple binary detection problem. Assume that the received signal r(t) is given as follows: H1:r(t)=√E s(t) + n(t), H0=r(t)=n(t), for 0≦t≦T, where s(t) is a known signal with unit energy for 0≦t≦T. n(t) is a zero mean nonwhite Gaussian noise with autocovariance function Kn(t1, t2)=(N0/2)*δ(t1-t2)+Kc(t1,t2), where Kc(t1,t2) is the autocovariance finction of nc(t). (a) Plot the detector system using the inverse kernel Qn(t1,t2) of Kn(t1,t2) based on the concept of a whitening approach.(5%) (b) Find the solution for Qn(t1,t2) of Part(a) using the concept of the Karhunen-Loveve (K-L) series expansion. You must justify your answer.(8%) (c) Find the unit ipmulse responde of the corresponding whitening filter hw(t1,t2). You must justify your answer.(7%) Good luck and Have a Nice Summer Recess!! --

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