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）︰104/4/14 考試時限（分鐘）：120 試題 : Problem 1:(20%) In this problem, we consider a detection with a binary hypothesis given as follows: H_0:pr(R) = exp{-R^2 /2}/√2π, H_1:pr(R)= exp{-|R|}/2, where r is the recieved random data, let C_ij be the cost of deciding H_i when H_j is acting, i=0,1. Moreover, the null hypothesis H_0 with probability P(H_0) and the alternative hypothesis H_1 with probability P(H_1). (a) Find the likelihood ratio test (LRT) and the threshold value η required for performing the Bayes test. You must justify your answer.(12%) (b) Based on part(a), decide the decision regions on the real line for the detection. You must justify your answer.(8%) Problem 2:(20%) In this problem, we consider a binary detection problem. Let the observed data r be a Gaussian random variable with probability density function (PDF) given by P_r|H_k(R|H_k)=exp{-(R-m_k)^2 /2(σ_k)^2}/√2π(σ_k), where -∞<R<∞:k=0,1;m_0=0,m_z=1;σ_0=σ_1=1. (a) Find the optimum decision rule uding the Neyman-Pearsom criterion. You must justify your answer. (8%) (b) Decide the corrisponding decision regions. You must justify your answer.(4%) (c) Describe how to find the threshold value in order to satisfy the preset constraint of the false-alarm probability P_F=0.3 You must justify your answer. (8%) Problem 3:(20%) In this problem, we consdier adetection with a binary hypothesis given as follows: exp{-R}, R>0 H_0:p_r(R)={ 0, otherwise αexp{-αR}, R>0 H_1:p_r(R)={ 0, otherwise where r is the recieved random data and α>1. (a) Find the optimum decision rule using the Neyman-Pearson criterion. You must justify your answer. (4%) (b) Find the false-alarm probability P_F corresponding to Part(a). You must justify your answer. (4%) (c) Find the probability of detection P_D corresponding to Part(a). You must justify your answer. (4%) (d) Plot some curves of the receiver operating characteristic (ROC) corresponding to Part(a). You must justify your answer. (4%) (e) Find the slope of any curve of the recevier operating characteristic (ROC) corresponding to Part (d). You must justify your answer. (4%) Problem 4:(20%) In this problem, we consider an estimation problem. A real parameter a is to be estimated by using N independent experiments. During the N independent experiments, we find the a specific event occurs r times eith the following probailitic transition mechanism: N Probability (r event|a)=( )a^r(1-a)^(N-r) , r=0,1,2,...,N. r ^ (a) Find the maxmimum likelihood (ML) estimate a_ml(N) of a. You must justify your answer.(8%) (b) Find the estimation error variance corresponding to Part(a). You must justify your answer. (6%) ^ (c) Is the ML estimate aml(N) efficient? Why? (6%) Problem 5 (20%) In this problem, ew consider the estimation of a random variable a. Assume that a is aGaussian random variable with N(0,(σ_a)^2) amd the received data samples are given by r_i=a+n_i, i=1,2,...,K, where the noise samples are independent Gaussian random variables with N(0,(σ_n)^2). ^ (a) Find the optimum estimate a_mse of a according to the mean-square error criterion. You must justify your answer. (7%) ^ (b) Find the optimum estimate a_map of a according to the maximum a posteriori criterion. You must justify your answer. (8%) (c) find ther mean square error corresponding to Part(a). You must justify your answer.(5%) -- 這世界偶爾殘酷過了頭 因此選擇用閱讀祛除迷惘 用編程面對寂寞 用寫作宣泄悲傷 --

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