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）︰104/4/14
考试时限（分钟）：120
试题 :
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%)