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）︰104/6/23
考试时限（分钟）：120
试题 :
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!!