ECE 561: Detection and Estimation Theory
Spring 2009
Issued: February 28, 2009
March 15, 2009
•1)
Binary frequency shift keying (FSK) on a Rayleigh fading channel can
be modeled in terms of a four-dimensional observation vector Y, where
Y=X+Z, and where Zis a zero-mean Gaussian vector, with covariance
matrix σ2I4×4. Under hypothesis H0, we have X=[X1X20 0], while
under hypothesis H1, we have X=[0 0 X3X4]. The Xiare independent,
identically-distributed, standard Gaussian random variables. Also, the
two hypotheses are equally likely.
a) Find the maximum likelihood rule for the receiver.
b) Find the probabilities of false alarm and miss for the maximum likeli-
hood decision rule.
•2)
Consider the composite binary hypothesis testing problem for which
fx(y) = xexp{−xy}, y ≥0
and
fx(y)=0, y < 0,
where x∈[1,∞).
a) For α∈(0,1), show that a UMP test of level αexists for testing the
hypotheses H0:X0= [1,2), and H1:X1= [2,∞). Express the test ratio
as a function of α.
b) Assume that the conditional distribution described above is Laplacian
instead, with mean x∈[0,∞) and parameter λ= 1. Does there exist a
UMP test for H0:X0={0}, and H1:X1= (0,∞)?
•3)
Consider the following two forms of generalized maximum likelihood tests
for two composite hypotheses:
a)maxx∈X1fx(y)
maxx∈X0fx(y);
1
