ECE 459 Fall 2000
Handout # 4 September 27, 2000
HOMEWORK ASSIGNMENT 3
Reading: Lecture notes (lectures 8-11), Proakis (Chapter 5), papers/books referenced in lecture notes.
Due Date: Thursday, October 12, 2000 (in class)
1. (15 pts) Estimation.
(a) (5 pts) Suppose the likelihood function for real-valued parameter λis given by
pλ(y)=e
−(y−λ)
11
{y≥λ}.
Find ˆ
λML(y). Now find ˆ
λMAP(y) under the assumption that pΛ(λ)=e
−λ
11
{λ≥0}
. Also find
ˆ
λMAP(y) under the assumption that Λ is uniformly distributed on [0,1].
(b) (10 pts) Suppose Y=[Y
1Y
2··· Y
n] is a vector of i.i.d. random variables with marginal pdf
given pλ(·) given in part (a). Find ˆ
λML(y). Also find ˆ
λMAP(y) under the assumption that
pΛ(λ)=e
−λ
11
{λ≥0}
.
2. (15 pts) Detection in Discrete-Time Colored Gaussian Noise. Consider the binary detection
problem (i.e., λ∈{0,1}) with likelihood function
pλ(y)= 1
π
n
det(Σ)exp n−(y−µλ)†Σ−1(y−µλ)o,λ=0,1.
(a) (5 pts) Show that for equal priors
ˆ
λMPE =ˆ
λML =(1ifB(y)≥d
2
0 otherwise
where
B(y) = 2Re[(µ1−µ0)†Σ−1(y−µ0)] ,and d2=(µ
1−µ
0
)
†
Σ
−1
(µ
1−µ
0
)
(b) (x pts) Show that Pefor the MPE detector equals Q(d/√2).
3. (10 pts) Noncoherent Demodulation of Linearly Modulated Signals. The received signal for one
symbol period for linear memoryless modulation on an ideal AWGN channel is given by:
y(t)=pE
m
e
jθmg(t)ejφ +w(t)
where the phase offset φis due to the delay introduced by channel. If φis known at the receiver,
we can correct for it (by say projecting y(t)ong(t)e
jφ to produce the sufficient statistic) and
suffer no loss in detection performance. However, if φis not known, we may project y(t)ong(t)
to get the sufficient statistic
y=pEmejθmejφ +w
where w∼CN(0,N
0). Since φis not of direct interest to the receiver, we treat it as a nuisance
parameter. As we saw in class, there are two ways to deal with such parameters.
c
V. V. Veeravalli, 2000 1
