University of Illinois
Spring 2011
ECE 361:
Problem Set 9
MaximumLikelihood Sequence Estimation; Passband Signals
Due:
Tuesday
April 26 at 9:30 a.m.
1.
[Maximumlikelihood sequence estimation]
The received signal in a communication system operating over a bandlimited channel is
Y
[
m
] =
X
[
m
] + 0
.
2
X
[
m

1] + 0
.
1
X
[
m

2] +
N
[
m
]
where
{
X
[
m
]
}
is a sequence of independent random variables taking on values
±
1 with equal probability,
and
{
N
[
m
]
}
is a sequence of
N
(0
,
1) random variables independent of each other and of the sequence
{
X
[
m
]
}
. Transmission begins at 0 and so
X
[

1] =
X
[

2] = 0.
Assume that the received sequence is
Y
[0]
,
Y
[1]
,
Y
[2]
,
Y
[3]
···
= 0
.
9
,

0
.
7
,

1
.
1
,
0
.
8
,
···
(a) Sketch the initial portion of the trellis that would be used for maximumlikelihood detection of
the received sequence and mark on each branch the
squared distance
that would be added as the
computations proceed.
(b) Find the best path from the start node (at depth 0) to each of the four nodes at depth 3 in the
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 Spring '09
 Probability theory, Passband Signals, maximumlikelihood sequence estimation

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