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Assignment 4
Problems:
1. Let
X
denote a random variable distributed on the set
A
=
{
a
1
,a
2
,...,a
N
}
with correspond
ing probabilities
{
p
1
,p
2
,...,p
N
}
. Let
Y
be another random variable deﬁned on the same set
but distributed uniformly. Show that
H
(
X
)
≤
H
(
Y
)
with equality if and only if
X
is also uniformly distributed. Hint: First prove the inequality
log
x
≤
x

1 with equality for
x
= 1, then apply this inequality to
N
X
n
=1
p
n
log
1
N
p
n
!
.
2. Determine the average energy of a set of
M
PAM signals of the form
s
m
(
t
) =
s
m
ψ
(
t
)
,
m
= 1
,
2
,...,M
where
s
m
=
p
E
g
A
m
,
m
= 1
,
2
,...,M
The signals are equally probable with amplitudes that are symmetric about zero and are
uniformly spaced with distance
d
between adjacent amplitudes as shownn in Figure 7.11.
3. Consider the four waveforms
s
1
= (2
,

1
,

1
,

1)
s
3
= (1
,

1
,
1
,

1)
s
2
= (

2
,
1
,
1
,
0)
s
4
= (1
,

2
,

2
,
2)
•
Determine the dimennsionality of the waveforms.
•
Determine the minimum distance between any pair of vectors.
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This note was uploaded on 11/13/2011 for the course ECEN 455 taught by Professor Staff during the Spring '08 term at Texas A&M.
 Spring '08
 Staff

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