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ECE1502F — Information Theory
Final Examination — Solutions
December 11, 2007
1.
(a) Let
X
and
Y
be independent Bernoulli(1/2) random variables and let
Z
=
X
⊕
Y
, where
⊕
denotes modulotwo addition. Since
X
and
Y
are independent, we have
I
(
X
;
Y
) = 0.
Since
X
is equally likely to be zero or one for each possible value of
Z
,
Z
provides no
information about
X
, i.e.,
I
(
X
;
Z
) = 0. However,
I
(
X
;
Y, Z
) = 1, since knowledge of
both
Y
and
Z
determines
X
uniquely. Thus the answer to the question is
no
.
(b) The same example as in the previous question provides a counterexample, i.e,
I
(
X
;
Y
) =
0, yet
I
(
X
;
Y

Z
) = 1.
(c)
True
. To see this, let
X
be a postive realvalued random variable over with
E
[
X
] =
m
and probability density function
p
(
x
). and let
Y
be an exponential random variable with
probability density function
f
(
x
) = (1
/m
)
e

x/m
. Then
0
≤
D
(
p

f
)
=
Z
∞
0
p
(
x
) ln[
p
(
x
)
/f
(
x
)]
dx
=
Z
∞
0
p
(
x
) ln
p
(
x
)
dx
+
Z
∞
0
p
(
x
) ln(1
/f
(
x
))
dx
=

h
(
X
) +
Z
∞
0
p
(
x
) ln(
me
x/m
)
dx
=

h
(
X
) +
1
m
Z
∞
0
xp
(
x
)
dx
+ ln(
m
)
Z
∞
0
f
(
x
)
dx
=

h
(
X
) + 1 + ln(
m
)
=

h
(
X
) + ln(
me
)
.
Thus we ﬁnd that
h
(
X
)
≤
ln(
me
) with equality achieved if and only if
p
(
x
) =
f
(
x
) a.e.,
i.e., if and only if
X
is an exponential random variable.
(d) To design a Huﬀman code over a quaternary alphabet it is necessary that the number
of probability masses be one larger than an integer multiple of 3. Thus, we must add
two dummy symbols. The resulting tree is shown in the ﬁgure, and corresponds to the
code given in the table.
3
1
0
3
2
1
0
2
/
8
4
/
8
0
1
2
0
0
1
/
8
1
/
8
1
/
8
1
/
8
1
/
8
1
/
8
1
/
8
1
/
8
0
1
2
3
4
5
6
7
d
d
dummy symbols
x
0
1
2
3
4
5
6
7
C
(
x
)
0
1
20
21
22
23
30
31
(e) The channel is symmetric, and so has capacity
C
= log
2
(3)

H
(1
/
2
,
1
/
4
,
1
/
4) = log
2
(3)

3
/
2
≈
0
.
085 bits per channel use.
1
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(a) The channel is neither symmetric nor weakly symmetric.
(b)
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This note was uploaded on 10/24/2011 for the course ELECTRICAL ECE 571 taught by Professor Kelly during the Spring '11 term at University of Illinois, Urbana Champaign.
 Spring '11
 Kelly

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