ECE 562
Fall 2011
October 12, 2011
EXAM 1 Solutions
1. (15 pts total)
True or False.
Determine if the following statements are True or False.
You need to provide a brief justification for your answer to get credit.
(a) The complex baseband representation of
√
2 cos(
πt
) sin(2
πf
c
t
) is cos(
πt
).
Ans:
False.
√
2 cos(
πt
) sin(2
πf
c
t
) =
√
2 cos(
πt
) cos(2
πf
c
t

π/
2). Thus, by the defini
tion, the complex baseband representation is cos(
πt
)
e

jπ/
2
=

j
cos(
πt
).
(b) If
X
and
Y
are zeromean proper complex random variables, then
Z
=
X
+
Y
is
also proper complex.
Ans:
False. Clearly
E
[
Z
] = 0. Therefore the pseudovariance of
Z
is given by
E
[
Z
2
],
which equals
E
[
X
2
]+
E
[
Y
2
]+2
E
[
XY
]. By properness of
X
and
Y
and the fact that they
are zero mean, we have that
E
[
X
2
] =
E
[
Y
2
] = 0. Thus
E
[
Z
2
] = 2
E
[
XY
]. Thus
E
[
Z
2
] = 0,
unless [
X Y
]
is a proper complex random vector or if
X
and
Y
are independent. As a
counterexample, consider
X
,
Y
which are
CN
(0
,
1) random variables, with
E
[
XY
] = 2.
(c) In a linear modulation constellation, if three nearest neighbors form an equilateral
triangle, it is impossible to Gray code the constellation.
Ans:
True. If we make both neighbors of a point differ by one bit from it, the neighbors
have to differ by two bits from each other.
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 Fall '09
 pts, Quadrature amplitude modulation, V.V. Veeravalli

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