ECE 562
Fall 2011
September 22, 2011
HOMEWORK ASSIGNMENT 3
Due Date: October 4, 2011
(in class)
1. For a complex random vector
Y
, show the following:
Σ = (Σ
I
+ Σ
Q
) +
j
(Σ
QI
−
Σ
IQ
)
and
ˇ
Σ = (Σ
I
−
Σ
Q
) +
j
(Σ
QI
+ Σ
IQ
)
2. Consider the signal
s
(
t
) = [sin(
πt
) +
j
cos(
πt
)]11
0
≤
t
≤
1
. Suppose this signal is corrupted
by complex WGN
w
(
t
) with PSD
N
0
= 2 to form the received signal
r
(
t
) =
s
(
t
) +
w
(
t
)
.
Further suppose we form the random variable
Z
as:
Z
=
integraldisplay
1
0
r
(
t
) sin(
πt
)
dt
(a) Find
P
{
Z
I
>
1
}
.
(b) Find
P
{
Z
I
+
Z
Q
>
1
}
.
(c) Find
P
(
{
Z
I
>
1
} ∩ {
Z
Q
>
1
}
)
3.
Unequal Priors.
Describe how the optimal decision regions for BPSK signaling get
modified when the priors on the messages are not the same, say
π
1
= 2
/
3 and
π
2
= 1
/
3.
4.
P
e
for PAM.
Compute (the exact) P
e
as a function of
E
s
for 4ary PAM. Note that
P
e
,m
is not the same for all
m
in this case.
5.
Performance of MPSK.
(a) Using the Intelligent Union Bound, show that the symbol error probability for
MPSK signaling in AWGN is bounded by
P
e
≤
2
Q
parenleftBigg
radicalbigg
2
E
s
N
0
sin
π
M
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 Fall '09
 Probability theory, bit error probability, Quadrature amplitude modulation

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