Digital Signal Processing Reference
In-Depth Information
Problems
16.1.
Let the error covariance
E
x
defined in Sec. 16.2.1 be given by
E
x
=
21
13
.
Find real unitary
U
such that
UE
x
U
†
has identical diagonal elements.
What are the element values in the resulting matrix
UE
x
U
†
?
16.2.
Let the error covariance
E
x
be given by
⎡
⎣
⎤
⎦
0
.
40 0 0
050 0
0
E
x
=10
−
3
.
0
60
0
0
0
0
100
1. Find a real unitary
U
such that
UE
x
U
†
has identical diagonal ele-
ments.
2. What are the element values in the resulting matrix
UE
x
U
†
?
3. Assuming 2-bit PAM and
σ
s
= 1, find the minimized error probability
(16.23) in the zero-forcing case.
4. If the unitary matrix
U
were not used to equalize the errors, the
average error probability would be as in Eq. (16.14). What is its
value?
16.3.
Let the error covariance
E
x
defined in Sec. 16.2.1 be given by
E
x
=
31
14
.
Find a real unitary
U
such that the error probability (16.14) is
worse
than
the case where no
U
is employed at all.
16.4.
Let
x
and
x
1
be random variables with identical mean
m
and cross corre-
lation
r
=
E
[
xx
1
]
.
Let
x
be the optimal linear estimate of
x
based on
x
1
,
that is,
x
=
ax
1
,
|
2
] is minimized.
where
a
is such that
E
[
|
x
−
x
1. Find an expression for the mean value
m
e
of the error
e
=
x.
2. Find specific numerical values for
r
and
m
and the mean square values
of
x
and
x
1
such that
m
e
is zero.
x
−
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