Databases Reference
In-Depth Information
In order to make
x
a WLQ minimum mean-squared error (WLQMMSE) estimator that
minimizes
E
2
, we apply the
orthogonality principle
of Result
5.2
. That is,
|
x
−
x
|
x
,
(
x
−
x
)
⊥
(5.154)
where
g
H
y
x
=
y
H
H
y
tr (
R
yy
H
)
+
−
(5.155)
is any WLQ estimator with arbitrary [
g
,
H
]. The orthogonality in (
5.154
) can be
expressed as
x
)
∗
x
}=
for all
g
and
H
.
E
{
(
x
−
0
,
(5.156)
We now obtain
E
(
x
∗
g
H
y
)
E
(
x
∗
x
)
E
(
x
∗
y
H
H
y
)
E
(
x
∗
)tr (
R
yy
H
)
=
+
−
,
(5.157)
and, because
x
has zero mean,
g
H
E
(
x
∗
y
)
E
(
x
∗
x
)
tr [
H
E
(
x
∗
y y
H
)]
=
+
[
g
,
E
(
x
∗
y y
H
)]
H
H
]
[
E
(
x
∗
y
)
=
,
,
.
(5.158)
Because
y
has zero mean, we find
=
g
H
E
y
g
T
y
∗
+
y y
T
H
∗
y
∗
E
(
x
x
∗
)
E
y
H
H
y
g
T
y
∗
+
y
H
H
y
tr (
R
yy
H
∗
)
.
y
H
H
y y
T
H
∗
y
∗
−
+
(5.159)
By repeatedly using the permutation property of the trace, tr (
AB
)
=
tr (
BA
), we obtain
g
H
E
y
g
T
y
∗
+
y y
T
H
∗
y
∗
E
(
x
x
∗
)
=
tr
H
E
(
y
g
T
y
∗
y
H
tr (
R
yy
H
∗
)
y y
H
)
y y
T
H
∗
y
∗
y
H
+
+
−
[
g
,
H
H
]
E
(
y y
T
H
∗
y
∗
)
=
,
[
R
yy
g
+
,
tr (
R
yy
H
∗
)
R
yy
]
E
(
y y
H
y
H
g
)
E
(
y y
H
y
T
H
∗
y
∗
)
+
−
(5.160)
[
g
,
H
]
H
H
]
=
,
K
,
.
[
g
(5.161)
In the last equation, we have introduced the positive definite operator
, which is defined
in terms of the second-, third-, and fourth-order moments of
y
.Itactson[
g
K
,
H
]asin
(
5.160
). We can find an explicit expression for this. Let
y
i
,
i
=
1
,...,
2
m
, denote the
i
th element of
y
, i.e.,
y
i
=
y
i
,
=
,...,
m
, and
y
i
=
y
i
−
m
,
=
+
,...,
i
1
i
m
1
2
m
.Also
,
let (
R
yy
)
ij
denote the (
i
j
)th element of
R
yy
and (
R
yy
)
i
the
i
th row of
R
yy
. Then
K
[
g
,
H
]
=
[
p
,
N
]
(5.162)
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