Databases Reference
In-Depth Information
Through the connection
y
∗
)
2
2
xy
)
E
|
x
(
y
,
−
x
|
=
R
xx
(1
−
ρ
¯
,
(4.14)
we obtain the corresponding squared correlation coefficient
2
2
ρ
xy
ρ
xy
ρ
yy
]
=
|
ρ
xy
|
+|
ρ
xy
|
−
2Re[
2
xy
ρ
¯
1
−|
ρ
yy
|
2
+|
R
xy
|
2Re(
R
xy
R
xy
R
yy
)
R
xx
(
R
yy
−|
R
yy
|
2
2
)
R
yy
−
(
|
R
xy
|
=
,
(4.15)
2
)
ρ
xy
=
ρ
yx
with 0
≤
ρ
¯
xy
≤
1. We note that the correlation coefficient ¯
ρ
xy
, unlike
and
xy
=
yx
.
ρ
xy
=
ρ
yx
, is not symmetric in
x
and
y
: in general, ¯
ρ
ρ
¯
The correlation coefficient ¯
ρ
xy
is bounded in terms of the coefficients
ρ
xy
and
ρ
xy
as
2
2
)
2
xy
2
2
,
|
ρ
xy
|
+|
ρ
xy
|
max(
|
ρ
xy
|
≤
ρ
¯
≤
min(
|
ρ
xy
|
,
1)
.
(4.16)
The lower bound holds because a WLMMSE estimator subsumes both the LMMSE
and CLMMSE estimators, so we must have WLMMSE
≤
CLMMSE. However, there is no general ordering of LMMSE and CLMMSE, which we
write as LMMSE
≤
LMMSE and WLMMSE
CLMMSE.
A common scenario in which the lower bound is attained is when
y
is maximally
improper, i.e.,
R
yy
=|
R
yy
|⇔|
ρ
yy
|
2
1, which yields a zero denominator in (
4.15
).
This means that, with probability 1,
y
∗
=
=
e
j
α
R
xy
.
In this case,
y
and
y
∗
carry exactly the same information about
x
. Therefore, WLMMSE
estimation is unnecessary, and can be replaced with either LMMSE or CLMMSE esti-
mation. In the maximally improper case, ¯
e
j
α
y
for some constant
α
, and
R
xy
=
ρ
xy
=|
ρ
xy
|
2
=|
ρ
xy
|
2
. Two other examples
of attaining the lower bound in (
4.16
) are either
R
xy
=
0 and
R
yy
=
0 (i.e.,
ρ
xy
=
0
0 and
R
yy
=
xy
2
,or
R
xy
=
and
ρ
yy
=
0), which leads to ¯
ρ
=|
ρ
xy
|
0 (i.e.,
ρ
xy
=
0 and
xy
2
;cf.(
4.15
).
ρ
yy
=
0), which yields ¯
ρ
=|
ρ
xy
|
2
is attained when the WLMMSE estimator is
the sum of the LMMSE and CLMMSE estimators:
x
(
y
xy
2
The upper bound ¯
ρ
=|
ρ
xy
|
+|
ρ
xy
|
x
(
y
∗
). In this case,
y
and
y
∗
carry completely complementary information about
x
. This is possible only for
uncorrelated
y
and
y
∗
, that is, a
proper y
. It is easy to see that
R
yy
=
y
∗
)
,
=
x
(
y
)
+
2
0
⇔|
ρ
yy
|
=
0
2
. The following example gives two scenarios in
which the lower and upper bounds are attained.
xy
2
in (
4.15
) leads to ¯
ρ
=|
ρ
xy
|
+|
ρ
xy
|
Example 4.2.
Attaining the lower bound
. Consider a complex random variable
e
j
α
(
u
y
=
+
n
)
,
where
u
is a
real
random variable and
is a fixed constant. Further, assume that
n
is a
real
random variable, uncorrelated with
u
, and
R
nn
=
α
Re (e
j
α
u
)
R
uu
.Let
x
=
=
1
)
e
−
j
α
y
and the CLMMSE estimator
cos(
α
)
u
. The LMMSE estimator
x
(
y
)
=
2
cos(
α
x
(
y
∗
)
)e
j
α
y
∗
both perform equally well. However, they both extract the
same
information from
y
because
y
is maximally improper. Hence, a WLMMSE estimator has
1
=
2
cos(
α
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