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As in the previous section, this breaks up the estimator into three blocks. However,
this time we use a
canonical coordinate system
rather than a
half-canonical coordi-
nate system
. We first transform the measurement
y
into white and proper canonical
measurement coordinates
G
H
R
−
1
/
2
yy
=
B y
using the
coder
B
=
. We then estimate
ˆ
ˆ
the signal in canonical coordinates as
has mutually
uncorrelated components but is generally
not white or proper
, the canonical signal
=
K
. While the estimator
is
ˆ
white and proper. Finally, the estimate
x
is produced by passing
through the
decoder
A
−
1
FR
1
/
2
xx
(where the inverse is a
right
inverse).
We now ask how to find a rank-
r
widely linear estimator that provides as much mutual
information about the signal as possible. Looking at (
5.86
), we are inclined to discard the
2(
p
=
−
r
) smallest canonical correlations. We would thus construct the rank-
r
estimator
x
r
=
W
r
y
=
R
1
/
2
xx
FK
r
G
H
R
−
1
/
2
yy
y
,
(5.88)
which is similar to the solution of the min-trace problem (
5.77
) except that it uses a
canonical coordinate system in place of a half-canonical coordinate system. The rank-
r
matrix
K
r
is obtained from
K
by replacing the 2(
p
−
r
) smallest canonical correlations
with zeros.
As in the min-trace case, our intuition is correct but it still requires a proof that
the reduced-rank maximum mutual information estimator does indeed use the same
canonical coordinate system as the full-rank estimator. The proof given below follows
Hua
et al
. (2001
).
Starting from (
5.60
), maximizing mutual information means minimizing
(
W
−
W
r
)
R
yy
(
W
−
W
r
)
H
)
det
Q
r
=
det(
Q
+
det
R
xx
det
I
yy
)
H
.
(5.89)
CC
H
R
−
1
/
2
xx
W
r
R
1
/
2
R
−
1
/
2
xx
W
r
R
1
/
2
=
−
+
−
−
(
C
yy
)(
C
For notational convenience, let
R
−
1
/
2
xx
W
r
R
1
/
2
X
r
=
C
−
.
(5.90)
yy
The minimum det
Q
r
is zero if there is at least one canonical correlation
k
i
equal to 1.
We may thus assume that all canonical correlations are strictly less than 1, which allows
us to write
det
R
xx
det
I
−
CC
H
+
X
r
X
r
CC
H
)det
I
CC
H
)
−
H
/
2
.
CC
H
)
−
1
/
2
X
r
X
r
(
I
=
det
R
xx
det(
I
−
+
(
I
−
−
(5.91)
CC
H
) are independent of
W
r
, they can be disregarded in the
minimization. The third determinant in (
5.91
)isoftheform
Since det
R
xx
and det(
I
−
2
p
YY
H
)
2
i
)
+
=
+
σ
,
det(
I
(1
(5.92)
i
=
1
2
p
i
=
1
CC
H
)
−
1
/
2
X
r
. This amounts to mini-
where
{
σ
i
}
are the singular values of
Y
=
(
I
−
mizing
with respect to an arbitrary unitarily invariant norm. In other words, we need
to find the rank-2
r
matrix (
I
Y
CC
H
)
−
1
/
2
R
−
1
/
2
xx
W
r
R
1
/
2
CC
H
)
−
1
/
2
C
.This
−
closest to (
I
−
yy
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