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Let us summarize. From the analysis model we say that every composite signal and
measurement vector [
x
T
y
T
]
T
may be modeled
as if
it were a virtual two-channel
experiment, wherein the signal is subtracted from a linearly filtered measurement to
produce an error that is orthogonal to the measurement. Of course, the filter and the
error covariance must be chosen just right. This model then decomposes the composite
covariance matrix for the signal and measurement, and its inverse, into block Cholesky
factors.
,
5.3.3
Filtering models
We might say that the models we have developed give us two alternative parameteriza-
tions:
R
xy
R
−
1
the
signal-plus-noise channel model
(
R
xx
,
H
,
R
nn
) with
H
=
and
R
nn
=
xx
R
xy
R
−
1
R
yy
−
xx
R
xy
; and
R
xy
R
−
1
the
measurement-plus-error channel model
(
R
yy
,
W
,
Q
) with
W
=
yy
and
Q
=
R
xy
R
−
1
yy
R
xy
.
R
xx
−
These correspond to the two factorizations (
A1.2
) and (
A1.1
)of
R
xy
, and
R
nn
and
Q
are the Schur complements of
R
xx
and
R
yy
, respectively, within
R
xy
. Let's mix the
synthesis equation for the signal-plus-noise channel model with the analysis equation of
the measurement-plus-error channel model to solve for the filter
W
and error covariance
matrix
Q
in terms of the channel parameters
H
and
R
nn
:
e
y
−
I0
HI
x
n
IW
0I
=
.
(5.36)
This composition of maps produces these factorizations:
Q0
0R
yy
−
I0
HI
R
xx
0
0
nn
IH
H
0I
−
IW
0I
I0
W
H
=
,
(5.37)
I
Q
−
1
0
0
−
1
yy
−
I
R
−
1
I 0
−
−
H
H
0I
I0
W
H
−
xx
0
0
−
1
nn
IW
0I
=
.
(5.38)
I
HI
We now evaluate the northeast block of (
5.37
) and the southwest block of (
5.38
) to obtain
two formulae for the filter
W
:
R
xx
H
H
(
HR
xx
H
H
R
nn
)
−
1
(
R
−
1
xx
H
H
R
−
1
nn
H
)
−
1
H
H
R
−
1
W
=
+
=
+
.
(5.39)
nn
In a similar fashion, we evaluate the northwest blocks of both (
5.37
) and (
5.38
) to get
two formulae for the error covariance
Q
:
R
xx
H
H
(
HR
xx
H
H
R
nn
)
−
1
HR
xx
=
(
R
−
1
xx
H
H
R
−
1
nn
H
)
−
1
Q
=
R
xx
−
+
+
.
(5.40)
These equations are Woodbury identities, (
A1.43
) and (
A1.44
). They determine the
LMMSE inversion of the measurement
y
for the signal
x
. In the absence of noise, if the
signal and measurement were of the same dimension, then
W
would be
H
−
1
, assuming
that this inverse existed. However, generally
WH
=
I
, but is approximately so if
R
nn
is
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