Digital Signal Processing Reference
In-Depth Information
q
y
s
M
x
P
J
F
H
channel
Figure 19.22
. A MIMO channel with a linear precoder.
Recall now that
1
E
k
1
E
k,br
+
1
σ
s
=
(19
.
132)
where
E
k,br
is the mean square error after bias removal (Sec. 16.3.2). In our
case,
E
k
=
E
ave
for all
k
,so
E
k,br
=
E
ave,br
for all
k,
where
1
E
ave
1
E
ave,br
1
σ
s
=
+
Thus Eq. (19.130) yields
1+
.
M−
1
σ
s
E
ave,br
I
(
x
;
y
)=
log
(19
.
133)
2
k
=0
This is the mutual information between the input and output of the channel. To
understand this expression in the proper light, observe that the reconstructed
vector
s
(at the detector input, after bias removal) can be written as
s
=
s
+
e
,
(19
.
134)
where the error
e
is zero-mean with covariance
R
ee
=
E
ave,br
I
.
(19
.
135)
s
as an (approx-
imately) AWGN channel with transfer matrix
I
,
and noise covariance
R
ee
=
E
ave,br
I
. This is demonstrated in Fig. 19.23. This is a set of identical, parallel,
independent channels. This explains in an independent way that the mutual in-
formation between
s
and
s
(with
s
chosen to be zero-mean circularly symmetric
Gaussian with covariance
σ
s
I
) is indeed of the form (19.133).
So, in the optimal system we can regard the path from
s
to
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