Digital Signal Processing Reference
In-Depth Information
q
(
n
)
0
/
H
0
1/
α
0
α
0
s
(
n
)
0
s
(
n
)
0
/
H
1
q
(
n
)
1
1/
α
1
α
1
s
(
n
)
1
s
(
n
)
1
/
H
M
−1
q
(
n
)
M
α
M
−1
1/
α
M
−1
−
1
s
(
n
)
M
−1
s
(
n
)
M
−1
Figure 11.4
. Redrawing the channel paths for further interpretation.
actual choice of
α
k
is as in Eq. (11.14), and this results in
σ
q
k
|
E
k
=
γ
H
k
|
2
for some constant
γ.
That is, the error variances are not equal, but their
original distribution
σ
q
k
/
H
k
|
2
is replaced by the square root distribution.
So the optimal choice of
α
k
is called a
half-whitening system
.Inshort,
under the zero-forcing constraint, half-whitening minimizes reconstruction
error. A similar phenomenon arises in certain data compression problems
[Jayant and Noll, 1984].
|
11.3 Minimizing MSE without ZF constraint
We now reconsider the optimization problem with the zero-forcing constraint
removed. This will lead to a smaller mean square error. Such systems are called
pure-MMSE systems to distinguish them from ZF-MMSE systems. Unlike in
a ZF-MMSE system, we will see that the pure-MMSE system forces an upper
bound on the error, independent of the channels (i.e., regardless of how large
σ
q
k
are, or how small
H
k
are).
The MMSE system is studied in considerable detail in Chap. 22 where we
develop it as an example of the application of the Karush-Kuhn-Tucker (KKT)
theory of optimization. In Chap. 22 we consider the example where
H
k
=1
for all
k
(with
σ
q
k
allowed to be different for different
k
). Theresultscanbe
modified readily for the case of arbitrary
H
k
as elaborated next.
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