Digital Signal Processing Reference
In-Depth Information
for optimizing channel capacity. The formulas are different because the first
term in Eq. (10.33) is not a constant. The construction in Eq. (10.33) can at
best be regarded as “pseudo” water filling.
Let
F
(
e
jω
)
|
2
is nonzero, and
F
be the set of frequencies in [
−
π, π
] for which
|
c
be the complementary set. From Eqs. (10.31) and (10.33) we have
p
0
=
γ
F
let
F
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
dω
2
π
−
S
qq
(
e
jω
)
dω
2
π
H
(
e
jω
)
|
2
|
H
(
e
jω
)
|
2
F
so that
p
0
+
S
qq
(
e
jω
)
|
dω
2
π
H
(
e
jω
)
|
2
F
γ
=
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|H
(
e
jω
)
|
2
,
(10
.
34)
dω
2
π
F
An expression for the minimized error
E
mse
can be obtained by substituting Eq.
(10.33) into Eq. (10.30) which yields
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
+
F
1
γ
dω
2
π
S
ss
(
e
jω
)
dω
E
mmse
=
2
π
,
H
(
e
jω
)
|
2
F
c
where the second term arises from the fact that when
F
(
e
jω
)=0,theintegrand
in Eq. (10.30) reduces to
S
ss
(
e
jω
). Substituting for
γ
from Eq. (10.34) yields
the final expression
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
2
dω
2
π
+
H
(
e
jω
)
|
2
S
ss
(
e
jω
)
dω
2
π
F
E
mmse
=
(10
.
35)
p
0
+
S
qq
(
e
jω
)
|
dω
2
π
F
c
H
(
e
jω
)
|
2
F
Since the mean squared error has been minimized without the zero-forcing con-
straint, this is called the
pure-MMSE
solution.
10.3.2 MMSE transceiver with zero forcing (ZF-MMSE)
The zero-forcing constraint in Fig. 10.5 implies
F
(
z
)
H
(
z
)
G
(
z
)=1
.
(10
.
36)
In this case the reconstruction error is simply the noise
q
(
n
) filtered by
G
(
e
jω
)
.
The error spectrum is therefore
S
qq
(
e
jω
)
S
ee
(
e
jω
)=
S
qq
(
e
jω
)
G
(
e
jω
)
|
2
=
|
|
F
(
e
jω
)
|
2
|
H
(
e
jω
)
|
2
The mean squared reconstruction error is
E
mse
=
π
−π
S
qq
(
e
jω
)
dω
2
π
(10
.
37)
|
F
(
e
jω
)
|
2
|
H
(
e
jω
)
|
2
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