Digital Signal Processing Reference
In-Depth Information
so that
0
]
Σ
h
U
h
U
h
Σ
h
Σ
f
=
Σ
f
[
Σ
h
]
M
,
(
HF
)
†
HF
=[
Σ
f
0
where [
Σ
h
]
M
is the
M
×
M
leading principal submatrix of
Σ
h
.
Thus the optimal
equalizer is
0
]
Σ
h
U
h
=
Σ
−
f
[
Σ
h
]
−
M
[
U
h
]
M×P
.
G
=[
Σ
h
]
−
M
Σ
−
2
[
Σ
f
(12
.
61)
f
The equalizer can be simplified a little bit by observing that
[
Σ
h
]
−
M
1
σ
f,k
σ
h,k
1
cσ
1
/
2
h,k
Σ
−
1
f
=
=
kk
where we have used Eq. (12.53). The equalizer matrix can therefore be written
in the form
0
]
U
h
,
G
=[
Σ
g
(12
.
62)
where
⎡
⎣
⎤
⎦
σ
−
1
/
2
h,
0
0
...
0
σ
−
1
/
2
h,
0
...
0
Σ
g
=
1
c
=
1
1
c
(
Σ
h
)
−
1
/
2
.
(12
.
63)
.
.
.
.
.
.
M
σ
−
1
/
2
h,M−
1
0
0
...
As a reminder, the optimal precoder is
F
=
V
h
Σ
f
0
,
(12
.
64)
where
Σ
f
is
M
×
M
diagonal with diagonal elements (12.53):
⎡
⎤
σ
−
1
/
2
h,
0
0
...
0
⎣
⎦
σ
−
1
/
2
h,
1
0
...
0
=
c
(
Σ
h
)
−
1
/
2
M
Σ
f
=
c
.
(12
.
65)
.
.
.
.
.
.
σ
−
1
/
2
h,M−
1
0
0
...
Summarizing the above results, we have proved the following:
Theorem 12.1.
Optimal ZF-MMSE transceiver.
The solution to the ZF-
MMSE transceiver optimization problem described in Eqs. (12.12)-(12.14) can
be summarized as follows:
♠
1.
Precoder.
The optimal precoder has the form (12.64), where (a)
V
h
is the
unitary matrix that occurs in the SVD of the channel Eq. (12.57) (i.e., it
diagonalizes
H
†
H
as in Eq. (12.58)), and (b)
Σ
f
is the diagonal matrix
shown in (12.65), where
σ
h,k
are the first
M
dominant singular values of
the channel
H
.
The constant
c
is such that the power constraint is satisfied.
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