Digital Signal Processing Reference
In-Depth Information
Table I.5. Linear transceivers with ZF and bit allocation (Chap. 14)
In the transceiver below assume
R
qq
=
σ
q
I
and that the components
s
k
(
n
)of
s
(
n
)are
zero-mean uncorrelated processes representing independent users with powers [
Λ
s
]
kk
=
P
k
.
Assume the
k
th user transmits
b
k
-bit symbols, and the desired error probability
is
P
e
(
k
)
.
Assume that zero forcing is in effect, that is
GHF
=
I
.
q
(
n
)
2
covar
.
I
σ
q
J
P
s
(
n
)
s
(
n
)
M
M
F
G
H
Λ
covar
.
s
equalizer
precoder
channel
Under the high-bit rate assumption (Sec. 14.2), the user power required to achieve the
error probability
c
k
2
b
k
[
GG
]
kk
,
where
c
k
=(
σ
q
/
3)[
≤P
e
(
k
)is
P
k
≈
Q
−
1
(
P
e
(
k
)
/
4)]
2
(assuming QAM). The total transmitted power is given by
M
−
1
c
k
2
b
k
[
F
†
F
]
kk
[
GG
†
]
kk
.
P
trans
=
(I
.
24)
k
=0
For fixed
b
=
k
b
k
/M,
the jointly optimal choices of bit allocation
{b
k
}
, precoder
F
,
and equalizer
G
which minimize this power are as follows (Sec. 14.4):
σ
q
3
2
Q
−
1
P
e
(
k
)
4
b
k
=
D −
log
2
+log
2
σ
h,k
,
(I
.
25)
[
Σ
h
]
−
M
,
G
=[
U
h
]
M×J
,
and
F
=
V
h
(I
.
26)
0
where
σ
h,k
are the
M
dominant channel singular values (nonzero owing to the ZF
assumption). The unitary matrices above come from the channel SVD, that is,
H
=
U
h
Σ
h
V
h
. The notation [
A
]
M×J
denotes the top-left
M
×
J
submatrix of
A
,
and
A
M
denotes the top-left
M
×
M
submatrix. The minimized transmitted power is given by
P
min
=
c
2
b
1
/M
1
M−
1
k
.
(I
.
27)
σ
h,k
=0
If the above
for arbitrary nonsingular
diagonal matrix
Σ
g
, the system continues to be optimal with the above bit allocation
(Sec. 14.5.1). The optimal solution is such that the channel eigenmodes receive the
same power for any diagonal
Σ
g
(Sec. 14.6.3.A). In particular, if we choose [
Σ
g
]
kk
=
1
/σ
h,k
then the precoder has orthonormal columns (that is,
F
†
F
=
I
), all equalization
is at the receiver, and
P
k
is identical for all users (
Λ
s
=
α
I
; see Ex. 14.1).
{
F
,
G
}
pair is replaced with
{
FΣ
−
g
,
Σ
g
G
}
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