Digital Signal Processing Reference
In-Depth Information
Table I.9. Circulant channel with bit allocation (Chap. 17)
The circulant channel with bit allocation at the transmitter arises in DMT sys-
tems. The optimal linear transceiver which minimizes transmitted power with
bit allocation under the
zero-forcing
condition was derived in Sec. 17.5. The
optimum solution is shown in the figure below, where
W
is the DFT matrix.
Thus the optimal precoder is automatically orthonormal in this problem set-up!
Since
H
is circulant, we have
H
=
W
−
1
Λ
c
W
.
The inverse
Λ
−
c
appears in the
optimal receiver. In what follows
C
[
k
] denotes the
k
th diagonal element of
Λ
c
.
q
(
n
)
σ
2
covar
.
I
s
(
n
)
s
(
n
)
W
M
W
M
Λ
−
1
H
Λ
s
covar
.
normalized
IDFT
circulant
channel
normalized
DFT
DFT domain
equalizers
The optimum bit allocation formula is
σ
q
3
2
+log
2
|
Q
−
1
P
e
(
k
)
4
|
2
,
b
k
=
D
0
−
log
2
C
[
k
]
(I
.
34)
where
P
e
(
k
) is the specified error probability for the
k
th component of
s
(
n
)
.D
0
is chosen such that
M−
1
k
=0
b
k
/M
=
b
for fixed
b.
The minimized power is
P
min
=
c
2
b
1
1
/M
,
(I
.
35)
M−
1
k
=0
|
C
[
k
]
|
2
where
c
=
M
(
k
c
k
)
1
/M
,
with
c
k
=(
σ
q
/
3)[
P
e
(
k
)
/
4)]
2
assuming a QAM
constellation. The optimal choice of user powers [
Λ
s
]
kk
happens to be iden-
tical for all
k,
and in fact
Λ
s
=2
D
0
I
.
Thus the precoder can be taken to be
orthonormal without loss of optimality, and furthermore
Λ
s
is proportional to
identity.
Q
−
1
(
Search WWH ::
Custom Search