Digital Signal Processing Reference
In-Depth Information
without compromising optimality seriously. In Eq. (17.47) the quantity
P
e
(
k
)
is the specified error probability for the
k
th symbol stream
s
k
(
n
)
.
The constant
D
0
is such that the bit rate constraint
M−
1
1
M
b
k
=
b
(17
.
48)
k
=0
is satisfied. In Fig. 17.11 the
k
th signal power [
Λ
s
]
kk
is such that the probability
of error is kept at the specified level
P
e
(
k
). As shown in Ex. 14.1 of Chap. 14,
the appropriate value of [
Λ
s
]
kk
(also denoted as
P
k
in Chap. 14) is independent
of
k
,thatis,
[
Λ
s
]
kk
=2
D
0
.
(17
.
49)
This is a consequence of the fact that the free matrix
Σ
g
was chosen to be such
that the precoder is orthonormal (review Ex. 14.1 here). The minimized value
of the transmitted power is given by Eq. (14.30), and is reproduced here with
appropriate substitutions:
P
min
=
c
2
b
1
1
/M
,
(17
.
50)
M−
1
k
=0
|
C
[
k
]
|
2
where
c
=
M
(
k
c
k
)
1
/M
with
Q
−
1
P
e
(
k
)
4
2
c
k
=
σ
q
3
,
(17
.
51)
assuming a QAM constellation. We are now ready to summarize the main results
on optimal DMT systems:
Theorem 17.3.
Optimal DMT system with zero-forcing constraint.
Con-
sider a cyclic-prefix transceiver system with a specified set of error probabilities
{P
e
(
k
)
♠
and average bit rate
b
. Assume the transmitter allows bit allocation
among the signals
s
k
(
n
), as in the DMT system. Assume the precoder
F
, equal-
izer
G
,
and bit allocation are optimized under the zero-forcing constraint such
that the transmitted power is minimized subject to the specified error probabil-
ity constraints
}
. This optimized system has the same transmitted power
as the DFT-based DMT system shown in Fig. 17.11. Thus the DMT system
shown in Fig. 17.11 is optimal for power. This system is designed as follows:
{P
e
(
k
)
}
1. The bit allocation is chosen as in Eq. (17.47), where
D
0
is a constant such
that the average bit rate constraint (17.48) is satisfied.
2. The diagonal elements of
Λ
s
are chosen as in Eq. (17.49), so that the
specified
P
e
(
k
)issatisfied.
The minimized value of the transmitted power is given by Eq. (17.50).
♦
A number of results in this section were first presented in [Lin and Phoong,
2001b]. As a final remark, it should be noticed that the results of this section
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