Digital Signal Processing Reference
In-Depth Information
h
g
l
, =, , ,
l
1
…L
(8.99)
l
γσ
2
l
l
the optimal broadcasting problem can be written as [83]
2
H
H
min
w
ww
subjectto
wg
≥
1
for all
l
= ,… .
1
L
(8.100)
l
According to (8.100), the optimal broadcast weight vector is designed by minimizing
the total transmit power under the constraints that the minimum acceptable QoS is
guaranteed for each user.
Unfortunately, the problem in (8.100) is NP-hard [83], and hence it cannot be solved
in polynomial time. To obtain a reasonably simple approximate solution to (8.100), it
has been proposed in [83] to reformulate this problem in terms of
Z
ww
H
and
Q
l
g
l
g
l
H
as
min{}
tr
Z
subjectto r
{
ZQ
}
≥
1
for all
l
= , , ,
1
…L
l
Z
(8 .101)
H
=,
Z
Z
rank{}
Z
=
1
and then to replace the nonconvex constraint rank{
Z
} = 1 by its relaxed convex version
Z
0.
The so-obtained problem is a convex SDP problem. However, in contrast to the prob-
lem in (8.83), the rank and semidefinite constraints on
Z
are not equivalent to each
other; that is, the matrix
Z
opt
obtained by solving the relaxed SDP problem is not rank
one in general [83]. Therefore, to recover the optimal value of weight vector from
Z
opt
,
the so-called
randomization
approach was adopted in [83].
Another alternative broadcast beamforming problem setting in the case of fixed
transmit power constraint is to maximize the minimum receiver SNR subject to this
constraint. This problem can be written as [83]
L
2
H
2
H
maxmin
w
wh
σ
subjectto
ww
≤
P.
(8.102)
l
l
l
l
=
1
This problem can also be shown to be NP-hard, but it can be relaxed to a convex SDP
form in nearly the same way as problem (8.100); see [
83
] for more details.
A useful worst-case design-based robust modification of problem (8.100) has been
discussed in [84]. The authors of [84] have assumed the norm-bounded channel errors
ν
l
˜
l
-
g
l
;
ν
l
≤ κ (with ˜
l
and
g
l
being the actual and presumed values of the normal-
ized channel vector, respectively), and modified problem (8.100) as
2
H
H
min
ww
subjectto
wg
(
+ ≥
ν
)
1
for all
ν
≤, =
κ 1, , .
l
…L
(8.103)
l
l
l
w
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