Adaptive Beamforming for Multiantenna Communications - Adaptive Signal Processing in Wireless Communications - page 220

2

H

wh

i ici , ()

SINR i = .

(8 . 81)

L

∑

2

2 H

σ + wh

i m icm , ()

mmi =≠ 1 ;

The traditional approach to optimizing the beamformer weight vectors is to minimize
the total transmit power under the constraints guaranteeing that an acceptable QoS is
provided for each user. Mathematically, this problem can be expressed as [11, 14]

L

∑

H

min
{}

ww subjectto ≥ γ for all i =, , 1 …L , (8.82)

SINR

l l i

i

l L

w

= 1

l = 1

where γ i is the minimum acceptable QoS for the i th user. This is a quadratic optimi-
zation problem with quadratic nonconvex constraints. Different algorithms have been
proposed in the literature to solve (8.82). For example, the algorithms of [11] and [70]
separate the problem into the following two subproblems: downlink power control and
downlink beamforming with norm-one weight vectors. Using this representation of the
problem at hand, the approaches of [11] and [70] propose to find the solution to (8.82) in
an iterative way, by finding the optimal norm-one weight vectors for given power levels,
and then updating the power levels based on these weight vectors.
Another powerful approach to solve (8.82) has been proposed in [71] (see also [ 14 ] )
using convex optimization. The key idea of this approach is to reformulate problem
(8.82) in a convex form as follows. Using the notation W i  w i w i H , problem (8.82) with
the user SINRs (8.80) can be transformed to [14, 71]

L

∑ ∑

2

min tr { W } subjecttotr { r W } − γ tr { rW } ≥ , γσ

l i ci , () i i icmm , () i i

l L

(8.83)

{} W

= 1

l = 1 mi ≠

H

WW W =,  0 for all i =, , 1 …L ,

i i i

where the last two constraints guarantee that the matrices W i are Hermitian and posi-
tive semidefinite for all i = 1, …, L . Note that the last constraint can be viewed as a con-
vex relaxation of the nonconvex rank-one constraint rank { W i } = 1.
The problem in (8.83) belongs to the class of convex semidefinite programming (SDP)
problems, and therefore, it can be efficiently solved using modern convex optimization
tools [50]. Moreover, it has been proved in [14] that for this problem, W i  0 is exactly
equivalent to rank { W i } = 1, and therefore, the replacement of the latter constraint by the
former one is not a relaxation but actually an equivalent reformulation of the problem.
After obtaining the optimal values for W i , i = 1, …, L , this property enables recovery of
the optimal weight vectors w i , i = 1, …, L in a simple way, from the principal eigenvectors
of W i , i = 1, …, L .