Adaptive Beamforming for Multiantenna Communications - Adaptive Signal Processing in Wireless Communications - page 223

and ξ determines the bound on the norm of downlink correlation matrix errors. To
account for the CSI mismatch, it is proposed in [14] and [80] to enforce the QoS con-
straints for all possible values of the downlink channel correlation matrices that satisfy
(8.95). Using this approach, the robust modification of (8.83) can be expressed as

L

∑ ∑

{ } − { } ≥,

2

min tr { W } subjecttotr ( r − ξ IW ) γ tr ( r + ξ I W ) γσ

l i ci , () i i icm , () m i i

L

{} W

l l = 1

l = 1 mi ≠

H

W ==, WW

i

0 forall i =, ,. 1 …L (8.97)

i i

Problem (8.97) is also a convex SDP problem that can be solved at the same complexity as
(8.83). It can be interpreted as the worst-case modification of the nonrobust problem (8.83).
A much simpler yet suboptimal approach to robust unicast downlink beamform-
ing has been recently proposed in [81]. The essence of this technique is, following the
ideas of the algorithms of [11], [70], and [82], to split the transmit beamforming problem
into the power control and weight vector optimization subproblems, and then to uti-
lize simple decentralized algorithms for the weight vector optimization, while using a
centralized worst-case optimization-based technique for adjusting the transmit powers.
This approach has been shown in [81] to have a performance comparable to that of [80],
while enjoying both a lower computational cost and a substantially reduced degree of
the required cooperation between the network BSs compared to the algorithm of [80].

8.4.2 Broadcast Transmit Beamforming
The traditional broadcasting strategy is to radiate the transmitted power isotropically or
using a fixed transmit beampattern. However, in such a case the transmit CSI about the
user channels is ignored, and as a result, the users with weak channels can experience a
severe QoS degradation. In future data broadcasting/multicasting applications, the CSI
for all the intended users is likely to be available at the transmitter. Therefore, this CSI
can be exploited to improve the performance with respect to the traditional broadcast-
ing scheme [83].
Let a single multiantenna transmitter with the weight vector w broadcast the signal
s ( t ) to L single-antenna users whose channels h l , l = 1, …, L , are known at the transmit-
ter. The signal received at the l th user can be modeled as (8.4), and the received SNR at
the l th user is given by

wh 2
2

H

l

SNR =

. (8.98)

l

σ

l

The SNRs in (8.98) can be viewed as the user QoS values. Let γ l be the minimum accept-
able QoS value for the l th user. Then, defining the normalized channel vectors