Adaptive Beamforming for Multiantenna Communications - Adaptive Signal Processing in Wireless Communications - page 212

and therefore, the solution to the RMV beamforming problem (8.46) can be expressed as

{ } .

P ( ˆ

− 1

w = r + γ Ir I )( − ε ) (8 . 51)

RMV s

From (8.49) and (8.51) it follows that, although ∆ i W + satisfies (8.48), the constraint (8.47)
is always violated with ∆ s WC . Therefore, as expected, problem (8.46) is an approximate
(conservative) version of the worst-case SINR optimization problem.
Taking into account that r s = σ 1 2 a 1 a 1 H and absorbing σ 1 2 in ε (that is, assuming without
any loss of generality that σ 1 2 = 1), equation (8.51) can be rewritten as

{ } .

P ( ˆ

− 1 H

w

RMV =

r + γ Iaa )( − ε I ) (8.52)

11

From (8.51) and (8.52) it can be seen that the approach of [44] is equivalent to a com-
bination of negative and positive diagonal loading. In the case of ε = 0, (8.51) and (8.52)
simplify to the DLMV beamformer weight vectors (8.24) and (8.25), respectively.
An interesting interpretation of these two types of diagonal loading has been estab-
lished in [44], where it has been obtained that (8.51) can be interpreted as a generalization
of the DLMV beamformer whose amount of positive diagonal loading is scenario-
adaptive rather than fixed. Similar adaptive diagonal loading interpretation of the RMV
beamformer (8.33) has been found in [38], where it has been shown that the amount of
diagonal loading in the latter RMV beamformer is optimally matched to the uncertainty
in the spatial signature of the user of interest.
Although the worst-case RMV beamformers are known to be very robust techniques,
they might be overly conservative because the actual worst operational conditions may
occur in practice with a very low probability. Thus, obtaining less conservative robust
alternatives to the worst-case techniques is of significant interest. This motivated the
authors of [53-55] to develop an alternative, more flexible approach to RMV beam-
forming that can provide the robustness against spatial signature errors with a certain
selected probability (i.e., using soft probabilistic constraints rather than deterministic
worst-case constraints). In particular, the probabilistically constrained counterpart of
problem (8.32) can be written as [53]

{ } >

ˆ

H H

min wrw subjecttoPr w ( a +≥ δ ) 1 p , (8.53)

1

w

where δ is assumed to be a random vector drawn from some known distribution, Pr{·}
is the probability operator whose explicit form can be obtained from the statistical
assumptions on the steering vector error, and p is some preselected probability thresh-
old value. It can be seen that, in contrast to the hard constraint used in (8.32) (that
requires the distortionless response to be maintained for all norm-bounded error vec-
tors in the uncertainty set), the sot constraint in (8.53) maintains the distortionless
response only for the error vectors δ whose probability is high enough, while skipping