Adaptive Beamforming for Multiantenna Communications - Adaptive Signal Processing in Wireless Communications - page 209

At first glance, the problem in (8.32) appears to be computationally intractable because
it involves minimization of a quadratic function subject to infinitely many nonconvex
quadratic constraints. However, it has been found in [38] that (8.32) can be converted to
a much simpler form:

ˆ

H H

min wrw subjectto w a ≥+ ε w 1 , (8.33)

1

w

where the constraint in (8.33) can be shown to hold with equality at the optimal point of
(8.33). The problem in (8.33) belongs to the class of convex second-order cone program-
ming (SOCP) problems [47-49], which can be easily solved using efficient interior point
methods [50] at complexity O ( M 3 ).
An alternative way to solve the problem in (8.33) is to resort to Newton-type algo-
rithms. Several approaches of that type (all with complexities O ( M 3 )) are described in
[46-52] to solve problem (8.33) and its extensions to the ellipsoidal uncertainty case
[46, 52], and to a more general class of beamformers or multiuser detectors [44, 51, 52].
In [45], the RMV beamformer of (8.33) has been generalized to account for interferer
nonstationarity in addition to the spatial signature errors. The essence of the approach
of [45] is, in addition to modeling the uncertainty in the spatial signature vector, to
model data nonstationarity by considering an uncertainty in the data matrix

Xx  [() 1 , …N x ( )] . (8.34)

The sample correlation matrix can be expressed through the data matrix (8.34) as

1
N

ˆ r X

=

H

. (8.35)

To take into account the nonstationarity of the array data, the so-called mismatch matrix

Δ x   X − (8.36)

was introduced in [45], where ~ and X stand for the actual and presumed data matrices,
respectively. Here, X is the data matrix acquired by the beamformer, while the actual
data matrix ~ may differ from X because of a nonstationary character of sensor array
snapshots. That is, the data samples in X can become irrelevant at the time when the
beamformer is used (and when the actual, yet unknown, data matrix is ~ rather than X ).
In such nonstationary scenarios, the actual sample correlation matrix is given by

ˆ r X 1 1

 

H

= = ( X + Δ x )( X + Δ x ) H . (8.37)

N N