Adaptive Beamforming for Multiantenna Communications - Adaptive Signal Processing in Wireless Communications - page 204

8.3.2.1 The SMI Algorithm
In practical applications, the true matrix r i+n is unavailable but can be estimated from
the receiver data* as [6, 26, 28]

N

1

∑

ˆ

H

r = x () t x () t , (8.18)

N

t = 1

where N is the number of snapshots available. The key idea of the sample matrix inver-
sion (SMI)-based version of the MV beamformer is to replace r i+n by (8.18) [28]. For the
MV beamforming problems in (8.14) and (8.15), the SMI weight vectors are given by
[26, 28]

P{ ˆ

− 1

w = r r }, (8.19)

MV s

ˆ

MV = −

1

wra , (8.20)

1

respectively.
The use of ˆ instead of r i+n in (8.19) or (8.20) is known to cause severe performance
degradation in the case when the desired signal component is present in the beamformer
snapshots used to estimate the sample correlation matrix (in the sequel, the latter case is
referred to as the signal-present case as opposed to the signal-free case). Although in the
signal-free case the output SINR of the SMI beamformer (8.20) rapidly converges to the
optimal SINR value with increasing N [28], in the signal-present case this convergence
becomes much slower [29], and the beamformer performance degrades severely even in
the presence of small errors between the presumed and actual spatial signatures of the
user of interest [27, 30]. Such signature errors can be caused by high user mobility and
wireless channel variability, as well as by a limited amount of training symbols and the
effect of multiuser interference that may prevent obtaining spatial signature estimates of
acceptable quality. The effect of the beamformer performance degradation due to spatial
signature errors is commonly known as signal self-nulling .
Another typical cause of the beamformer performance degradation in mobile com-
munications is a highly nonstationary behavior of the propagation channel and, in par-
ticular, high mobility of interfering users [31]. The effect of such nonstationarity on the
performance of receive adaptive beamformers is that the array weights may not be able
to adapt fast enough to compensate for the interfering user motion [30, 31]. This phe-
nomenon is usually referred to as interference undernulling .

* These data can be either information-bearing or training symbols. Therefore, there is no need
to increase the amount of training data when using the SMI-based MV beamformers in multi-
antenna wireless communication systems.