ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 115

The beamformer weight vector w is chosen with an aim that it is statistically opti-
mum in some sense. Naturally, different design objectives lead to different beamfor-
mer weight vectors. For example, the weight vector for the classic beamformer is

a
a H a

w BF W

where a ¼ a ( ~ u ) denotes the array response for fixed look direction u . The classic
Capon's [7] MVDR beamformer chooses w as the minimizer of the output power
while constraining the beam response along a specific look direction u of the SOI
to be unity

mi w w H C ( z ) w subject to w H a ¼ 1 :

The well-known solution to this constrained optimization problem is

w C ( z ) W C ( z ) 1 a
a H C ( z ) 1 a :

(2 : 29)

Observe that Capon's beamformer weight vector is data dependent whereas the classic
beamformer weight w BF is not, that is, w C ( ) is a statistical functional as its value
depends on the distribution F of z via the covariance matrix C ( F ). The spectrum
(2.28) for the classic and Capon's beamformers can now be written as

P BF ( u ) W a ( u ) H C ( z ) a ( u ) (2 : 30)

P CAP ( u ) W [ a ( u ) H C ( z ) 1 a ( u )] 1

(2 : 31)

respectively. (See Section 6 in [55]). Note that MVDR beamformers do not make any
assumption on the structure of the covariance matrix (unlike the subspace-methods of
the next section) and hence can be considered as a “nonparametric method” [55].
In practice, the DOA estimates for the classic and Capon's beamformer are calcu-
lated as the d highest peaks in the estimated spectrums P BF ( u ) and P CAP ( u ), where the
true unknown covariance matrix C ( z ) is replaced by its conventional estimate, the
SCM C . An intuitive approach in obtaining robust beamformer DOA estimates is to
use robust estimators instead of the SCM in (2.30) and (2.31), for example, the
M -estimators of scatter. Rigorous statistical robustness and efficiency analysis of
MVDR beamformers based on M -estimators of scatter is presented in Section 2.7.

2.6.2 Subspace Methods

A standard assumption imposed by subspace methods is that the additive noise n is
spatially white, that is C ( n ) ¼ s 2 I . We would like to stress that this assumption
does not imply that n is second-order circular, that is, n can have non-vanishing