Robust Reduced-Rank Adaptive LCMV Beamforming Algorithms Based on Joint Iterative Optimization of Parameters - Advances in Heuristic Signal Processing and Applications - page 21

A reduced-rank algorithm must extract the most important features of the pro-
cessed data by performing dimensionality reduction. This mapping is carried out by
a M

× D rank-reduction matrix S D on the received data as given by

S D r (i), ¯ r (i) = (2.6)

where, in what follows, all D -dimensional quantities are denoted with a “bar”. The
resulting projected received vector

r (i) is the input to a beamformer represented by ¯

T . The filter output is

the D × 1 vector w ¯ =[¯ w 1 ¯ w 2 ... w D ] ¯

w H

x(i) ¯ = ¯ r (i). ¯ (2.7)

In order to design a reduced-rank beamformer w we consider the following opti- ¯

mization problem

minimize E ¯ r (i)

= ¯

2

w H w H

R

¯ w ¯

(2.8)

w H subject to ¯ a (θ k ) ¯ = 1 .

The solution to the above problem is

R − 1

¯

a (θ k )

w opt =

¯

a H (θ k ) R − 1 ¯ a (θ k ) ¯

( S D RS D ) − 1 S D a (θ k )

=

a H S D (θ k )( S D RS D ) − 1 S D a (θ k ) ,

(2.9)

where the reduced-rank covariance matrix is R r H (i) S D RS D and the

= E [¯ r (i) ¯ ]=

S D a (θ k ) . The above development shows
that the choice of S D to perform dimensionality reduction on r (i) is very impor-
tant, and can lead to an improved convergence and tracking performance over the
full-rank beamformer. A key problem with the full-rank and the reduced-rank beam-
formers described in ( 2.5 ) and ( 2.9 ), respectively, is that their performance is dete-
riorated when they employ the presumed array steering vector a p (θ k ) .Inthesesit-
uations, it is fundamental to employ a robust technique that can mitigate the effects
of the mismatches between the actual and the presumed steering vector.

reduced-rank steering vector is a (θ k ) ¯ =

2.3.2 Robust Adaptive LCMV Beamformers

An effective technique for robust beamforming is the use of diagonal loading strate-
gies [ 7 , 9 , 12 , 13 ]. In what follows, robust full-rank and reduced-rank LCMV beam-
forming designs are described. A general approach based on diagonal loading is
employed for both full-rank and reduced-rank designs.