Robust Reduced-Rank Adaptive LCMV Beamforming Algorithms Based on Joint Iterative Optimization of Parameters - Advances in Heuristic Signal Processing and Applications - page 22

A robust full-rank LCMV beamformer represented by an M × 1 vector w can be

designed by solving the following optimization problem
minimize E w H r (i)

2 + ε 2

2

w H Rw + ε 2 w H w

w =

(2.10)

w H a (θ k )

=

subject to 1 ,

where ε 2 is a constant that needs to be chosen by the designer. The solution to the
problem in ( 2.10 ) is given by

( R + ε 2 I M ) − 1 a p (θ k )
a p (θ k )( R

w opt =

ε 2 I M ) − 1 a p (θ k ) ,

(2.11)

+

where a p (θ k ) is the presumed steering vector of the SoI and I D is an M -dimensional
identity matrix. It turns out that the adjustment of ε 2 needs to be obtained numeri-

cally by an optimization algorithm.
In order to design a robust reduced-rank LCMV beamformer

¯

w , we follow a sim-
ilar approach to the full-rank case and consider the following optimization problem
minimize E ¯ w H S D r (i)

2 +

w H S D RS D ¯

ε 2

2

S D ¯ w = ¯ w

ε 2 w H S D S D ¯

(2.12)

+ ¯ w

w H S D a p (θ k ) = subject to ¯ 1 .

The solution to the above problem is

( S D RS D + ε 2 I D ) − 1 S D a p (θ k )

w opt =

a p S D (θ k )( S D RS D + ε 2 I D ) − 1 S D a p (θ k ) ,

(2.13)

where the tuning of ε 2 requires an algorithmic approach as there is no closed-form
solution and I D is a D -dimensional identity matrix.

2.4 Robust Reduced-Rank Beamforming Based on Joint
Iterative Optimization of Parameters

In this section, the principles of the robust reduced-rank beamforming scheme based
on joint iterative optimization of parameters, termed RJIO, are introduced. The RJIO
scheme, depicted in Fig. 2.2 , employs a rank-reduction matrix S D (i) with dimen-
sions M

× D to perform dimensionality reduction on a data vector r (i) with dimen-

sions M × 1. The reduced-rank beamformer w (i) with dimensions D ¯ × 1 processes

the reduced-rank data vector r (i) in order to yield a scalar estimate ¯ x(i) . The rank- ¯

reduction matrix S D (i) and the reduced-rank beamformer w (i) are jointly optimized

in the RJIO scheme according to the MV criterion subject to a robust constraint that

¯