Robust Reduced-Rank Adaptive LCMV Beamforming Algorithms Based on Joint Iterative Optimization of Parameters - Advances in Heuristic Signal Processing and Applications - page 29

and P (i) (only for the RLS) for the maximum allowed rank D max and then the rank
adaptation algorithm determines the rank that is best for each time instant i using
the cost function in ( 2.32 ). The rank adaptation algorithm is then given by

D min ≤ d ≤ D max C S D (i

1 ) , D opt = arg min − 1 ), w D (i ¯ − (2.34)

where d is an integer, D min and D max are the minimum and maximum ranks allowed
for the reduced-rank beamformer, respectively. Note that a smaller rank may provide
faster adaptation during the initial stages of the estimation procedure and a greater
rank usually yields a better steady-state performance. Our studies reveal that the
range for which the rank D of the proposed algorithms has a positive impact on the
performance of the algorithms is limited, being from D min = 8forthe

reduced-rank beamformer recursions. These values are rather insensitive to the sys-
tem load (number of users) and the number of array elements, and work very well for
all scenarios and algorithms examined. The additional complexity of the proposed
rank adaptation algorithm is that it requires the update of all involved quantities with
the maximum allowed rank D max and the computation of the cost function in ( 2.32 ).
This procedure can significantly improve the convergence performance and can be
relaxed (the rank can be made fixed) once the algorithm reaches steady state. Choos-
ing an inadequate rank for adaptation may lead to performance degradation, which
gradually increases as the adaptation rank deviates from the optimal rank.

3to D max =

2.6 Simulations

In this section, the performance of the RJIO and some existing beamforming algo-
rithms is assessed using computer simulations. A sensor-array system with a ULA
equipped with M sensor elements is considered for assessing the beamforming al-
gorithms. In particular, the performance of the RJIO scheme with SG and RLS al-
gorithms is compared with existing techniques, namely, the full-rank LCMV-SG [ 4 ]
and LCMV-RLS [ 6 ], and the robust techniques reported in [ 9 ], termed WC-SOC,
and [ 10 ], called Robust-ME, and the optimal linear beamformer that assumes the
knowledge of the covariance matrix and the actual steering vector [ 2 ]. In particular,
the algorithms are compared in terms of the signal-to-interference-plus-noise ratio
(SINR), which is defined for the reduced-rank schemes as

w H (i) S D (i) R s S D (i) ¯

SINR (i) = ¯

w (i)
w H (i) S D (i) R I S D (i) w (i) ,

(2.35)

where R s is the covariance matrix of the desired signal and R I is the covariance
matrix of the interference and noise in the environment. Note that for the full-rank
schemes the SINR (i) assumes S D (i) I M . For each scenario, 200 runs are used

to obtain the curves. In all simulations, the desired signal power is σ d =

=

1, and