Stability analysis of the RC-PLMS adaptive beamformer using a simple transfer function approximation

In this paper, we propose a discrete time transfer function approximation for the reduced complexity parallel least mean square (RC-pLMS) adaptive beamforming algorithm. The RC-pLMS is built using a single least mean square (LMS) stage whose inputs are obtained as a linear combination of the present and past sample. Thus, in order to numerically assess the RC-pLMS stability and to determine the approximate maximum parametric value of the step size for which it remains stable, we derive its discrete time transfer function approximate. In this approximation, the input uniform linear antenna array is remodeled as a finite impulse response (FIR) fractional delay Farrow filter. Computer simulations, presented by the mean square error and beam radiation pattern, demonstrates the validity of the transfer function approximate. Additionally, the RC-pLMS stability is evaluated, with respect to the pole-zero plot, for different step sizes and the approximate upper bound value of the step size is determined.