Improved Proportionate Constrained Normalized Least Mean Square for Adaptive Beamforming

Abstract

A minimum-disturbance description of adaptive beamforming algorithms opens up the derivation of novel methods for linearly constrained settings. Under this context, this paper proposes two algorithms, the IPCNLMS and the IPCNLMS-$\ell _{0}$, and discusses their application for adaptive beamforming using a uniform rectangular array. These algorithms combine both proportionate and norm constraint strategies that are inserted in the traditional CNLMS algorithm in a smooth and rigorous way. The idea of jointly implementing CNLMS and IPNLMS algorithms is used to achieve faster convergence and the ability to attenuate interfering signals from multiple directions while using adaptive beamforming. The IPCNLMS updates each filter coefficient independently by adjusting the adaptation step size proportionally to the magnitude of the estimated filter coefficient. It is based on the $\ell _{1}$-norm penalty to exploit the convergence speed of the system. To take this idea further, the $\ell _{0}$-norm penalty is also considered in the IPCNLMS-$\ell _{0}$ proposition. Simulations demonstrate that the proposed algorithms present faster convergence, under equivalent conditions of asymptotic performance, even when more realistic coupling effects between the array elements are taken into account. Beam pattern results show that the proposed algorithms are capable of achieving the optimum solution from LCMV as well as the CNLMS.