Shrinkage Linear and Widely Linear Complex-Valued Least Mean Squares Algorithms for Adaptive

Abstract—In this paper, shrinkage linear complex-valued least mean squares (SL-CLMS) and shrinkage widely linear com-plex-valued least mean squares (SWL-CLMS) algorithms are devised for adaptive beamforming. By exploiting the relationship between the noise-free a posteriori and a priori error signals, the SL-CLMS method is able to provide a variable step size to update the weight vector for the adaptive beamformer, significantly enhancing the convergence speed and decreasing the steady-state misadjustment. On the other hand, besides adopting a variable step size determined by minimizing the square of the augmented noise-free a posteriori errors, the SWL-CLMS approach exploits the noncircular properties of the signal of interest, which consid-erably improves the steady-state performance. Simulation results are presented to illustrate their superiority over the CLMS, complex-valued normalized LMS, variable step size, recursive least squares (RLS) algorithms and their corresponding widely linear-based schemes. Additionally, our proposed algorithms are more computationally efficient than the RLS solutions though they may have a slightly slower convergence rate. Index Terms—Complex-valued least mean squares (CLMS), convergence speed, shrinkage, steady-state, variable step size, widely linear. I