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

In this paper, shrinkage linear complex-valued least mean squares (SL-CLMS) and shrinkage widely linear complex-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 considerably 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.

I. Introduction

In adaptive filtering applications for modeling, equalization, control, echo cancellation and beamforming, the complex-valued least mean squares (CLMS) algorithm is a well-known adaptive estimation and prediction technique which is capable of converging to the optimal Wiener solution [1]. The application of the CLMS algorithm to the beamforming and its analysis have been extensively studied [1]–[4]. The weight vector of the adaptive beamformer can be computed based on different kinds of design criteria. The most promising criteria include the minimum mean-squared error (MMSE) [3], minimum variance [5] and constant modulus [6]. In this paper, we focus on the scenario where the MMSE criterion is applied to the adaptive beamforming system because it only requires the training sequence of the desired signal.