Robust Adaptive Beamforming

Introduction

In this chapter, we first review the background, basic principle and structure of adaptive beamformers. Since there are many robust adaptive beamforming methods proposed in literature, for easy understanding, we organize them into two categories from the mathematical point of view: one is based on quadratic optimization with linear and nonlinear constraints; the another one is max-min optimization with linear and nonlinear constraints. With the max-min optimization technique, the state-of-the-art robust adaptive beamformers are derived. Theoretical analysis and numerical results are presented to show their superior performance.

Background

The array signal processing has been studied for some decades as an attractive method for signal detection and estimation in hash environment. An array of sensors can be flexibly configured to exploit spatial and temporal characteristics of signal and noise and has many advantages over single sensor. It has many applications in radar, radio astronomy, sonar, wireless communication, seismology, speech acquisition, medical diagnosis and treatment (Tsoulos, 2001) (Krim & Viberg, 1996) (Van Veen & Buckley, 1988), etc.

There are two kinds of array beamformers: the fixed beamformer and the adaptive beamformer. The weight of the fixed beamformer is pre-designed and it does not change in applications. The adaptive beamformer automatically adjusts its weight according to some criteria. It significantly outperforms the fixed beamformer in noise and interference suppression. The typical adaptive beamformer is the linearly constrained minimum variance (LCMV) beamformer (Compton, 1988) (Hudson, 1981) (Johnson & Dudgeon, 1993) (Monzingo & Miller, 1980) (Naidu, 2001). A famous representative of the LCMV is the Capon beamformer (Capon, 1969). In ideal cases, the Capon beamformer has high performance in interference and noise suppression provided that the array steering vector (ASV) is known. However, the ideal assumptions of adaptive beamformer may be violated in practical applications (Vural, 1979) (Jablon, 1986a) (Jablon, 1986b) (Cox, Zeskind, & Owen, 1988) (Chang & Yeh, 1993). The performance of the adaptive beamformers highly degrades when there are array imperfections such as steering direction error, time delay error, phase errors of the array sensors, multipath propagation effects, wavefront distortions. This is known as the target signal cancellation problem. Tremendous work has been done to improve the robustness of adaptive beamformer (Nunn, 1983) (Er & Cantoni, 1983) (Buckley & Griffiths, 1986) (Er & Cantoni, 1986c) (Er, 1988) (Thng, Cantoni, & Leung, 1993) (Thng, Cantoni, & Leung, 1995) (Zhang & Thng, 2002) (Er & Cantoni, 1986b) (Er & Cantoni, 1990) (Cox, Zeskind, & Owen, 1987) (Vorobyov, Gershman, & Luo, 2003) (Lorenz & Boyd, 2005) (Li, Stoica, & Wang, 2003) (Stoica, Wang, & Li, 2003) (Affes & Grenier, 1997) (Er & Ng, 1994).