A fast adaptive reduced rank transformation for minimum variance beamforming

Abstract

In conventional reduced rank minimum variance beamformer (RRMVB), a rank-reducing transformation is usually obtained from eigenvectors of the estimated sample covariance matrix, while the eigenvectors are usually obtained via eigen-decomposition. To alleviate the computational burden caused by eigen-decomposition, a fast reduced rank minimum variance beamformer (FRRMVB) is proposed in this paper. In the estimated covariance matrix case, a set of receive data vectors are taken as a rough and fast estimate of the true interference subspace, and the rank-reducing transformation is chosen as the augmentation of the estimated interference subspace with the steering vector of the desired signal. As FRRMVB performs without eigen-decomposition, it requires less computational load and is easier to be executed in practical applications compared with the conventional RRMVB. Moreover, it has good performance even with small sample size. Simulation results demonstrate the efficiency of the proposed method. The proposed method can be used for real-time adaptive array processing.

Introduction

The adaptive array processing technique is widely used in radar, wireless communications, sonar, acoustics, and medicine. It has received considerable attention in the past decades. Adaptive beamforming is an effective technique for removing directional interference from the data of an array of sensors, as in radar or sonar. To obtain a high spatial resolution and good beamforming performance, an antenna array with a large number of antenna elements should be used in radar. The dimension of the adaptive weight vector for space time adaptive processing (STAP) radar also can be large. For a large dimension adaptive array processing, computational burden and high-rate data transmission are two bottlenecks in the implementation of an adaptive beamforming system.

Many techniques have been proposed to alleviate these bottlenecks [1], [2], [3]. Among them, a widely considered one is the partially adaptive processing technique, which utilizes a fraction of the available adaptive dimensions of an array for adaptation. This results in reducing the required computational load and increasing the convergence speed. Many partially adaptive processing methods have been proposed [4]. They can be classified into two main architectures: the reduced rank minimum variance beamformer (RRMVB) and the reduced rank generalized sidelobe canceler (RRGSC) [4]. For both of the RRMVB and RRGSC, with different rank-reducing transformation T, adaptation can take place in the subspace spanned by the principal components (PCs) of the transform [5], [6], [7], or in the noise subspace (eigen-canceler) [8], [9], or based on cross-spectral metric (CSM) [10], [11], [12].

The principal components inverse (PCI) method exploits the low rank structure of the interference for rapid adaptive nulling of interference [8]. In the PCI method, the weight vector $w=(I_N-{\hat{U}}_L{\hat{U}}_L^H)s$ is orthogonal to the interference subspace (where s is the steering vector of the desired signal, and the columns of ${\hat{U}}_L$ span the interference subspace of the sample covariance matrix $\hat{R}$). It is equivalent to the rank-reducing transformation $T={\hat{U}}_n$ (where the columns of ${\hat{U}}_n$ span the noise subspace of $\hat{R}$). Eigencanceler based RRMVB method is equivalent to the PCI method. It is derived from the interference subspace constraint, and is referred to as principal components sample matrix inversion (PC-SMI) [6]. In PC-SMI method, the weight vector is constrained to the signal-plus-interference subspace, the rank-reducing matrix T is obtained by $T={\tilde{U}}_L$ (${\tilde{U}}_L$ consists of the principal components (PCs) of covariance matrix ${\tilde{R}}_1=Ps{s}^H+\hat{R}$, where the desired signal is present, and P is the power of the desired signal). Considered that if the look direction is in the noise subspace of the transform T, i.e., T^Hs=0, there is no solution that meets the linear constraint. To alleviate this problem, rank-reducing transformation T is designed as $T=\left[{\hat{U}}_{L}s\right]$ in [7], where ${\hat{U}}_L$ is r PCs of the estimated covariance matrix $\hat{R}$. The columns of T can also be designed by using the CSM approach [10]. In CSM based RRMVB method, the eigenvectors with the greatest impact on the output SINR of MVB are chosen as the columns of the transformation matrix T. All of these reduced rank methods are eigen-based methods. The computational cost of eigen-based methods is an issue due to the high computational cost associated with the eigen-decomposition (O(N³)), where N is the number of adaptive degrees of freedom, especially when N is large.

The family of Krylov subspace methods has been investigated in recent years, including the multistage Wiener filter (MWF) [13], [14] and the auxiliary-vector filter (AVF) with orthogonal auxiliary vectors [15], [16]. This family of methods projects the observation onto a lower dimensional Krylov subspace, and it can avoid the high computational cost associated with eigen-decomposition. However, how to obtain the block matrix efficiently and fast is still an open problem for the multistage Wiener filter (MWF) method.

The Hung–Turner projection (HTP) [17], [18] has been proposed as a computationally efficient eigen-canceler method for computing adaptive array weights. The HTP is a simple and fast method to estimate the interference subspace without eigen-decomposition. The computational load of the HTP is about O(L³+2L²N+(L+1)N²) (L is the number of snapshot vectors which are used to estimate the interference subspace). Since part of the noise subspace is also used to suppress the interference, the performance of this method is inferior to that of an eigen-analysis based method, and is affected by overestimating the rank of the interference subspace. In [20], the proposed scheme works on an instantaneous basis, selecting the best suited set of basis functions at each instant to minimize the squared error. However, the performance of this method is restricted by the sets of basis functions. Recently, several adaptive reduced-rank filtering schemes based on joint iterative optimization of adaptive filters with a low-complexity implementation using iterative adaptive algorithms are proposed [1], [3], [21], [22]. The computational complexity of the system is reduced substantially to O(Nr) for each iteration, where r is the dimension of the reduced-rank filter. For this kind of method, how to obtain appropriate convergence factor is still a problem.

In this paper, a fast reduced rank minimum variance beamformer (FRRMVB) is proposed to estimate the interference subspace in a computationally efficient manner. First, when the actual covariance matrix is known, we prove that if the rank-reducing transformation T is chosen as [U_Ls] (U_L is the real interference subspace), the RRMVB has the same output SINR as the full rank beamformer. In practice, the covariance matrix is estimated from a limited sample size. The FRRMVB uses a set of receiving data vectors as a fast estimate of the interference subspace. It is a direct form analog to the HTP method. The FRRMVB rank-reducing matrix is estimated interference subspace augmented by the steering vector of the desired signal. The computational load of FRRMVB is on the order of O((L+1)³)+O(N(L+1)), where L is the number of snapshot vectors which are used to estimate the interference subspace and usually $L\ll N$ for large scale array. As FRRMVB performs without eigen decomposition, it requires less computational load and is easier to execute in practical applications compared with the conventional RRMVB. Moreover, it has good performance even with small sample size, and is less affected by overestimating the rank of interference subspace than the HTP method. Simulation results demonstrate the efficiency of this proposed method. FRRMVB can be used for real-time adaptive array processing. Here, we just consider fast reduced rank adaptive beamforming based on the minimum variance beamformer (MVB). As MVB is a direct form perspective of the generalized sidelobe canceler (GSC) [10], one can also consider fast reduced rank adaptive beamforming method based on GSC.

The rest of this paper is organized as follows. Section 2 contains background material of the reduced rank minimum variance beamformer. In Section 3, the new proposed fast adaptive reduced rank transformation for minimum variance beamforming is discussed in detail. Some numerical studies are presented in Section 4 to illustrate the effectiveness of the proposed fast reduced rank beamforming algorithm. Finally, a brief conclusion is given in Section 5.