Elsevier

Signal Processing

Volume 92, Issue 1, January 2012, Pages 163-169
Signal Processing

Robust forward backward based beamformer for a general-rank signal model with real-valued implementation

https://doi.org/10.1016/j.sigpro.2011.07.001Get rights and content

Abstract

A robust forward backward (FB) beamformer for an incoherently scattered general-rank signal model with real-valued implementation is proposed based on the uniform linear array (ULA) structure by introducing a preprocessing transformation matrix. With the preprocessing, the computational complexity is reduced significantly due to a real-valued close-form solution for the optimum weight vector; furthermore, the proposed algorithm can achieve a higher output SINR (signal-to-interference-plus-noise ratio) and has lower sensitivity to the diagonal-loading parameters especially for low input SNR (signal-to-noise ratio) scenarios.

Highlights

► A robust beamformer for an incoherently scattered general-rank model is proposed. ► It is based on the uniform linear array structure with forward backward processing. ► A real-valued close-form solution is provided by introducing a preprocessing transformation. ► Computational complexity of the system is reduced significantly. ► It has achieved a higher output signal-to-interference-plus-noise ratio.

Introduction

Beamforming is a technique for receiving the signal of interest from some specific directions while suppressing the interfering signals from other directions using an array of sensors (Note that the same technique can also be used for signal transmission) [1], [2]. One of the most commonly used beamformers is the linearly constrained minimum variance (LCMV) beamformer [3]. However, in practice, the performance of an LCMV beamformer degrades significantly when the sample size of the received signal is small or the steering vector of the desired signal has a mismatch. Various robust algorithms have been proposed in the past decades, such as the diagonal-loading-based beamformer and the eigenvector-based beamformer [4], [5]. A drawback of the diagonal-loading-based beamformer is that it is not clear how to choose the optimum diagonal-loading factor, while the eigenvector-based beamformer suffers from the subspace swap phenomenon in low signal-to-noise ratio (SNR) environments. In [6], the worst-case optimization-based robust beamformer was proposed. By estimating the real steering vector while maximizing the beamformer output power, a robust Capon beamformer was proposed in [7] and then implemented in a recursive form in [8]. By generalizing the signal covariance matrix into a higher-rank (non-point source) one [9], a robust approach for general-rank signal models was derived, which was further developed in [10] with positive semidefinite constraints.

One key issue in many of the beamforming algorithms is the estimation of data covariance matrix based on finite samples. Based on the specific structure of uniform linear arrays (ULAs) and the resultant persymmetric structure of their covariance matrices, a forward backward (FB) averaging method was proposed for covariance matrix estimation with a significant performance improvement [11], [12]. It was then exploited in different applications such as directional of arrival (DOA) estimation [13], [14], spectrum estimation [15], and adaptive beamforming [16]. It is further shown in [17] that the optimum weight vector of a ULA (or the more general case: symmetrically distributed arrays) has a generalized conjugate symmetric property, which is employed to form a constrained beamforming problem and leads to improved performance. However, the conjugate symmetric property of the optimum weight vector will be destroyed in the presence of model perturbations, such as array mutual coupling, sensor position errors, discrepancies in sensor responses, etc., which leads to a degraded performance. Therefore, there is a need to develop a robust FB algorithm against all kinds of model errors.

In this paper, we will first propose a robust FB beamformer, based on a general-rank signal model and a ULA structure; then by introducing a transformation matrix to preprocess the received array data, the original complex-valued optimum weight vector will be reduced to a real-valued one, which reduces computational complexity of the system significantly.

Section snippets

Beamforming based on general-rank signal models

Consider a linear array with M sensors, the nth snapshot vector x[n] of the received array signals can be expressed asx[n]=s[n]+i[n]+n[n],where s[n], i[n] and n[n] are the desired signal, interference and noise vectors, respectively. By applying a set of coefficients wi, i=0,,M1 to the received array signals x[n], we obtain the beamformer outputy[n]=wHx[n],where {·}H denotes the Hermitian transpose operation and w is the weight vector.

In the absence of model errors, the optimum solution of

Robust forward backward beamformer

The general-rank signal model in Section 2 is based on an arbitrary array structure. Now we focus on the ULA structure and suppose there are M omnidirectional sensors with an adjacent sensor spacing d and a signal wavelength λ0. We can see that DOA angle mismatch and incoherent scattering of the desired signal do not destroy the centrohermitian structure of Rs, i.e.JRsJ=σ02π/2π/2ρ(θ)Ja(θ)aT(θ)Jdθ=σ02π/2π/2ρ(θ)a(θ)aH(θ)dθ=Rs,where {·}T is the transpose operation, a(θ)=[1,ejψ(θ),,ej(M1)ψ

Simulations and results

In this part we will examine the performance of the proposed algorithms and the original one in terms of output SINR. The optimum output SINR is given bySINR=woptHRswoptwoptHRinwoptwith Rin being the correlation matrix of interference plus noise, and wopt=P{R1Rs} being the optimum weight vector.

Our simulations are based on a ULA with 10 elements and an adjacent sensor spacing λ0/2. There is an interfering signal arriving from the DOA angle θ1=30° and the desired signal is an incoherently

Conclusion

A forward backward-based robust beamformer with real-valued implementation has been proposed for the incoherently scattered general-rank signal model, by introducing a unitary transformation matrix for preprocessing. As a result, computational complexity of the original general-rank robust beamformer is reduced by at least 50%. Moreover, simulation results have shown that the proposed beamformer is less sensitive to the involved parameters and has achieved a higher output SINR at low input SNR

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