Elsevier

Signal Processing

Volume 93, Issue 9, September 2013, Pages 2543-2547
Signal Processing

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Adaptive beamforming based on covariance matrix reconstruction by exploiting interferences' cyclostationarity

https://doi.org/10.1016/j.sigpro.2013.02.009Get rights and content

Abstract

Adaptive beamforming is known to be sensitive to array system mismatch, especially when the sample covariance matrix is used and the desired signal is present in the training snapshot. To alleviate the sensitivity, in this paper, the sample covariance matrix is replaced by the interference-plus-noise covariance matrix (INCM), which is reconstructed by exploiting the cyclostationarity of interference signals. In contrast to the existing INCM reconstruction methods, the proposed technique is based on the knowledge of the interferences' cycle frequencies and needs no information of the array structure, thus it can deal with unknown perturbations in the array. The numerical simulations show that the proposed method improves the robustness of adaptive beamformers and has superior performance to the existing INCM reconstruction methods especially for strong interferences.

Highlights

► The interference-plus-noise covariance matrix (INCM) is reconstructed for adaptive beamforming. ► The INCM is reconstructed by exploiting the cyclostationarity of interferences. ► The proposed method needs no information of the array structure. ► The proposed method can deal with unknown array perturbations.

Introduction

Adaptive beamforming has been used in many areas, such as radar, sonar, microphone array speech processing and wireless communications [1]. Beamformers are designed for desired signal enhancement and interference suppression. The well-known minimum variance distortionless response (MVDR) beamformer is obtained by preserving a unity gain for the desired signal while minimizing the power of interference and noise [1]. Theoretically, MVDR beamformer's weight is a function of the interference-plus-noise covariance matrix (INCM). In practice, the INCM is unavailable, it is then replaced by the sample covariance matrix [2]. However, since the desired signal component is usually included in the sample covariance matrix, the resultant beamformer is known to be sensitive even to a slight mismatch between the presumed system model and the actual one [2], [3]. Such model mismatch is common in practice due to direction of arrival (DoA) mismatch, imperfect array calibration, source spreading, etc. [3].

During the past decade, many approaches by exploiting the properties or prior information of the array have been proposed to improve the performance of adaptive beamforming [2], [3], [4], [5], [6], [7], [8], [9], [10]. For example, the reduced-rank technique is used in [4] to achieve a fast convergence rate. However, the system mismatch issue is not considered in [4]. Thus, a blind and robust interference suppression algorithm is proposed in [5] which relies on the constant modulus property. In [6], [7], [8], several robust approaches against model mismatch are proposed by processing the steering vector of the desired signal. Nevertheless, the problem in the covariance matrix is not well addressed.

Besides the steering vector processing based approaches, an alternative robust beamforming design is based on the processing of the sample covariance matrix. One popular technique is the diagonal loading [2], where a scaled identity matrix is added to the sample covariance matrix. However, the desired signal is still not removed from the sample covariance matrix. Most recently, Mallipeddi et al. [9] and Gu and Leshem [10] addressed this problem by reconstructing the INCM rather than finding a diagonal level for the sample covariance matrix. Since the desired signal is removed from the reconstructed INCM, the resultant beamformers are shown to have a near optimal performance in case of steering vector mismatch of the desired signal. However, these methods need to know the array structure. When the array has unknown gain and phase errors, the interferences will be less suppressed by the beamformer, which lead to a performance penalty, especially in the case of strong interferences.

In this paper, we consider the INCM reconstruction for adaptive beamforming by exploiting the cyclostationarity of interference signals. Different from the constrained cyclic adaptive beamforming (CCAB) [12] which exploits the cyclostationarity of the desired signal, we take into account the cyclostationarity of the interferences, and use this property to estimate the steering vectors of the interferences. By further estimating the power of the interferences and noise, the INCM can be reconstructed. In contrast to the previous INCM reconstruction methods, no knowledge of array structure is needed for the proposed technique, thus, it can deal with the unknown perturbations in the array. The effectiveness of the proposed method is demonstrated by numerical simulations.

Notation: Boldfaced capital and lowercase letters denote matrices and the column vectors, respectively. (·)H is the Hermitian transpose, (·)T is the transpose, (·) is the complex conjugate. · represents the Frobenius norm.

Section snippets

Problem formulation

Consider an array of N elements. Suppose P+1 (P+1<N) far-field signals impinging on the array, the received signal at time k is given asx(k)=p=0Papsp(k)+n(k)where ap is the steering vector of the pth source signal sp(k), n(k) is the noise. Here, s0(k) is considered as the desired signal while sp(k),p=1,,P are interferences. The output signal-to-interference-plus-noise ratio (SINR) of the beamformer is given asSINR=σ02|wHa0|2wHRinwwhere w is the beamformer weight, Rin is the INCM, σ02 is the

Cyclostationarity and CCAB

A signal s(k) is said to be cyclostationary if its cyclic conjugate or cyclic correlation function is defined respectively asrss(α,τ)=s(k)s(k+τ)ej2παkrss(α,τ)=s(k)s(k+τ)ej2παkis nonzero at cycle frequency α for some time shift τ [12], [11], where ·=limK(1/K)k=1K(·). Most man-made signals exhibit cyclostationarity with cycle frequency equal to the twice of the carrier frequency, multiples of the baud rate, or combinations of these [12].

For the array signal model (1), the cyclic

Simulations

To examine the performance of the proposed algorithm, simulations have been done based on a uniform linear array with N=10 sensors and inter-element spacing is half the wavelength of the desired signal. We consider one desired signal, two interferences which are all binary phase-shift keying (BPSK) signals with rectangular pulse. The baud rates for the desired signal and the interferences are respectively 1/10, 1/30 and 1/10 with respect to the 30 MHz sampling frequency. The carrier frequencies

Conclusion

An adaptive beamforming algorithm based on INCM reconstruction is proposed by exploiting the cyclostationarity of interferences. With the reconstructed INCM, the robustness of adaptive beamformer can be improved since the desired signal is removed. The proposed reconstruction method needs no information of array structure and can deal with the unknown perturbations in the array. The proposed method is shown to have a superior performance to the existing INCM reconstruction based beamformers in

References (17)

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This work was supported by the National Natural Science Foundation of China (61201349, U1035003), Natural Science Foundation of Guangdong Province (s2011040001426) and Science and Technology Project of Guangzhou (2012J2200005).

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