Robust beamforming via alternating iteratively estimating the steering vector and interference-plus-noise covariance matrix

https://doi.org/10.1016/j.dsp.2019.102620Get rights and content

Abstract

To develop an adaptive beamformer against the steering vector mismatch of the signal of interest (SOI), a novel robust algorithm is proposed to estimate the steering vector of the SOI and interference- plus-noise covariance matrix (INCM) in an alternative and iterative way. That is, via determining a convex optimization problem, which forces the steering vector moving towards the signal-plus- interference subspace (SIS) but getting away from the interference subspace (IS), the actual steering vector of the SOI is estimated. To proceed, the suitable SIS is easy to obtain through applying eigendecomposition on the sample covariance matrix while the appropriate IS is hard to estimate because of the array perturbations. Given this, a novel INCM reconstruction method, which utilizes a blocking matrix to eliminate the SOI from the training samples, is provided to realize the preferable estimate of the IS. More specifically, the abovementioned processes are carried out in an alternative iteration scheme, which leads to the SOI steering vector and INCM converging to the theoretical ones sufficiently, respectively. Unlike the conventional algorithms which are vulnerable to the various mismatches, the proposed beamforming algorithm is insensitive to the SOI steering vector mismatch arisen from the DOA error and array perturbations, numerous theoretical analysis and simulation experiments are presented to demonstrate the superiority of the proposed adaptive beamformer.

Introduction

Adaptive beamformer, which can adjust the weight vector in real time according to the signal environment, has drawn widespread attentions and been applied in several fields, such as radar, sonar, remote sensing, wireless communication, and satellite navigation [1], [2], [3], [4], [5], [6]. The classical standard Capon beamformer (SCB), as a famous adaptive beamformer, has remarkable resolution and interference suppression capability upon the assumption that the signal of interest (SOI) is absent from the training data and the steering vector of the SOI is known perfectly [7]. However, the SCB will suffer from degradation dramatically in case of the SOI involved in the array received data, which is called the signal self-nulling, when the steering vector mismatch of the SOI is present due to some factors, such as direction-of-arrival (DOA) error, antenna position displacement, and antenna gain and phase perturbations [8], [9], [10]. Therefore, the study for improving the robustness against the SOI steering vector mismatch of the SCB becomes fairly important.

During the past decades, numerous robust beamforming methods have been developed. For instance, the loading methods in [11], [12], [13], [14], [15], [16] and the weight norm constraint algorithms in [17], [18], [19] are known as popular techniques, including the quiescent diagonal loading (DL) method in [11] which involves adding a fixed identity matrix to the sample covariance matrix (SCM). The variable diagonal loading (VDL) method in [15] loads a variable matrix that just provides a loading factor to the minor eigenvalues of the SCM. The white noise gain constrained beamformer (WNGCB) in (19) improves the output performance using a noise power minimization-based weight vector optimization processing. Nevertheless, the loading methods and weight norm constraint algorithms cannot alleviate the steering vector mismatch of the SOI. To tackle this problem, the steering vector projection methods in [20], [21], [22] are promoted. The eigenspace-based beamformer (ESB) in [21] directly projects the SOI steering vector onto the signal-plus-interference subspace (SIS) to lessen the mismatch. But the subspace swap effect occurred at low signal-to-noise ratio (SNR) case leads to the increased steering vector mismatch of the SOI. Owning to the shortages aforesaid approaches, a class of beamformers based upon the steering vector of the SOI optimization estimation develops rapidly [23], [24], [25], [26], [27], such as the robust Capon beamforming (RCB) method in [23] whose core concept is to estimate the steering vector of the SOI in a user-defined uncertainty set by maximizing the array output power. Note that the RCB method also belongs to the family of loading methods, thus it cannot alleviate the steering vector mismatch. The sequential quadratic programming (SQP) beamformer in [24] aims to revise the steering vector of the SOI iteratively by maximizing the array output power with strong constraints. However, this method only suits for the situation of DOA error. Upon this, the array response control approaches are provided to maintain nearly flat response over the region of the SOI [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. For example, the worse-case performance optimization (WCPO) approach is present in [28], which forces the unity magnitude responses to the vectors around the nominal SOI steering vector, to ensure that the actual steering vector of the SOI can enjoy unabated gain. Unfortunately, in case of the indeterminate steering vector error bound, the situation, that the constrained area always involves the actual steering vector of the SOI, cannot be guaranteed. The mainbeam control beamformer (MBCB) in [34] applies the gain and phase constraints in the mainlobe area to widen the undistorted response region of the SOI, thereby improves the robustness against the SOI steering vector mismatch. But it is inevitable that the beam resolution of this robust approach will be significantly reduced.

Actually, the training data, which is contaminated by the SOI, causes performance deterioration due to the SOI steering vector mismatch at high SNR case. In other words, obtaining the SOI-free sample data [38], [39], [40], [41], [42], [43], [44] or reconstructing the interference-plus-noise covariance matrix (INCM) [45], [46], [47], [48], [49], [50], [51], [52], [53], [54] has the potential to significantly improve the performance of the SCB although the steering vector mismatch of the SOI exists. Under this condition, the two layers beamforming (TLB) method in [42] constructs the DOA extension-based SOI blocking matrix to converts the training data into SOI-free samples under sub-array level, which can significantly overcome the large DOA error at the cost of reduced degrees of freedom (DOF). In [44], the multiple constrained l2-norm minimization algorithm removes the SOI by forming the blocking matrix with a tiny power adjust factor and the presumed SOI steering vector. Even this algorithm has low complexity, its SOI blocking performance at strong SNR case cannot be of assurance, which causes the output SINR drop. Different from the SOI elimination based beamformers above, the INCM-quadratically constrained quadratically programming (INCM-QCQP) algorithm in [45] reconstructs the INCM with the Capon spectrum estimator in the spatial region outside the region of the SOI, which can acquire good performance without array perturbations. To reduce the high computational load of processing the spectrum estimation in the INCM-QCQP algorithm, the INCM reconstruction via spatial power spectrum sampling (INCM-SPSS) beamformer is investigated in [48] to give a rapid and easy INCM estimation way, but its performance is a little worse than that of the INCM-QCQP method. In [50], a novel subspace algorithm for INCM reconstruction (NS-INCM) provides to pre-estimate the steering vectors of the interferences through Capon spectrum estimator in the known small spatial regions, and then utilizing the subspace projection technique to yield the enhanced ones, which somewhat improves the accuracy on estimating the interference steering vectors. But the decreased ability in rejecting strong interferences under the defective array structure circumstance is still unchecked. To further handle the INCM reconstruction problem at array geometry mismatch case, the INCM-steering vector estimation (INCM-SVE) algorithm in [51] reconstructs the INCM with each steering vector of the interference estimated by the iterative RCB (IRCB) method in [25], and then estimates the steering vector of the SOI with a convex optimization problem, whose main purpose is to obtain the steering vector nearest the orthogonal space of the interference subspace (IS). However, the negative aspects, including the INCM mismatch resulted from the disadvantage of the RCB method and the steering vector imprecision derived from the nonorthogonality between the actual SOI steering vector and the IS, severely hit the output signal-to- interference-plus-noise ratio (SINR). To proceed, the interference steering vector and power estimation-INCM (ISVPE-INCM) approach in [52] utilizes the similar reconstruction idea as INCM-SVE method to obtain the INCM, then forms a different steering vector of the SOI optimization model via maximizing the array output power, together with keeping the steering vector parallel to the subspace spanned by the mainlobe steering vectors. Apart from the INCM mismatch, a distinct weakness of this algorithm is that in the event of array perturbations, the actual steering vector of the SOI is rarely contained in the ideal manifold-based mainlobe subspace. Given all these unsolved problems, an INCM reconstruction method based on subspace bases transition (INCM-SBT) is put forward to counter the antenna position displacement in [53], which derives an optimization problem to obtain the SIS bases to finish the INCM construction. Whereas the INCM-SBT algorithm does not fit other common circumstances like the antenna gain and phase perturbations and mutual coupling effect.

It's noteworthy that in general the aforesaid methods are vulnerable to the steering vector mismatch of the SOI arisen from both DOA error and array geometry perturbations. Even worse, the methods, which determinate the weight vector through the SCM, would improperly reject the true SOI during the interference suppression at high SNR case. In light of these shortcomings, we prepare to design a new beamformer, which is robust against both DOA and array imperfections, insensitive to the presence of the SOI in the training data, and thereby able to significantly improve the output SINR.

In this paper, a novel robust beamforming algorithm is proposed to cope with the steering vector mismatch of the SOI by presenting an effective scheme for alternating iteratively estimating the steering vector of the SOI and INCM. This algorithm determines the actual steering vector of the SOI by searching for the steering vector which is closest to the SIS but away from the IS. In order to estimate the accurate IS with considering the array imperfections, an original INCM reconstruction method which employs a blocking matrix to separate the SOI from the training data is introduced in details. Therefore, through initializing the steering vector of the SOI as the principal eigenvector of the Capon spectrum-based SOI matrix, this algorithm achieves both the estimates of the SOI steering vector and INCM converging to the actual by applying alternative iteration technique. Combining the above steps together, this algorithm achieves significant improvements on estimating the steering vector of the SOI and INCM underlying the scenarios, where both the DOA of the SOI and array structure information are not precisely given. Numerical experiments are executed to demonstrate that the proposed robust beamformer can outperform other existing beamformers and it can almost attain the optimal performance.

The paper contributes to the field of adaptive beamforming in the following aspects,

(1) We propose a SOI steering vector optimization problem, via compelling the steering vector nearing the SIS but standing off the IS with significant objective function and constraints, which is manifestly different from the existing subspace projection or uncertainty set methods and fundamentally a problem of determining the faithful signal subspace (SS). Therefore, the steering vector mismatch of the SOI can be conquered by solving the given optimization problem.

(2) We devise an original INCM reconstruction method to cope with array perturbations by separating the SOI component from the training data with a blocking matrix, where the blocking matrix is formed with the estimated SOI steering vector and its pre-defined power. Thus, the quasi INCM is calculated by means of the SOI-absent data. And then, the dominant eigenvectors related to the quasi INCM are processed by the inversion matrix of the blocking matrix, which results in the INCM reconstruction. Therefore, the signal self-nulling at strong SNR case and anti-interference performance loss at array imperfection status can be simultaneously avoided.

(3) We provide an iteration scheme to alternate estimating the steering vector of the SOI and INCM in terms of that the theoretical IS employed in the optimization problem is difficult to achieve, especially in the array deficiency circumstance. That is, via constructing a blocking matrix and performing some concise matrix transitions, the accurate INCM is obtained to realize the IS. Then, the steering vector of the SOI is renewed through solving the optimization problem. It is remarkable that the aforementioned steps are repeated until the steering vector of the SOI and INCM trend towards the theoretical ones, which leads to the noteworthy output SINR enhancement.

(4) We exhibit the analysis on the convergence feature and computational complexity of the proposed approach. And we also give the performance comparisons of the proposed and relevant beamforming algorithms through typical experiments. Apparently, the proposed robust adaptive beamformer acquires prominent improvement on estimating the steering vector of the SOI and INCM.

The remainder of this paper is organized as follows. In Section 2 and Section 3, the signal model and state-of-art beamforming methods are introduced, respectively. The proposed adaptive beamforming algorithm is specified in Section 4. In Section 5, numerical simulation experiments are carried out to verify the performance of the proposed beamformer. Section 6 concludes the entire paper at length.

Section snippets

Signal model

Assume a linear array with M antenna elements, receiving narrowband far-field signals including one SOI from θ0 and J interferences from θi, i=1,2,,J. The array observation complex vector at discrete time t can be modeled as:x(t)=a0s0(t)+i=1Jaisi(t)+n(t) where ai, i=0,1,,J and si(t), i=0,1,,J denote the steering vector and waveform of the ith source, respectively, n(t) is the additive Gaussian white noise. Here the SOI, the interferences, and the noise are assumed to be statistically

Conventional optimization algorithm

The SCB is quite sensitive to the mismatch between the presumed and actual steering vectors of the SOI. To address this issue, Hassanien et al. have created the SOI steering vector optimization problem as below [24]:mine(a˜0+e)HRˆ1(a˜0+e)s.t.P(a˜0+e)=0(a˜0+e)HC¯(a˜0+e)a˜0HC¯a˜0||a˜0+e||22Ma˜0He=0 where P=ILLH is the orthogonal projection, L denotes the subspace spanned by the major eigenvectors of C=ΘaaHdθ (Θ is the SOI angular region, a denotes the steering vector from θ), and C¯

Proposed algorithm

In this section, a robust beamforming algorithm based on iteratively estimating the steering vector of the SOI and INCM is elaborated. We first establish a subspace-based convex optimization problem to estimate the steering vector of the SOI. And then, we achieve the INCM reconstruction by introducing a signal blocking matrix to eliminate the SOI from the training data, which follows the estimate of the IS indeed. Finally, through repeating the abovementioned steps in turn, the proposed robust

Simulation

In this section, representative simulations are put into action to testify the effectiveness of the proposed AIERB algorithm. A uniform linear array (ULA) composed of eight omnidirectional antenna elements spaced half a wavelength apart is considered (i.e. M=8). There are one SOI and two interferences. Two strong interferences with interference-to-noise ratios (INR) of 30 dB impinge on the ULA from the directions 25 and 45, respectively.

To complete the proposed AIERB algorithm, the SOI

Conclusion

The problem of adaptive beamforming in the situation of the steering vector mismatch of the SOI is fully considered in this paper. Through reconstructing the INCM with the constructed signal blocking matrix, we achieve the estimation on the IS. With that, we estimate the actual steering vector of the SOI by solving the proposed subspace-based convex optimization problem upon an alternative and iterative scheme. Moreover, the convergence feature and computational complexity of the proposed

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Zhiwei Yang was born in Sichuan, China, in 1980. He received the Ph.D. degree in electric engineering from Xidian University, Xi'an, China, in 2008. He is currently a professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include adaptive array signal processing, space-time-polarmetric processing, and the design of ground moving target indication system.

References (58)

  • A.F. Liu et al.

    An eigenvector based method for estimating DOA and sensor gain-phase errors

    Digit. Signal Process.

    (August 2018)
  • I.S. Reed et al.

    Rapid convergence rate in adaptive arrays

    IEEE Trans. Aerosp. Electron. Syst.

    (November 1974)
  • R. Lorenz et al.

    Robust minimum variance beamforming

    IEEE Trans. Signal Process.

    (January 2005)
  • J.L. Yu et al.

    Generalized eigen-subspace based beamformers

    IEEE Trans. Signal Process.

    (February 1995)
  • A.B. Gershman et al.

    Experimental performance of adaptive beamforming in a sonar environment with a towed array and moving interfering sources

    IEEE Trans. Signal Process.

    (January 2000)
  • F.H. Zhou et al.

    Robust AN-aided beamforming and power splitting design for secure MISO cognitive radio with SWIPT

    IEEE Trans. Wirel. Commun.

    (April 2017)
  • J. Capon

    High-resolution frequency wave number spectrum analysis

    Proc. IEEE

    (August 1969)
  • S. Shahbazpanahi et al.

    Robust adaptive beamforming for general-rank signal models

    IEEE Trans. Signal Process.

    (September 2003)
  • A. Khabbazibasmenj et al.

    Robust adaptive beamforming based on steering vector estimation with as little as possible prior information

    IEEE Trans. Signal Process.

    (June 2012)
  • H. Ruan et al.

    Robust adaptive beamforming based on low-rank and cross-correlation techniques

    IEEE Trans. Signal Process.

    (August 2016)
  • J. Li et al.

    On robust Capon beamforming and diagonal loading

    IEEE Trans. Signal Process.

    (July 2003)
  • X. Mestre et al.

    Finite sample size effect on minimum variance beamformers: optimum diagonal loading factor for large arrays

    IEEE Trans. Signal Process.

    (January 2006)
  • M. Zhang et al.

    Robust adaptive beamforming based on conjugate gradient algorithms

    IEEE Trans. Signal Process.

    (November 2016)
  • L. Du et al.

    Fully automatic computation of diagonal loading levels for robust adaptive beamforming

    IEEE Trans. Aerosp. Electron. Syst.

    (January 2010)
  • J. Zhuang et al.

    Low-complexity variable loading for robust adaptive beamforming

    Electron. Lett.

    (March 2016)
  • H. Cox et al.

    Robust adaptive beamforming

    IEEE Trans. Acoust. Speech Signal Process.

    (October 1987)
  • Z. Tian et al.

    A recursive least squares implementation for LCMP beamforming under quadratic constraints

    IEEE Trans. Signal Process.

    (June 2001)
  • L. Chang et al.

    Performance of DMI and eigenspace-based beamformers

    IEEE Trans. Antennas Propag.

    (November 1992)
  • D.D. Feldman et al.

    A projection approach for robust adaptive beamforming

    IEEE Trans. Signal Process.

    (April 1994)
  • Cited by (0)

    Zhiwei Yang was born in Sichuan, China, in 1980. He received the Ph.D. degree in electric engineering from Xidian University, Xi'an, China, in 2008. He is currently a professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include adaptive array signal processing, space-time-polarmetric processing, and the design of ground moving target indication system.

    Pan Zhang was born in Shaanxi, China, in 1993. He received the M.S. degree in electric and communication engineering from Xidian University, Xi'an, China, in 2019. He is currently an associate engineer with the Beijing Institute of Radio Measurement. His research interests includes adaptive array signal processing and space-time adaptive processing.

    Guisheng Liao was born in Guilin, China, in 1963. He received the Ph.D. degrees in signal and information processing from Xidian University, Xi'an, China, in 1992. He is currently a professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include synthetic aperture radar, space-time adaptive processing, ground moving target indication, and distributed small satellite synthetic aperture radar system design.

    Chongdi Duan was born in Shaanxi, China, in 1972. He received the M.S. degree in signal and information processing from Xidian University, Xi'an, China, in 2005. He is currently a professor with the National Key Laboratory of Science and Technology on Space Microwave, China Academy of Space Technology. His research interests include radar signal waveform design, adaptive array signal processing, and mobile target detection.

    Huajian Xu was born in Fujian, China, in 1990. He received the Ph.D. degree in signal and information processing from Xidian University, Xi'an, China, in 2018. He is currently an engineer with the Nanjing Electronic Equipment Institute. His research interests include synthetic aperture radar, ground moving target indication, and space-time adaptive processing.

    Shun He was born in Hunan, China, in 1980. She received the Ph.D. degree signal and information processing from Xidian University, Xi'an, China, in 2016. She is currently an associate professor with the Communication and Information Engineering Collage, Xi'an University of Science and Technology. Her research interests include adaptive array signal processing and wideband signal processing.

    This work was supported in part by the National Natural Science Foundation of China under grant 61671352, the National Science Foundation for Young Scientists of China under grants 61801373, 61701395, and 61501471, and the Foundation of Key Laboratory of Cognitive Radio and Information Processing (Guilin Science and Technology University), Ministry of Education under grant CRKL160206.

    View full text