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

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Zhiwei Yang

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, Pan Zhang , Guisheng Liao

a, b

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, Chongdi Duan , Huajian Xu , Shun He

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a

National Laboratory of Radar Signal Processing, Xidian University, Xi’an, China b Collaboration Sensing Centre of Information and Understanding, Xidian University, Xi’an, China c Beijing Institute of Radio Measurement, Beijing, China d National Key Laboratory of Science and Technology on Space Microwave, China Academy of Space Technology, Xi’an, China e Nanjing Electronic Equipment Institute, Nanjing, China f Communication and Information Engineering College, Xi’an University of Science and Technology, Xi’an, China

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Article history: Available online xxxx Keywords: Array signal processing Robust beamforming Steering vector mismatch Interference-plus-noise covariance matrix reconstruction Alternative iteration

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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- plusnoise 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. © 2019 Published by Elsevier Inc.

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1. 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 [1–6]. The classical standard Capon beamformer (SCB), as a famous adaptive beamformer, has remarkable resolution and interference suppression capability upon the assumption

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✩

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. Corresponding author at: National Laboratory of Radar Signal Processing, Xidian University, Xi’an, China. E-mail addresses: [email protected] (Z. Yang), [email protected] (P. Zhang), [email protected] (G. Liao), [email protected] (C. Duan), [email protected] (H. Xu), [email protected] (S. He).

*

https://doi.org/10.1016/j.dsp.2019.102620 1051-2004/© 2019 Published by Elsevier Inc.

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 [8–10]. 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 [11–16] and the weight norm constraint algorithms in [17–19] 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)

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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 [20–22] are promoted. The eigenspace-based beamformer (ESB) in [21] directly projects the SOI steering vector onto the signal-plusinterference 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 [23–27], 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 [28–37]. 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 [38–44] or reconstructing the interferenceplus-noise covariance matrix (INCM) [45–54] 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 (INCMSBT) 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 contributions to the ﬁeld of adaptive beamforming in the following aspects,

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

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σi2 , i = 1, 2, · · · , J and σn2 represent the power of the ith interfer-

ing signal and noise power, respectively. The output of array can be written as y (t ) = w H x(t ), where w represents the complex weight vector of the adaptive beamformer. Thus, the output SINR is deﬁned as:

SINR =

wH R IN w

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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:

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(3)

where | · | denotes the operator that returns the absolute value. The optimal adaptive beamformer can be taken into realization through maximizing the array output SINR, or equivalently by solving the following mathematical problem:

min

wH R IN w

s.t.

w H a0 = 1

w

ai si (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 independent. The array covariance matrix can be expressed as:

R = E {x(t )xH (t )} = R S + R I N

(2)

where E {·} is the statistical expectation operator, (·)H stands for the conjugate transpose. R S = σ02 a0 aH 0 is the actual SOI covariance

J

matrix, with σ02 being the power of the SOI. R I N = i =1 σi2 ai aH i + σn2 I denotes the theoretical INCM, in which I is an identity matrix,

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w OPT =

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−1

85

R I N a0

(5)

−1 aH 0 R I N a0

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where (·)−1 denotes the inversion matrix of a positive deﬁnite square matrix. By substituting (5) back into (3), the optimal output SINR can be calculated as: −1

SINROPT = σ02 aH 0 R I N a0

(6)

Since the theoretical INCM R I N is always unavailable even in the SOI-free application, it is usually replaced by the array covariance matrix R, which results in the equivalent problem of (4) as:

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min

wH R w

s.t.

w a0 = 1

w

99

(7)

H

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In practice, only the maximum likelihood estimation of the array covariance matrix R is available (i.e. the SCM), and it can be obtained from the training data x(t ) as:

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ˆ = R

1 L

L

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x(t )xH (t )

(8)

t =1

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ˆ w SCB = α R

−1

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a0

(9)

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−1

ˆ a0 ) denotes the normalization factor that where α = 1/(aH 0R does not affect the output SINR.

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(1)

i =1

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(4)

with the optimal weight vector:

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]:

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min

−1 (a˜ 0 + e ⊥ ) Rˆ (a˜ 0 + e ⊥ )

s.t.

P ⊥ (a˜ 0 + e ⊥ ) = 0 ¯ ˜0 (a˜ 0 + e ⊥ )H C¯ (a˜ 0 + e ⊥ ) ≤ a˜ H 0 Ca ||a˜ 0 + e ⊥ ||22 ≤ M H a˜ 0 e ⊥ = 0

e⊥

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wH R IN w

3. Conventional optimization algorithm

J

x(t ) = a0 s0 (t ) +

=

σ02 | w H a0 |2

where L denotes the number of snapshots. This results in the solution of the SCB as:

2. Signal model

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wH R S w

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3

H

124 125

(10)

where P ⊥ = I − L L H is the orthogonal projection, L denotes the subspace spanned by the major eigenvectors of C = aaH dθ ( is

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the SOI angular region, a denotes the steering vector from θ ), and ¯ is the complementary of the region ). a˜ 0 and C¯ = ¯ aaH dθ ( e ⊥ denotes the presumed SOI steering vector and the mismatch vector orthogonal to a˜ 0 . || · ||2 and 0 stand for the l2 norm and all zero column vector, respectively. The main weakness of this technique is the fact that the subspace L has to be known precisely, which in practice is impossible [9]. In terms of the shortcomings in (10), Gu et al. have provided to estimate the steering vector of the SOI by the following optimization problem [45]:

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min e⊥

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s.t.

(a˜ 0 + e ⊥ )

H

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−1

(a˜ 0 + e ⊥ ) R I N (a˜ 0 + e ⊥ ) H

H R I N (a˜ 0 + e ⊥ ) ≤ a˜ 0 R I N H a˜ 0 e ⊥ = 0

a˜ 0

(11)

−1

ˆ a)dθ . Although the inequality conwhere R I N = ¯ aa /(a R straint in (11) keeps the steering vector away from the interference directions, the objective function in (11) cannot guarantee the compensating steering vector of the SOI a˜ 0 + e ⊥ converging to the actual direction. To further alleviate the steering vector mismatch, Zheng et al. have formulated the steering vector of the SOI optimization model as [52]: min e⊥

s.t.

H

(a˜ 0 + e ⊥ ) Rˆ H

H

−1

(a˜ 0 + e ⊥ )

(a˜ 0 + e ⊥ )H U U H (a˜ 0 + e ⊥ ) ≤ 0 H a˜ 0 e ⊥ = 0

(12)

where U denotes the subspace spanned by the minor eigenvectors related to the matrix C . As we can observe, the simpliﬁed optimization model in (12) faces the similar drawback as (12), where the subspace U is hard to be precisely estimated.

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4. Proposed algorithm

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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 adaptive beamformer acquires the weight vector with provable performance improvement.

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4.1. Optimization problem formulation

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To strengthen the robustness against the SOI steering vector mismatch, we propose to search for the steering vector which has the shortest spatial distance to the SS. As we all know, the classical multiple signals classiﬁcation (MUSIC) algorithm has provided to estimate the directions of the signals according to the orthogonality between the steering vectors of the signals and the noise subspace (NS) [55,56], which is also termed as the subspace theory:

span(a0 , a1 , · · · , a J ) = span(U S I )⊥span(U N )

(13)

where U S I and U N denote the theoretical SIS and NS, respectively. Enlightened by the aforementioned subspace orthogonality theory, one has:

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H H 2 aH 0 U N U N a0 = || U N a0 ||2 = 0

(14)

Without loss of generality, the relationship between the actual and presumed steering vectors of the SOI can be represented as a0 = a˜ 0 + e ⊥ if the norm inconsistency is unconsidered. That is to say, the orthogonality between presumed steering vector of the SOI a˜ 0 and NS U N must be lower than that between the actual steering vector of the SOI a0 and NS U N :

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||U HN a0 ||22 = ||U HN (a˜ 0 + e )||22 ≤ ||U HN a˜ 0 ||22

(15)

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Clearly, an indirect way to estimate the actual steering vector 2 ˜ of the SOI a0 through minimizing the term ||U H N (a0 + e ⊥ )||2 with respect to the mismatch vector e ⊥ is behind (15). However, the 2 ˜ minimum value of the term ||U H N (a0 + e ⊥ )||2 may not result in the desired consequence (a˜ 0 + e ⊥ ) a0 since the interference steering vectors ai , i = 1, 2, · · · , J are also orthogonal to the NS U N . Therefore, the steering vector of the SOI optimization problem is formulated as:

76

min

||U HN (a˜ 0 + e ⊥ )||22

s.t.

||U HI (a˜ 0 + e ⊥ )||22 ≤ ||U HI a˜ 0 ||22 H a˜ 0 e ⊥ = 0

e⊥

M

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(16)

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where U I denotes the IS. The objective function in (16) is introduced to force the corrected steering vector a˜ 0 + e ⊥ getting close to the orthogonal subspace of the NS U N (i.e. the SIS U S I ) suﬃciently. Besides, the inequality constraint is imposed to prevent the revised steering vector a˜ 0 + e ⊥ from converging to the directions of the interferences and their linear combinations, and the equality constraint is added to ensure that the presumed SOI steering vector a˜ 0 is orthogonal to the mismatch vector e ⊥ . Noting that here the norm constraint on the steering vector a˜ 0 + e ⊥ is unnecessary because the norm of the steering vector of the SOI ||a˜ 0 + e ⊥ ||2 does no effect on changing the output SINR. Comparing the newly constructed optimization problem (16) to the existing ones (10), (11), and (12), we note that the major differences between them are, i) The objective function in (16) is more effective to search for the true SS or IS than that in (10) and (12), because the MUSIC spectrum estimator enjoys higher resolution than of the Capon spectrum estimator in practice. Moreover, the solution to the objective function in (16) should be the exact SS or IS, while after solving the objective function in (10), one can only acquire the IS. ii) The constraint with the IS in (16) can perform better in avoiding the revised steering vector converging to the interference directions than that with the INCM in (16), which can also be explained by the resolution superiority of the MUSIC spectrum estimator. Besides, using the constraint related to the IS in (16) to constrain the steering vector can provide suﬃcient chance to seek the desired steering vector of the SOI from any non-interference direction, while the constraints based on the orthogonal matrix in (10) and subspace in (12) only offer the chance to gain the steering vector from the settled SOI region. iii) The number of constraints in the derived optimization problem (16) is much less than that in (10), thus the proposed optimization method is computationally eﬃcient. However, it’s not easy to address the optimization problem (16) since the NS U N and IS U I are unknown. As is acknowledged in the MUSIC method, if the SCM inaccuracy due to the ﬁnite training samples is ignored, a reliable estimate of the NS can be obtained by eigendecomposing the SCM as:

ˆ = R

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H i i

γi η η

(17)

i =1

where γi , i = 1, 2, · · · , M and η i , i = 1, 2, · · · , M represent the eigenvalues in descending order and corresponding eigenvectors,

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5

respectively. Thus, the well-estimated NS under arbitrary array geometry errors is given as:

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Uˆ N = (η J +2 , η J +3 , · · · , η M )

(18)

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To proceed, the existing INCM reconstruction approaches have been attested not the proper choices to estimate the IS in the presence of array imperfections. In the next subsection, an advanced INCM reconstruction approach will be presented to achieve the IS estimation.

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4.2. INCM reconstruction

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Since the training data always reﬂects the real array structure information, we prepare to estimate the INCM by detaching the SOI component from the sample data by means of constructing a blocking matrix. In order to obtain the blocking matrix, the deﬁnition of the signal-plus-noise covariance matrix (SNCM) is foremost given as:

˜ S N = σ˜ 02 a˜ 0 a˜ 0 + σ˜ n2 I R H

where σ˜ 02 stands for the pre-deﬁned power of the SOI, and σ˜ n2 represents the estimation on noise power. Evidently, the SNCM can be eigendecomposed as:

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˜ SN = R

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−1

˜ SN = R

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(20)

μi

=

p1 pH 1 ˜ 02 + ˜ n2

σ

σ

+

M pi pH i i =2

(21)

σ˜ n2 M

H In view of that both the conditions σ˜ 02 σ˜ n2 and i =1 p i p i = I are satisﬁed, an approximate expression of the inversion matrix of the SNCM can be formed as:

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μ

H i pi pi

M pi pH i i =1

1 ∼ ˜− R SN =

M pi pH i i =2

σ˜ n2

=

I − p1 pH 1

σ˜ n2

(22)

−1

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Fig. 1. Function || Ba||2 versus DOA for number of snapshots L = 80 and input SNR=25 dB in the scenario of exactly known SOI steering vector.

Based on the discussions above, the blocking matrix B in (23) is utilized herein to pre-process the sample data x(t ) as:

J

x˜ (t ) = Bx(t ) = Ba0 s0 (t ) +

(24)

Bai si (t ) + Bn(t )

(25)

L

H

x˜ (t )˜x (t ) =

t =1

σi2 Bai aHi B H + σn2 B B H

(26)

i =1

where the blocking matrix currently performs the characteristic B a˜ 0 = 0. To conﬁrm the performance of the blocking matrix B with considering the SOI power σ˜ 02 selection problem, the function || Ba|| versus the DOA under different SOI power settings σ˜ 02 ∈

(27)

{10σ˜ n2 , 102 σ˜ n2 , 103 σ˜ n2 , 102 trace ( Rˆ )} is plotted in Fig. 1, where a and trace (·) represent the steering vector corresponding to the DOA θ and the trace of matrix. In this example, the simulation

parameters in section 5 are used. As we can see from Fig. 1, if the SOI power satisﬁes σ˜ 02 > 102 σ˜ n2 , the performance of the blocking matrix B seems unchanged and nearly excellent, which means that the qualiﬁcation σ˜ 02 σ˜ n2 is now met. Noting that, on ac-

ˆ always contains strong interferences and count of that the SCM R ˆ ) > σ˜ n2 is tenable, we have 102 trace ( Rˆ ) σ˜ n2 . Therefore, trace ( R for general cases, a reasonable selection on the SOI power is given ˆ ). as σ˜ 02 = 102 trace ( R

Subsequently, the quasi INCM after modiﬁcation R I N can be eigendecomposed as:

RIN=

91 93 94 95 96 97 99

J

λi v i v H i =

i =1

102 103 105 106 107

In terms of that the noise component in (26) has not been the ˜ I N needs to be modiideal white one (i.e. σn2 I ), the quasi INCM R ﬁed as:

M

89

104

J

(23)

88

101

Hence, the covariance matrix named as the quasi INCM can be computed as:

L

87

100

i =1

1

86

98

J

˜ IN = R

85

92

It should be pointed out that the SOI component a0 s0 (t ) has been greatly blocked or weakened as well as the gap between the steering vectors a˜ 0 and a0 is negligible, and then the term Ba0 s0 (t ) can be omitted, which follows the approximate form of (24) as:

x˜ (t ) ∼ =

84

90

Bai si (t ) + Bn(t )

˜ I N − σ˜ n2 B B H + σ˜ n2 I RIN= R

Hence, we can deﬁne the signal blocking matrix as:

˜ SN B=R

81

i =1

where μi , i = 1, 2, · · · , M and p i , i = 1, 2, · · · , M represent the eigenvalues in descending order and corresponding eigenvectors, respectively. Thus, the inversion matrix of the SNCM can be expressed as follows if the pre-deﬁned power of the SOI is relatively stronger than that of the noise (i.e. σ˜ 02 σ˜ n2 ):

38 39

M i =1

29 30

(19)

80

108 109 110 111 112 113 114 115 116

˜ n2 I v i (λi − σ˜ n2 ) v H i +σ

(28)

117 118

i =1

where λi , i = 1, 2, · · · , M and v i , i = 1, 2, · · · , M represent the eigenvalues in descending order and corresponding eigenvectors,

119

respectively. Since the revised quasi INCM R I N can also be denoted as:

122

RIN=

J

σ

˜ n2 I

+σ

(29)

123

J

127

i =1

128 129 130

J

σi2 Bai aHi B H =

125 126

We can easily draw the following conclusion according to the equivalence between (28) and (29):

i =1

121

124

2 H H i Bai ai B

i =1

120

v i (λi − σ˜ n2 ) v H i

(30)

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Obviously, if we left-multiply and right-multiply both sides of (30) by the inversion matrix of the blocking matrix B −1 (i.e. the ˜ S N ) and the inversion matrix of the conjugate transpose of SNCM R the blocking matrix ( B H )−1 , the recovery of the actual interference component can be attained, which leads to the INCM reconstruction in this paper as:

67 68 69 70 71 72 73

7 8 9

ˆ IN = R

12 13 14 15 16 17 18 19 20 21 22 23 24

H −1 B −1 v i (λi − σ˜ n2 ) v H + σ˜ n2 I i (B )

74

(31)

75

i =1

10 11

J

76 77

ˆ I N , a trustworthy estimate of the IS After obtaining the INCM R can be accomplished as follows: Uˆ I = B

−1

(v 1, v 2, · · · , v J )

(32)

What should be emphasized is that, in this paper, the prior knowledge of the source number J + 1 are always precisely given or estimated, thus the principal eigenvectors related to the SCM ˆ or INCM Rˆ I N are simple to recognized. Moreover, as compared R to the existing reconstruction-based methods, our INCM estimation is data-dependent and thus robust against array imperfections. Meanwhile, the given INCM estimation approach can guarantee the DOF and provide better SOI component removal result than those of the existing SOI blocking-based algorithms.

78

Fig. 2. The AIERB algorithm ﬂow chart.

79 80

To reduce the computational complexity, we replace the integral with the summation over a discrete angular sector. In this way, the steering vector of the SOI can be initialized as follows in view of that there only exists one signal in the region :

a˜ 0 =

√

M

|| P { Q }||2

(34)

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

66

84 86

where P {·} denotes the operator which returns the principal eigenvector of a matrix (i.e. the eigenvector that corresponds to the largest eigenvalue).

88 89 90 91

4.3. Steering vector and INCM alternative iteration scheme

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4.5. Summary of the proposed algorithm

93

As we can notice, the proposed INCM reconstruction procedure is based upon the assumption that the mismatch between the presumed and actual steering vectors of the SOI is small. In other words, the reconstructed INCM in subsection 3.2 cannot offer desired estimation on the IS in case of signiﬁcant SOI steering vector mismatch, which means that the actual steering vector of the SOI may not be reached after solving the optimization problem (16) once (please see Fig. 4). To tackle this drawback, we determine to alternate estimating the steering vector of the SOI and INCM in an iterative strategy. That is, ﬁrst, using the initial steering vector of the SOI a˜ 0 to construct the blocking matrix B; Second, realizing ˆ I N and IS Uˆ I by processing the training data x(t ) with the INCM R the signal blocking matrix B and performing some brief matrix transitions; Third, solving the formulated optimization problem to obtain the mismatch vector e ⊥ , and then renew the steering vector of the SOI a˜ 0 ; At last, repeating the above steps with the revised versions of the steering vector of the SOI and INCM, which results in both the steering vector of the SOI and INCM converging to the preferred ones. From what has been described above, the devised steering vector optimization problem, which is based upon the subspace theory but visibly different from the ESB method, can also be viewed as an iterative procedure of ﬁnding the exact SS. For convenience, the proposed alternative iteration estimation robust beamformer is referred to as the AIERB in the sequel. Moreover, the main alternative iteration process of the proposed algorithm is exhibited in Fig. 2. To accomplish the AIERB algorithm showed in Fig. 2, only the initial value of steering vector of the SOI is unknown, so we intend to pre-estimate it in the next subsection. 4.4. Steering vector initialization Starting from the point that the actual DOAs of the array received signals always lie in some small angular sectors, the SOI matrix can be obtained with the Capon spectrum estimator as [57]:

64 65

83

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27 28

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P{Q }

25 26

81

Q =

aa

ˆ aH R

H

−1

dθ a

(33)

The basic idea of the proposed algorithm is, making use of the initial steering vector of the SOI to estimate the IS by reconstructing the INCM, and then ﬁguring out the designed convex optimization problem in an alternative iteration way. The proposed AIERB algorithm is summarized in Table 1.

94 95 96 97 98 99 100

4.6. Convergence analysis

101

In this part, we brieﬂy analyze the convergence of the proposed algorithm. As we can observe, the quadratic objective function in (16) is similar to that in (10) because of the relationship ˆ ˆH ˆ −1 = R U N U N , which means that the objective function value in

(16) is positive and always nonzero. Also, the solution to (16) is guaranteed to reach a global minimum as a result of convexity [24]. Therefore, the solution to (16) can be of guarantee H ||Uˆ N (a˜ 0

+ e ⊥ )||22 ≤

H ||Uˆ N a˜ 0 ||22

and the objective function value in (16) can be reduced further by updating the steering vector of the SOI at any iteration. For that reason, the steering vector of the SOI will converge to the actual SS gradually, together with the rising INCM estimation precision. It is worth remarking that the mathematical convergence proof of the AIERB is hard to give, so we evaluate the objective function H

||Uˆ N a˜ 0 ||22 , the correlation coeﬃcient of the estimated steering vector of the SOI a˜ 0 and the actual steering vector of the SOI a0 , and ˆ I N and the the correlation coeﬃcient of the reconstructed INCM R actual INCM R I N versus the number of iterations in Figs. 3–8 to make it certain that the proposed AIERB is not divergent. Here we deﬁne the correlation coeﬃcients of vectors and matrices as:

˜ |aH 0 a0 | Cor v (a0 , a˜ 0 ) = ||a0 ||2 ||a˜ 0 ||2

(35)

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126

and: H ˆ ˆ I N ) = | vec [ R I N ] vec [ R I N ]| Corm ( R I N , R || vec [ R I N ]||2 || vec [ Rˆ I N ]||2

102

127 128

(36)

where vec [·] denotes the operation which gets a vector by stacking the columns of a matrix on top of each other [50].

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ˆ with (8); Calculate the SCM R Initialize the steering vector of the SOI a˜ 0 with (33) and (34); Estimate the NS Uˆ N with (17) and (18); Construct the signal blocking matrix B with (19) and (23); Acquire the SOI-free data x˜ (t ) with (24);

69 70 71 72

˜ I N with (26), and get the modiﬁed quasi INCM R I N with (27); Calculate the quasi INCM R ˆ I N with (28) and (31); Reconstruct the INCM R Achieve the estimate of the IS Uˆ I with (32); Solve the steering vector mismatch e⊥ with (11) by replacing U N and U I with the estimated Uˆ N and Uˆ I , respectively; H

73 74 75 76 77

Judge the tolerance δ by measuring the relative change of the objective functions ||Uˆ N a˜ 0 ||22 and

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||Uˆ N (a˜ 0 + e ⊥ )||22 , once ||Uˆ N a˜ 0 ||22 − ||Uˆ N (a˜ 0 + e ⊥ )||22 > δ is reached, complete the compensation of the √ SOI steering vector as a˜ 0 = M (a˜ 0 + e ⊥ )/||a˜ 0 + e ⊥ ||2 and then repeat steps 4-10. Otherwise, go to

79

step 11;

81

H

13

16

67

Table 1 Steps of the proposed AIERB algorithm.

3

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7

H

H

−1

−1

ˆ I N a˜ 0 , where β = 1/(a˜ 0 Rˆ I N a˜ 0 ). Attain the adaptive beamforming weight vector as w AIERB = β R H

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35 36 37

Fig. 3. Objective function value versus number of iterations for number of snapshot L = 80 in the scenario of DOA error and antenna position displacement.

Fig. 4. Coeﬃcient of steering vectors versus number of iterations for number of snapshot L = 80 in the scenario of DOA error and antenna position displacement.

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

In this example, both the DOA error and antenna position displacement are considered, where the simulation settings here are chosen as those in subsection 5.3. Apparently, at low SNR case,

105 106 107

H

following the declining of the objective function value ||Uˆ N a˜ 0 ||22 , both the correlation coeﬃcient of the estimated and actual steering vectors of the SOI Cor v (a0 , a˜ 0 ) and the correlation coeﬃcient of ˆ I N ) get closer and the reconstructed and actual INCMs Corm ( R I N , R closer to the ideal values (i.e. 1). Similar to the low SNR situation,

108 109 110 111 112

H

the objective function value ||Uˆ N a˜ 0 ||22 , the correlation coeﬃcient of the steering vectors Cor v (a0 , a˜ 0 ), and the correlation coeﬃcient ˆ I N ) can also verge on ﬁxed values by of the matrices Corm ( R I N , R solving the optimization problem (16) three times at high SNR case. All in all, the objective function value at each iteration process is always less than that of the previous iteration. Meanwhile, the estimations on the SOI steering vector and INCM likewise converge to the actual ones step by step, which signiﬁcantly validates the feasibility and effectiveness of the proposed AIERB algorithm.

56 57

4.7. Complexity comparison

113 114 115 116 117 118 119 120 121

Fig. 5. Coeﬃcient of matrices versus number of iterations for number of snapshot L = 80 in the scenario of DOA error and antenna position displacement.

60 61 62 63 64 65 66

122 123 124

58 59

103 104

38 39

102

From a complexity point of view, the computational complexity of the proposed AIERB algorithm is dependent on calculating the SCM with O ( LM 2 ), initializing the steering vector of the SOI with O ( M 3 ) + O ( S 1 M 2 ), where S 1 denotes the number of sample points in the SOI region, estimating the NS with O ( M 3 ), constructing the signal blocking matrix with O ( M 2 ) + O ( M 3 ), calculating and modifying the quasi INCM with O ( LM 2 ) + O ( M 3 ), realizing the INCM reconstruction and IS estimation with O ( M 3 ) + O ( J ( M 2 +

M 3 )) + O ( J M 2 ), solving the provided optimization problem with O ( M 3.5 ) + O (( J + 1) M 2 ) + O ( J M 2 ), and acquiring the weight vector with O ( M 3 ) + O ( M 2 ). Hence, the overall computational complexity of the proposed algorithm is O ( K 1 M 3.5 ) + O ((2K 1 J + 3K 1 + 3) M 3 ) + O ((2K 1 J + K 1 L + L + S 1 + K 1 + J + 2) M 2 ), where K 1 denotes the number of iterations. For the existing adaptive beamformers, the computational complexities of the SCB method, the ESB method, and the RCB method are O ( M 3 ) + O ( LM 2 ),

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Table 2 Complexities of different algorithms.

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Algorithm

Complexity

SCB ESB RCB SQP INCM-QCQP ISVPE-INCM MCLM AIERB

O ( M ) + O ( LM ) O (3M 3 ) + O (( L + J + 1) M 2 ) O (2M 3 ) + O (( L + 1) M 2 ) O ( K 2 M 3.5 ) + O (2M 3 ) + O (( L + S 2 + S 3 + N 1 + 1) M 2 ) O ( M 3.5 ) + O (2M 3 ) + O (( L + S 4 + 1) M 2 ) O ( M 3.5 ) + O ((3 + J + 1) M 3 ) + O (( L + S 5 + N 2 + J + 1) M 2 ) O (4M 3 ) + O (( L + 4 J + 4) M 2 ) + O ((4 J 2 + 1) M ) + O ( J 3 ) O ( K 1 M 3.5 ) + O ((2K 1 J + 3K 1 + 3) M 3 ) + O ((2K 1 J + K 1 L + L + S 1 + K 1 + J + 2) M 2 ) 3

2

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

73 74 75 76

5. Simulation

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

Fig. 6. Beampattern versus DOA for number of snapshots L = 80 and input SNR=5 dB in the scenario of exactly known SOI steering vector.

96 97 98

32 33

72

78

O (3M 3 ) + O (( L + J + 1) M 2 ), and O (2M 3 ) + O (( L + 1) M 2 ), respectively. For the SQP method, it owns the computational complexity of O ( K 2 M 3.5 ) + O (2M 3 ) + O (( L + S 2 + S 3 + N 1 + 1) M 2 ), where K 2 represents the number of iterations, S 2 and S 3 stand for the numbers of sample points in the SOI region and interference region, respectively, N 1 is the number of major eigenvectors of the matrix C . Besides, the INCM-QCQP algorithm has a computational complexity of O ( M 3.5 ) + O (2M 3 ) + O (( L + S 4 + 1) M 2 ), where S 4 denotes the number of sample points in the interference region. In addition, the ISVPE-INCM algorithm has a computational cost of O ( M 3.5 ) + O ((3 + J + 1) M 3 ) + O (( L + S 5 + N 2 + J + 1) M 2 ), where S 5 represents the number of sample points in the SOI region, N 2 is the number of minor eigenvectors of the matrix C . Furthermore, the MCLM method’s computational complexity is O (4M 3 ) + O (( L + 4 J + 4) M 2 ) + O ((4 J 2 + 1) M ) + O ( J 3 ). The computational complexities of the proposed AIERB method and other relevant approaches are listed in Table 2.

30 31

71

77

11 12

70

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 region is set as [θ˜0 − 5◦ , θ˜0 + 5◦ ], of which the number of grid points is ﬁxed to S 1 = 100. θ˜0 is the prior DOA of the SOI. The noise power σ˜ n2 is selected to be the minimum eigenvalue of the SCM. Besides, the termination tolerance δ = 0.00001 is ﬁxed. For comparison purpose, the performances of the SCB method [7], the ESB method [21], the RCB method [23], the SQP method [24], the INCM-QCQP method [45], the ISVPE-INCM method [52], and the MCLM method [44] are presented as well. The signal number in the ESB method is always precisely known and the norm bound in the RCB approach is taken as 0.3M. Moreover, the uncertain region of the SOI in the SQP method is set the same as that in the AIERB algorithm, of which the number of grid points is ﬁxed to S 2 = S 1 , the interference region is set as [−90◦ , θ˜0 − 5◦ ) ∪ (θ˜0 + 5◦ , 90◦ ], of which the number of grid points is ﬁxed to S 3 = 860, and the number of major eigenvectors related to the matrix C is ﬁxed to N 1 = 5. The interference region in the INCM-QCQP method is set as that in the SQP approach, of which the number of grid points is ﬁxed to S 4 = S 3 . For the ISVPE-INCM algorithm, the uncertain region of the SOI is set the same as that in the AIERB algorithm, of which the number of grid points is ﬁxed to S 5 = S 1 , and the norm bound and number of minor eigenvectors related to the matrix C are set as 0.1 and N 2 = 3, respectively. In Addition, the tiny power adjustment factor in the MCLM is set ˆ )}. All simulation results are obtained by as min{0.02, M /trace ( R averaging 300 Monte-Carlo runs, and convex optimization toolbox CVX [58] is used to solve these optimization problems.

5.1. Beampattern under different SOI power pre-deﬁnition

99 100

In this subsection, we further explore the inﬂuence of different SOI power pre-deﬁnitions, i.e. σ˜ 02 ∈ {10σ˜ n2 , 102 σ˜ n2 , 103 σ˜ n2 ,

101 102

ˆ )}, on the output performance the proposed AIERB ap10 trace ( R proach. Figs. 6–7 exhibit the beampatterns versus the DOA in the situation of weak and strong SOI scenarios. From these ﬁgures, we can see that the AIERB algorithm has same beampattern under different σ˜ 02 when the input SNR is 5 dB. However, when the input SNR turns to 25 dB, only the proposed algorithm usˆ ) has the stable and best ing σ˜ 02 = 103 σ˜ n2 and σ˜ 02 = 102 trace ( R beampattern. That is, when the pre-deﬁned SOI power satisﬁes σ˜ 02 > 102 σ˜ n2 , the proposed beamformer will have a desired beampattern for the large dynamic input SNR changing environment. ˆ ) > σ˜ n2 is still rational, so we can directly set Recalling that trace ( R 2 ˆ ) to construct the blocking mathe SOI power as σ˜ 0 = 102 trace ( R trix to ensure that the proposed AIERB method can be applied to general scenarios.

103

5.2. Output SINR in the scenario of DOA error

118

2

104 105 106 107 108 109 110 111 112 113 114 115 116 117 119

In this subsection, we just think about the DOA error. The presumed θ˜0 is ﬁxed to 7◦ , whereas the actual θ0 is 10◦ . Fig. 8 displays the average output SINR of different methods against the input SNR. Obviously, the robust adaptive beamformers behave better than the SCB in a large range of −10 dB to 30 dB. Moreover, the output SINR losses of the ESB approach, the RCB approach, and the SQP approach are because the SOI is suppressed as interference due to the imperfect estimation on the SOI steering vector. In contrast, the proposed AIERB algorithm and the INCM-QCQP approach can hold outstanding performances due to the effective reconstructions of the INCM. Even the INCM and SOI- free samples are also reached in the ISVPE-INCM method and MCLM method, respectively, the existing interference steering vector mismatches and SOI

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Fig. 7. Beampattern versus DOA for number of snapshots L = 80 and input SNR=25 dB in the scenario of exactly known SOI steering vector.

Fig. 9. Output SINR versus number of snapshots for input SNR=5 dB in the scenario of DOA error.

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Fig. 8. Output SINR versus input SNR for number of snapshots L = 80 in the scenario of DOA error.

41 42 43 44 45 46 47 48 49

component imperfect removal still lead to the output SINR reduction. Additionally, the different beamformers’ output SINR versus the number of snapshots is also exposed in Fig. 9. It can be concluded from Fig. 9 that except for the SCB and SQP methods, other compared beamformers enjoy fast convergence rates. Due to the precise estimations on the SOI steering vector and INCM, the proposed AIERB algorithm also presents enough robustness in case of ﬁnite sample size.

50 51 52

5.3. Output SINR in the scenario of DOA error and antenna position displacement

53 54 55 56 57 58 59 60 61 62 63 64 65 66

In this subsection, we consider both the DOA error and antenna position displacement, where the DOA error is set as that in subsection 5.2, and the position displacement of each antenna element is assumed to be far away from its theoretical location to follow uniform distribution of the set [−0.05, 0.05] measured in wavelength. Following the example in subsection 5.2, the performances of the proposed AIERB algorithm and other compared approaches in terms of the output SINR versus the input SNR, the output SINR versus the number of snapshots are shown in Figs. 10–11, respectively. Because of the occurrence of signal self-nulling, the performances of the SCB method, the ESB method, the RCB method, and the SQP method decrease rapidly at high SNR case. Though the INCM-QCQP algorithm, the ISVPE-INCM algorithm, and the MCLM

Fig. 10. Output SINR versus input SNR for number of snapshots L = 80 in the scenario of DOA error and antenna position displacements.

104 105 106

algorithm can combat the signal self-nulling using the INCM or SOI-free data, the terrible interference rejection ability still results in the robustness drops. Yet the proposed AIERB algorithm is able to outperform other methods, which is mainly on account of the novel way of iteratively estimating the SOI steering vector and INCM. Nevertheless, the AIERB algorithm has a slight decrease in output SINR when the input SNR is stronger than 25 dB, this may be ascribed to the inaccurate IS estimated by the SOI-involved quasi INCM. Besides, when the number of snapshots is bigger than 20, the proposed and other compared approaches but the SCB and SQP can provide stable performances, which means that our method can be applied to number of snapshots deﬁciency circumstance. 5.4. Output SINR in the scenario of DOA error and antenna gain and phase perturbations

107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

In this subsection, we examine the performances of the proposed AIERB method and other relevant approaches in the scenario of the DOA error and antenna gain and phase perturbations, where the DOA error is set as that in subsections 5.2 and 5.3. In particular, the gain mismatch and phase mismatch of each antenna element are set to obey norm distributions of [−0.5 dB, 0.5 dB] and [5◦ , 5◦ ], respectively. Following the cases in examples 5.2 and 5.3, the performances of the AIERB algorithm and other compared methods in view of the output SINR versus the input SNR

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Fig. 11. Output SINR versus number of snapshots for input SNR=5 dB in the scenario of DOA error and antenna position displacements.

Fig. 13. Output SINR versus number of snapshots for input SNR=5 dB in the scenario of DOA error and antenna gain and phase perturbations.

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Fig. 12. Output SINR versus input SNR for number of snapshots L = 80 in the scenario of DOA error and antenna gain and phase perturbations.

Fig. 14. Number of iterations versus input SNR for number of snapshots L = 80 in different scenarios.

and the output SINR versus the number of snapshots are depicted in Figs. 12–13, respectively. As we can see, the output SINR of the AIERB approach remains an outstanding level while other compared methods cannot maintain desired output performances. However, at SNR>25 dB case, the output SINR of the derived method decreases slightly and it’s due to the SOI component insuﬃcient removal resulted from the steering vector of the SOI initialized by the ideal array manifold. To continue, we can observe from Fig. 13 that the devised AIERB algorithm also suits for limited number of snapshots scenario, and shows enough superiority when compared to the listed beamformers.

6. Conclusion

43 44 45 46 47 48 49 50 51 52 53 54

5.5. Convergences in different scenarios

55 56 57 58 59 60 61 62 63 64 65 66

106 107

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In this subsection, we test the convergence rate of the proposed beamformer in different scenarios. Accordingly, the number of iterations versus the input SNR is exhibited in Fig. 14. It is straightforward to discover that at any case, along with the increase of input SNR, the number of iterations decreases. And the main reason is that, at low SNR case, the SIS is dominantly occupied by the interference steering vectors, so multiple searching procedures have to be taken to obtain the actual SS. Oppositely, when the SOI power is strong enough, the difference between the SIS and SS becomes extremely small, which causes the prompt achievement on estimating the actual steering vector of the SOI.

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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 method are also elaborated. Extensive simulation results have illustrated that the proposed adaptive beamforming algorithm has signiﬁcant performance improvement when facing the DOA error, the antenna position displacement, and the antenna gain and phase perturbations, especially at high SNR case. 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. References [1] I.S. Reed, J.D. Mallett, L.E. Brennan, Rapid convergence rate in adaptive arrays, IEEE Trans. Aerosp. Electron. Syst. 10 (6) (November 1974) 853–863.

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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.

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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 the Beijing Institute of Radio Measurement. His research interests includes adaptive array signal processing and space-time adaptive processing.

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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.

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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.

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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.

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