Robust and Efficient Adaptive Beamforming Using Nested Subarray Principles

In this paper, we propose an adaptive beamforming algorithm for large uniform linear arrays (ULAs), where only a nested subarray is utilized to calculate the beamforming coefficients for the original ULA. In this algorithm, the steering vectors and powers of the signal-of-interest (SOI) and interferences are firstly estimated using the Capon spatial spectrum and known array structure, and the interference-plus-noise covariance matrix (INCM) is then constructed. Subsequently, an augmented INCM is formed via vectorization and spatial smoothing operations. Finally, the beamformer weight vector is determined by the augmented INCM and the estimated SOI steering vector. Our proposed algorithm exploits the enhanced degrees of freedom of the nested array, and thus can be applied to a large ULA to reduce the implementation complexity. Moreover, it fundamentally eliminates the SOI component. Numerical results demonstrate that the proposed algorithm performs better than the existing approaches.


I. INTRODUCTION
In array signal processing, adaptive beamforming is a fundamental technology due to wide applications, e.g., in radar, sonar, wireless communications, cognitive radio networks, medical imaging [1]- [3]. The minimum variance distortionless response (MVDR) beamformer is an optimal adaptive filter, which provides outstanding spatial resolution and interference rejection capability [4]. Nevertheless, the standard MVDR beamformer is quite sensitive to the mismatch between the actual steering vector and presumed one, which is caused by various imperfections such as look direction errors and local scattering effects. The standard MVDR beamformer will suffer significant performance deterioration with steering vector mismatches, especially when the training data contains the signal-of-interest (SOI). In the past decades, many robust adaptive beamformers have been suggested to enhance the robustness of the standard MVDR beamformer [5]- [37].
The associate editor coordinating the review of this manuscript and approving it for publication was Liangtian Wan . Degrees of freedom (DOFs) is a key factor affecting the performance of adaptive beamforming. Generally, a higher number of DOFs means higher spatial resolution and better interference rejection capability [38]. Traditional adaptive beamformers are mostly designed for the uniform linear array (ULA), where the element spacing is not more than half a wavelength to avoid spatial aliasing. However, the number of DOFs of ULAs increases linearly as the number of sensors increases. To enhance the number of DOFs within the ULA configuration, more sensors are required, thus bringing about a high complexity that may be impractical or uneconomical. Fortunately, nonuniform linear arrays (NLAs) provide an effective solution to this issue [39]. By generating the difference coarray [40], a NLA can provide a significantly higher number of DOFs than the traditional ULAs.
The nested array, as a NLA, have attracted intensive attention for years. It can achieve O(M 2 ) DOFs using only M physical sensors [41]. In particular, the two-level nested array can yield a hole-free difference coarray. Recently, lots of works have been reported on direction-of-arrival (DOA) estimation based on nested arrays [42]- [50]. But only a few works on nested array adaptive beamforming are presented. Nested array adaptive beamforming was firstly proposed in [38]. The beamformer of [38] is actually the MVDR beamformer with increased DOFs, where an augmented covariance matrix called spatially smoothed matrix (SSM) is constructed to compute the beamformer weight vector. In [51], another augmented covariance matrix was constructed for adaptive beamforming with enhanced DOFs. The new covariance matrix is obtained by forming a Toeplitz matrix with the observed signal of the virtual array, but it is essentially the square root of the SSM [52]. The beamformers of [38] and [51] do not consider various imperfections such as look direction error and coherent local scattering, and thus they are very sensitive to the model mismatch. To tackle this problem, Yang et al. [53] proposed a robust adaptive beamformer, where the interference-plus-noise covariance matrix (INCM) is reconstructed by projecting the SSM into the interference subspace, while the SOI steering vector is estimated by solving a convex optimization problem. The algorithm of [53] is more robust than the methods of [38] and [51] in the presence of various imperfections. However, its performance suffers some degradation as compared to the optimal value, especially at high signal-to-noise ratios (SNRs) because the SOI component is not thoroughly eliminated. These abovementioned methods utilize the SSM or its derivation to design the adaptive beamformer. However, the SSM needs to take a large number of samples to converge to a full-rank covariance matrix. Accordingly, the resulting beamformers have a slow convergence rate.
In this paper, a novel algorithm based on nested subarray principles is proposed for adaptive beamforming in largescale ULAs. Unlike the traditional schemes, the proposed algorithm only utilizes a small number of array sensors to determine the weighted vector of the original ULA. Hence, the implementation complexity can be significantly reduced. By constructing an augmented INCM, the proposed algorithm not only exploits the enhanced DOFs of the nested array, but also fundamentally removes the SOI component. Therefore, it achieves better performance than the existing schemes in high SNR regions. Numerical results verify the advantage of the proposed algorithm over the existing algorithms.
The rest of this paper is organized as follows. Some necessary preliminaries including signal model and problem background are described in Section II. An adaptive beamforming algorithm using nested subarrays is suggested in Section III. Numerical examples are provided in Section IV. Finally, conclusions are made in Section V.
Notations: We use bold lower-case and upper-case characters to denote vectors and matrices, respectively. (·) T , (·) * and (·) H represent the transpose, complex conjugate and conjugate transpose, respectively. E{·} represents the statistical expectation operator, · F denotes the Frobenius norm, and vec(·) stands for the vectorization operator that turns a matrix into a vector by stacking all columns on top of the another.
[·] i,j denotes the (i, j)th entry of a matrix, and [·] i denotes the ith element of a vector; diag(a) denotes a diagonal matrix that uses the elements of a as its diagonal elements. I m stands for the m × m identity matrix, and 0 m×n means the m × n zero matrix. The symbol denotes the Khatri-Rao product. Suppose that one SOI and L interference signals impinge on the nested array from the distinct directions θ 1 , . . . , θ N . The array observed vector at time instant k is given by where x s (k) = a 0 s 0 (k), x i (k) = L l=1 a l s l (k), and x n (k) are the SOI, interference, and noise components, respectively. Moreover, these components are assumed to be mutually independent. s l (k) is the lth source signal waveform, and a l = a(θ l ) is the corresponding steering vector. x n (k) is assumed to be the additive spatially white Gaussian noise with zero mean and variance σ 2 n . The steering vector for the lth source is expressed as a(θ l ) = 1, e j 2π λ z 2 d 1 sin θ l , . . . , e j 2π λ z M d 1 sin θ l T .
The beamformer output is given by where w = [w 1 , . . . , w M ] T ∈ C M ×1 is the beamformer weight vector. The output signal-to-interference-plus-noise ratio (SINR) of the beamformer is defined as where σ 2 0 = E{|s 0 (k)| 2 } is the SOI power, and R i+n is the M × M INCM, whose definition is given by where σ 2 l denotes the lth interference power. VOLUME 8, 2020 Maximizing the output SINR (4) is equivalent to the minimization problem: whose solution is Since the exact R i+n is unavailable in practice, it is usually replaced by the sample covariance matrix (SCM): where K denotes the number of training snapshots. Then the resulting beamformer is which is the so-called sample matrix inversion (SMI) beamformer.

B. ADAPTIVE BEAMFORMING WITH ENHANCED DOFS
To exploit the enhanced DOFs of the nested array, some approaches, such as [51], construct an augmented covariance matrix using the coarray output, and further generate the weighted vector of the beamformer. However, the dimension of the generated weighted vector is more than that of the physical array. Therefore, the adaptive beamforming with enhanced DOFs can only be applied to a large array to reduce the implementation complexity. The covariance matrix of x(k) is expressed as Vectorizing R xx yields T with e i being a column vector of all zeros except one 1 at the ith location. The vector z can be regarded as a single snapshot data from the equivalent signal vector p, and σ 2 n 1 n is a deterministic noise vector. The distinct rows of A * A correspond to the manifold of a longer array whose sensor locations are represented by the difference set of S.
Removing the redundant rows from the vector z and rearranging the remaining rows gives [41]: where is a manifold matrix, which is constructed by removing the repeated rows from A * A and rearranging the remaining rows, and e ∈ R (2M 2 (M 1 +1)−1)×1 means an all-zero vector except one 1 at the M 2 (M 1 + 1)th location.
Dividing z 1 into M 2 (M 1 + 1) subvectors, we can construct the following full-rank matrix [41]: (13) where z 1i is the ith subvector corresponding to the ( The matrix R ss can be represented as [41]: appears the same form as the covariance matrix of the observed signal from a longer ULA with M 2 (M 1 + 1) elements, whose manifold matrix is denoted by A 11 consisting of the last M 2 (M 1 + 1) rows of A 1 . Therefore, the matrixR, also referred to as the SSM, can be used to perform adaptive beamforming.
Employing the matrixR, the adaptive beamformer with enhanced DOFs is given by [51]: where d(θ 0 ) denotes the SOI steering vector of an equivalent ULA with M 2 (M 1 + 1) elements. However, the adaptive beamformer is very sensitive to the model mismatch and has a slow convergence rate.

III. PROPOSED ALGORITHM
In this section, we utilize the nested array to develop a new adaptive beamforming algorithm with enhanced DOFs. Unlike the existing schemes, our algorithm not only exploits the enhanced DOFs of nested arrays, but also thoroughly removes the SOI component. First, we construct the INCM of the nested array via steering vector and power estimation. With the known array structure, the steering vectors of the incident signals can be uniquely determined by their DOAs. When the number of signals is unknown, we utilize the Capon spatial spectrum to estimate the DOAs of the SOI and interferences. The Capon spectrum estimator is [54]: where a(θ ) is the steering vector associated with a direction θ ∈ [−90 • , 90 • ]. From (17), we can get multiple peaks.
Since there may be some spurious peaks, the number of the peaks is usually greater than the number of sources. Using the minimum eigenvalue ofR xx as the threshold, we can eliminate most of the spurious peaks. Assume there are still L + 1 peaks above the threshold, we can derive the corresponding DOA estimates,θ 0 ,θ 1 , . . . ,θ L by the positions of the peaks. Further, we can determine the DOAs of the SOI and 4078 VOLUME 8, 2020 interferences from different angular sectors. As [18], [53], we have assumed that is the angular sector in which the SOI lies and¯ is the out-of-sector of . Suppose thatθ 0 is located at , whileθ 1 , . . . ,θ L belong to¯ . Then, the estimates of interference steering vectors, a(θ 1 ), a(θ 2 ), . . . , a(θ L ), can be obtained using the known array geometry and the estimated DOAs. Additionally, the interference powers can be estimated as P(θ 1 ), P(θ 2 ), . . . , P(θ L ). The INCM of the nested array is reconstructed aŝ whereσ 2 n is the estimate of the noise variance, which is computed as the minimum eigenvalue ofR xx .
Proof: According to (22), the matrixR can be written as  (27) where On the other hand, we havê Therefore, (23) is equivalent to Namely, we only need to prove: The left-hand side of (33) can be expressed as where v l = e −j 2π λ d 1 sin(θ l ) . The right-hand side of (33) can be expressed as Comparing (34) and (35) results in (33). Consequently, the following equation holds:  (16) to result in the proposed beamformer (38).
To design the adaptive beamformer with enhanced DOFs, we need to consider anM -element equivalent ULA. With the estimated SOI DOAθ 0 , the SOI steering vector corresponding to the large ULA is given by SubstitutingR and d(θ 0 ) in (16), we get a new adaptive beamformer whose weight vector is For clarity, the proposed algorithm is summarized in Algorithm 1.
Remark 1: Unlike the traditional methods, the proposed approach is used for implementing low-complexity beamforming in large-scale array systems. Specifically, for a large ULA, our algorithm only selects a small number of sensors to form a nested subarray, and uses its coarray output to generate an augmented weighted vector. Then, the generated augmented weighted vector is applied to the whole ULA to realize beamforming. By doing so, the full aperture of the original ULA can be exploited, while the implementation complexity of beamforming is significantly reduced.

A. CONVERGENCE RATES OF COVARIANCE MATRICES
In the first experiment, we examine the convergence rates of the SCM, SSM and AINCM. To evaluate the similarity between two matrices, we introduce the following correlation coefficient: where R andR denote the true matrix and the estimated matrix, respectively. Clearly, when cor(R, R) is closer to 1, the similarity betweenR and R is higher. Fig. 2 shows the correlation coefficients of the covariance matrices versus number of snapshots, while Fig. 3 displays the correlation coefficients of their inverse matrices versus number of snapshots. It is observed that the convergence rate of the AINCM is obviously faster than those of the SCM and the SSM, and the latter two converge to their theoretical values only when K > 10 4 . On the other hand, we see clearly from Fig. 3 that the inverse of the AINCM also shows a faster convergence rate than that of the SCM, and especially that of the SSM, which still cannot converge to its theoretical value even when K = 10 5 . It can be attributed to the fact that the AINCM can be expressed as the covariance matrix form (23), and its model error is only determined by the accuracy of DOA and power estimates.

1) EXAMPLE 1-EXACTLY KNOWN SIGNAL STEERING VECTOR
Firstly, we consider the situation where the steering vectors of the SOI and interferences are accurately known. It is noteworthy that even in the absence of steering vector mismatches, the SOI component contained in the training data can seriously slow down the convergence rates of adaptive beamformers. The output SINR of the examined beamformers versus input SNR is shown in Fig. 4, from which we see that the proposed beamformer achieves almost the same performance as the reconstruct-estimate-based beamformer [18]. The two beamformers provide nearly optimal SINR from -30 dB to 50 dB and significantly outperform the remaining beamformers at high SNRs because the SOI component is thoroughly eliminated. However, the proposed algorithm is computationally more efficient than [18] because it uses fewer sensors to calculate the weight vector as [51], [53]. The output SINR of the beamformers against the snapshot number is plotted in Fig. 5. It is observed that both the proposed  beamformer and the beamformer of [18] enjoy a much faster convergence rate than the other beamformers because the convergence properties of the INCM and the AINCM are distinctly better than those of the SCM and the SSM.   output SINR of the examined beamformers versus SNR. But there is still a slight gap between some beamformers and the optimal SINR, as shown in Fig. 7. It is observed that the proposed beamformer and the reconstruct-estimate-based beamformer [18] have similar output performance, and both of them perform better than the remaining beamformers at high SNRs. Moreover, the proposed beamformer achieves about 1 dB gain over the beamformer of [18] when the SNR is more than 0 dB. The output SINR of the examined beamformers against number of snapshots is illustrated in Fig. 8. It can be seen that the proposed beamformer and the beamformer of [18] achieve almost the same and fast convergence rate, while the other beamformers experience a slow convergence rate.

3) EXAMPLE 3-SOI STEERING VECTOR MISMATCH DUE TO COHERENT LOCAL SCATTERING
Thirdly, the SOI steering vector is affected by coherent local scattering effects and expressed as [7]: where p corresponds to the direct path while a(θ p ) (p = 1, 2, 3, 4) refers to the coherently scattered paths. The angles θ p (p = 1, 2, 3, 4) are independently and uniformly distributed in [θ 0 − 4 • , θ 0 + 4 • ] in each trial. The phase parameters ψ p (p = 1, 2, 3, 4) are independently and uniformly drawn from [0, 2π ] in each trial. Note that θ p and ψ p change from trial to trial while keeping unchanged from snapshot to snapshot. Fig. 9 demonstrates the output SINR of the  examined beamformers versus the SNR. Their deviations from the optimal SINR are displayed in Fig. 10. Note that in this example, the SNR is defined by considering all signal scattering paths. As it can be observed, the proposed beamformer and the reconstruct-estimate-based beamformer [18] achieve almost the same output SINR, and they are superior to the remaining beamformers. More specifically, when the SNR is greater than -5 dB, the proposed beamformer performs better than [18] because it eliminates the SOI component and gets a more accurate INCM. When the SNR is greater than -5 dB, there is about 1 dB performance loss for the beamformer of [18] and our proposed beamformer, because there may be no obvious peak in for the Capon spectrum and the presumed DOA is used to derive the SOI steering vector. The output SINR of the examined beamformers against number of snapshots is shown in Fig. 11.
We observe clearly that our proposed beamformer and the beamformer of [18] show significantly better convergence performance than other beamformers being compared.

V. CONCLUSION
In this paper, we have developed a new adaptive beamforming approach for large-scale ULAs based on nested subarray principles. Typically, the proposed method is applied to a large ULA to reduce the implementation complexity. Compared with the existing methods, our approach not only exploits the enhanced DOFs of the nested array, but also thoroughly removes the SOI component. Therefore, its performance is more stable in the presence of various non-ideal factors. Numerical results show that our proposed approach significantly outperforms the existing approaches with nested arrays, and its output SINR is almost equal to the optimal value across a wide range of SNR.