Robust adaptive beamforming based on the direct biconvex optimization modeling

In general, robust adaptive beamforming (RAB) is modeled as a nonconvex optimization problem. The most of existing methods solve it indirectly by approximating the nonconvex problem to the convex optimization problem, which will cause the approximation error. Different from the existing methods, a novel method, which reformulates RAB as the biconvex form directly by use of an auxiliary variable, is proposed. Thus, the errors caused by the approximation can be avoided and the performance will be improved. Then the alternating direction methods of multiplier (ADMM) algorithm is applied to solve it. Compared with the existing methods, the results of the simulation experiments indicated that the proposed method has better performance encountered with types of signal steering vector mismatches.


I. Introduction
IN recent years, the robust adaptive beamforming (RAB) has been widely used in wireless communication, radio astronomy, radar, sonar [1]. Despite great development, solving the nonideal factors, such as, antenna manifold distortion, incoherent local scattering, imprecise knowledge of the expected signal steering vector [2], in practical engineering application is still a challenge. Those non-ideal factors will cause the mismatch between the actual and the presumed signal steering vector, which will seriously degrade the performance of traditional Capon beamforming [3]. Therefore, RAB has received extensive attention. Some examples of the existing RAB approaches are diagonal loading of the sample covariance matrix (LSMI) methods [4], spatial projection methods [5], indirect convex optimization modeling methods [6]- [10]. However the two former have worse performance at low signal-to-noise ratio (SNR), so the indirect convex optimization modeling methods is particularly concerned.
Typically, the method in [6] notices that the influence of various errors can be boiled down to the signal steering vector mismatch, and model RAB as the nonconvex optimization problem. They further establishes the uncertainty set by upper bounding the mismatch between the actual and the presumed signal steering vector, then approximate the nonconvex problem to the convex second-order cone program (SOCP) to solve it [7]. However, it is difficult to determine the upper bound properly in practice. In [8], the signal steering vector is modified by minimizing the projection of the desired signal steering vector on the interference subspace, where the nonconvex constraint on steering vector is approximated to convex. In order to further improve the performance, the target signal steering vector and the interference steering vector are constructed by the eigenvectors of the constructed matrix subspace [9], then RAB is approximated as the semi-positive definite program (SDP) neglecting the constraint on the unknown variables. The method in [10] establish the nonconvex amplitude response constraints with the prior knowledge, and approximate the problem to the convex SOCP.
Noteworthily, to obtain the convex optimization problem in the above methods, an approximate transformation is performed, which will introduce errors inevitably. In this paper, a novel RAB method is proposed to avoid the approximating errors. The proposed method reformulates RAB as the biconvex optimization directly, then solve it with alternating direction methods of multiplier (ADMM) algorithm. Compared with the existing methods, the simulation results show that the proposed method has better performance and validate its effectiveness. The paper is organized as follows. The background is given in Section II. We introduce the proposed method in Section III. Computer simulation results are presented in Section IV and conclusions are drawn in Section V.

II. Problem Fomulation
We assume that L + 1 narrowband signals impinge on a uniform linear array (ULA) with M omnidirectional sensors. The far-field narrowband signal received by the ULA at the time instant k can be expressed as where A (θ) = [a (θ 0 ) , a (θ 1 ) , · · · , a (θ L )] ∈ C M×L+1 is the direction matrix, θ= [θ 0 , θ 1 · · · , θ L ] is the direction-ofarrivals (DOAs) vector of the signal sources. Assume that θ 0 is the DOA of the expected signal, a (θ) = [1, e j2πd sin θ/λ , · · · , e j2πd(M−1) sin θ/λ ] T , d is inter-element space, λ is wavelength. s (k) ∈ C (L+1)×1 is signal waveform, n (k) is a Gaussian white noise. (·) T and (·) H stand for the transpose and Hermitian transpose, respectively. The traditional Minimum Variance Distortionless signal Response (MVDR) beamforming method obtains the weight vector by solving the following problem : whereã (θ 0 ) is the presumed signal steering vector, the sample covariance matrixR = K n=1 x (n)x H (n) /K, K is the number of training snapshots. The solution to (2) is: However, there is mismatch between the presumed steering vector and actual steering vector in practical applications. Let a (θ 0 ) =ã (θ 0 )+e , where a (θ 0 ) is the actual steering vector, e is the mismatch vector. Specifically, the mismatch will seriously affect the beamformer performance, which motivates the RAB to develop rapidly.
Let a = a (θ 0 ),ã =ã (θ 0 ), then a =ã + e. Assuming that the uncertainty set A (ε) ∆ = {a|a =ã + e, e 2 ≤ ε} contains the actual steering vector , ε is the upper bound of the norm of e. Then, the RAB problem can be written as The constraint in (4) is nonlinear nonconvex. There is currently no effective solution to tackle such problems. Therefore, the most of existing methods approximate it to a convex optimization problem, then apply the corresponding optimization toolkit to solve it.
The method in [6] approximate the constraint in (4) to w Hã + w H e ≥ w Hã − w H e ≥ w Hã − ε w by the Cauchy inequality to derive w H a = w Hã − ε w approximately. Eventually, the nonconvex problem (4) can be approximated as the following convex optimization problem: The value of ε is determined by the mismatch between the actual and the presumed signal steering vector. The above methods have the common characteristic-approximation which will introduce error into the original problem, resulting in the degradation of performance.

III. The Proposed Beamformer Method
Aiming at the above problems, the errors caused by approximation, a novel RAB method based on direct biconvex optimization modeling is proposed. In the proposed method, RAB is reformulated as a biconvex form with an auxiliary variable and solved by ADMM.

A. The proposed optimization model
Different from the uncertainty set established in (4), we establish an ideal signal response constraint to ensure the gain of the signal steering vector in the mismatch range and suppress the other signals out of the range: where λ (θ n ) = 1, θ n ∈ Θ i , Θ j 0, else , Θ i , Θ j is the priori error range in which the expected signal is located. α is the optimization scaling parameter [11], varying with the average value of P (θ n ) within Θ in each iteration. Then the optimization problem can be modeled as: where J (α, w) is a Lagrange function, ϕ n is weighting coefficients, which can be adjusted according to the actual requirements; N is the number of the angular intervals. In order to simplify (8), we rewrite it as: Thus, the optimization problem of RAB can be rewritten as: It is obvious that the optimization problem in (11) is a fourth-order nonconvex problem, and there is no effective solution to solve such issues directly. Thus, we introduce an auxiliary variable y ∈ C (M+1) to convert (11) into the biconvex optimization problem.
Through the above derivation of reformulating the nonconvex problem of (8) as the convex problem of (12) without approximating process, we avoid the approximation errors brought in [6]- [10].

B. Solution to the direct-biconvex optimization problem
We notice that the variable y, x are convex to the problem (12), which is suitable for the dual ascent and the augmented Lagrangian method. In addition, the ADMM algorithm is intended to blend the dual ascent with the augmented Lagrangian method, possessing the decomposability of dual ascent and the numerical robustness of the augmented Lagrangian method.Therefore, the ADMM algorithm is applied to solve the problem by alternating iterations.
To solve and accelerate the convergence of the problem (12), we introduce the dual variable z ∈ C (M+1) , obtaining the augmented Lagrange function expressed as: where ρ is a penalty parameter. The larger value of ρ will enforce the constraint in (12) more strongly, resulting in the smaller dual residuals in (22) but the larger primal residuals in (23), which can be adjusted as practical requirements. The coefficient selection of ϕ n is related to the signal power, because the latter term of the objective function in (12) contains the signal-interference-noise energy, due to the imprecise estimation about interference-noise matrix. When the target signal energy increases, the energy of echo signal and the coefficient must change accordingly. From the equation (10), in order to balance the value of the former against the latter in (12), the value of the coefficient ϕ n should be proportional to σ 2 n with scale factor κ, which make it is suitable to the condition of various echo signal power. Since the sample covariance matrix contains all the echo information, including signal energy, we define σ 2 n ∆ = 1/a(θ n ) HR−1 a (θ n ).
Furthermore, the function in (13) can be rewritten as we denote u = 1 ρ z, the cost function is simplified as Obviously, the variable y,x are convex to the problem in (15). Therefore, the ADMM can be utilized to estimate the variable y, x iteratively. At the (k + 1)th iteration, the forms of the closed-form solutions consists of y k+1 := arg min y L y, x k , u k x k+1 := arg min To solve (16), we need calculate the derivation of the Lagrange function in (15) with respect to y and x in each iteration, respectively.
(y + u) (18) Therefore, the closed-form solution of (16) is expressed as: At last, we define as the dual residuals at kth iteration and as the primal residual at kth iteration, respectively. The reasonable termination criterions are [12]: where ε abs > 0 is the absolute error and ε rel > 0 is the relative error. Based on the above derivation, the complete algorithm is as follows: The solution applied to the problem (13) 1)Initialize: y 0 , x 0 , u 0 , ρ, ε abs , ε rel ; 2)While the residuals are not satisfied the termination criterions in (25) do 3)Applying (19) to update y k+1 ; 4)Applying (20) to update x k+1 ; 5)Applying (21) to update u k+1 ; 6)k = k + 1; 7)End While

IV. Simulation Results
A ULA of 10 sensors with half-wavelength inter-element spacing is considered in this simulations. There are two interferers imping on the antenna from −30 • and 35 • . The interference-to-noise ratio(INR) is fixed at 30 dB. The additive noise is complex Gaussian white distributed with zero mean and unit covariance. The desired signal comes from the presumed direction θ = 3 • . The SMI is calculated by K = 30 snapshots. This section compares the proposed method with the following methods: SMI beamformer, LSMI beamformer, beamformer [6] and [8]. In the proposed method, it is assumed that the direction form −80 • to 80 • by step 1 • , N = 161, and the angular error range is Θ = [θ − 5 • , θ + 5 • ], which is same as the beamformer in [8]. In the LSMI beamformer, the diagonal loading factor is twice the noise energy. ε = 3 is used in the beamformer [6]. All simulation results are subjected to 200 independent Monte-Carlo independent experiments.

A. Setting of the ρ and κ
In the first experiment, assuming the steering vector mismatch due to the look direction mismatch and antenna manifold distortion, we will examine the performance of the proposed with different values of ρ and κ. The actual DOA of the expected signal is assumed to be random and uniformly distributed in [θ − 5 • , θ + 5 • ]. The phase distortions are independently chosen from the Gaussian generator with zero-mean and 0.04 standard deviation.
The output SINR performance against input SNR with ρ = 10, 50, 100, 300 is displayed in Fig.1. The results indicate that there is a range of ρ , which can provide stability and well performance to the proposed, e.g.,50 ≤ ρ ≤ 100 .
Next, how the scale factor κ affect the performance of the proposed method is revealed. With the same condition and fixing ρ at 50, the Fig.2 shows the output SINR performance against input SNR with κ = 10, 50, 100, 300, 500. The results demonstrate that, at low SNR, the smaller is, the better the performance is, contrarily, the larger κ provides better performance at high SNR. From the results, the performance can be satisfied with κ = 100 on the whole range of SNR. Output SINR(dB) OPTIMAL SMI Proposed beamformer LSMI Beamformer of [6] Beamformer of [8] (a) Output SINR(dB) OPTIMAL SMI Proposed beamformer LSMI Beamformer of [6] Beamformer of [8] (b)

B. Signal steering vector mismatch
In this experiments, we will examine the performance and robustness of the proposed when the signal steering vector is mismatched. From the above results, we set κ = 100, ρ = 50.
Case one: We assume the steering vector mismatch caused by look direction mismatch and antenna manifold distortion. In Fig.3, the mean output SINR are illustrated versus the input SNR as well as the number of snapshots. The actual DOA of the expected signal is assumed to be random and uniformly distributed in [θ − 5 • , θ + 5 • ]. The phase distortions are independently chosen from the Gaussian generator with zero-mean and 0.04 standard deviation.
Case two: The actual steering vector isã (k) = s 0 (k) a +    Fig.4 show the performances versus the input SNR and the number of snapshots with the incoherent local scattering mismatch. The results from the figures indicate that the proposed method possesses better performance and stronger robustness than the aforementioned methods. The improvement is attributed to avoiding the approximation process where errors are introduced.

C. Convergence
At last, we will test the convergence of the proposed method. In this experiment, The actual DOA of the expected signal is assumed to be random and uniformly distributed in [θ − 5 • , θ + 5 • ], INR=30 dB and SNR=-10 dB. The primal and dual residual norms, as well as the corresponding stopping criterion limits ε pri and ε dual versus the iteration number are shown in Fig.5. The results indicate that the proposed method will converge after about 60 iterations.

V. Conclusion
In this paper, we proposed a method which reformulates the RAB in a biconvex form directly making use of an auxiliary variable and apply the ADMM to solve it. The proposed avoids the error caused by the approximation process on which the original nonconvex problem is converted to be convex. The simulation results show that the proposed possesses better performance and stronger robustness with signal steering vector mismatch.