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A high performance robust beamforming scheme is proposed to combat model mismatch. Our method lies in the novel construction of interference-plus-noise (IPN) covariance matrix. The IPN covariance matrix consists of two parts. The first part is obtained by utilizing the Capon spectrum estimator integrated over a region separated from the direction of the desired signal and the second part is acquired by removing the desired signal component from the sample covariance matrix. Then a weighted summation of these two parts is utilized to reconstruct the IPN matrix. Moreover, a steering vector estimation method based on orthogonal constraint is also proposed. In this method, the presumed steering vector is corrected via orthogonal constraint under the condition where the estimation does not converge to any of the interference steering vectors. To further improve the proposed method in low signal-to-noise ratio (SNR), a hybrid method is proposed by incorporating the diagonal loading method into the IPN matrix reconstruction. Finally, various simulations are performed to demonstrate that the proposed beamformer provides strong robustness against a variety of array mismatches. The output signal-to-interference-plus-noise ratio (SINR) improvement of the beamformer due to the proposed method is significant.

Adaptive beamforming is one of the important aspects of array processing, which has been widely used in radar, sonar, mobile communications, radio astronomy, and other fields [

To improve the robustness of adaptive beamformers, many robust adaptive beamforming methods have been developed over the past several decades [

Robust adaptive beamforming based on steering vector estimation has been proposed in [

Recently, robust adaptive beamforming based on interference-plus-noise (IPN) covariance matrix reconstruction and ASV estimation has been proposed in [

In this paper, we propose a novel IPN covariance matrix reconstruction method. The estimated IPN covariance matrix consists of two parts. The first part is obtained by utilizing the method proposed in [

Simulation results show that the output signal-to-interference-plus-noise ratio (SINR) of the proposed adaptive beamforming is closer to the optimal value than other previously developed robust beamforming methods in the presence of various array imperfections, especially when the array calibration error exists. Performance improvement due to the proposed method approach is significant.

We consider a uniform linear array (ULA) with

We assume that the signal and noise are statistically independent. The output of the beamformer

The minimum variance distortionless response (MVDR) beamformer is formulated as the following linearly constrained quadratic optimization problem:

The solution (

Recently, in [

In this section, we propose a novel method to reconstruct the IPN matrix. The estimated IPN matrix consists of two parts. The first part is

The sample covariance matrix

The projections

As well known, the maximum of the projections

Since the SOI component has been removed from

Since the actual steering vector of the desired signal is difficult to obtain in practical applications and the mismatches between the presumed and actual ASVs cause significant performance degradation, here, we propose a new ASV estimation method based on the orthogonal constraint. As we all know, the actual ASVs of the signal and interference should be orthogonal to the noise subspace, which means that we can obtain accurate ASV of the desired signal by using orthogonal constraint under the condition where the estimate does not converge to any of the interference steering vectors. Taking into consideration the mismatch norm constraint, the problem of estimating the ASV of the desired signal based on orthogonal constraint can be formulated according to the following optimization problem:

This optimization problem can be efficiently solved by convex optimization toolbox [

As we all know, the signal and noise subspaces of the eigencomposition cannot be accurately separated in practice, especially when the SNR is low. As a result, the performance of the basic proposed beamformer is not so good when

We can use the parameter

As we all know, if the input SNR is very small,

We consider a ULA of

Values of

We can observe from Figure

Output SINR of beamformers versus input SNR for the case of perturbations in antenna array geometry.

We can see that the improved reconstruction method outperforms the original method significantly in low SNRs, which can get the high output SINR performance in both of them at low and high SNR.

The main computational complexity of the proposed method is the IPN matrix reconstruction problem and the QCQP problem. The computational complexity of the former is

In this section, the basic simulation conditions are the same as above unless otherwise is specified. The proposed beamformer is investigated and compared with the diagonally loaded SMI (LSMI) [

In this example, the performance of the proposed beamformer against the random desired signal direction error is investigated. We assume that the random DOA estimation mismatch of the SOI is uniformly distributed in

Figure

Figure

Output SINR versus the input SNR in the case of look direction mismatch.

Output SINR versus the number of snapshots in the case of look direction mismatch.

In this simulation, we consider that the desired signal ASV is distorted by the effects of wave propagation in an inhomogeneous medium which is used in [

It can be observed from these figures that the output SINR of LSMI decreased sharply with an increment of the input SNR. The proposed beamformer enjoys best performance compared to other beamformers in the case of wavefront distortion in the whole SNR range. In particular, the deviation from the optimal is only 0.3 dB when SNR = 10 dB. As described above, the performance of the proposed method degraded in small number of snapshots due to the inaccurate estimation of the SOI component.

Output SINRs versus input SNR in the case of wavefront distortion.

Output SINRs versus the number of snapshots in the case of wavefront distortion.

In this simulation, we consider that the desired ASV is distorted by local scattering effects. The actual ASV is formed by several signal paths, which can be modeled as

Figure

Output SINR of beamformers versus input SNR in the case of coherent local scattering.

Output SINR versus the number of snapshots in the case of coherent local scattering.

In this simulation, we study the effect of the error in the knowledge of antenna array geometry on the performance of the tested beamformer. The difference between the presumed and actual positions of each antenna element is modeled as a uniform random variable distributed in the interval

Figure

Output SINR of beamformers versus input SNR for the case of perturbations in antenna array geometry.

Output SINR versus the number of snapshots for the case of perturbations in antenna array geometry.

In this simulation, we investigate the performance of the proposed beamformer when arbitrary ASV errors are considered. Here, the ASV mismatch is comprehensive and arbitrary-type, which may be caused by direction errors, calibration errors, gain and phase perturbations, and so on. The actual ASVs of are modeled as

The output SINR of the beamformers versus input SNR for

In general, fewer snapshots mean worse performance for certain beamformer, and some algorithms may outperform the proposed method with a small number of snapshots. We can find that the proposed beamformers enjoy the best performance

Output SINR of beamformers versus input SNR with arbitrary ASV error.

Output SINR of beamformers versus snapshots with arbitrary ASV error.

The main idea of the proposed method is to estimate the IPN matrix by using a weighted summation of two parts (

First, we investigate the performance of

In this section, we consider the effect of the error in the knowledge of antenna array geometry on the performance of the beamformers. The basic simulation conditions are the same as the simulation experiment 4 unless otherwise is specified. The INR of the interference is 20 dB unless otherwise is specified.

Figure

Output SINR versus the input SNR with different reconstructed IPN matrixes.

Figure

Values of

This is the initial idea. We need to investigate the reconstruction method through the same simulation experiment, and the simulation results were shown in Figure

Output SINR versus the input SNR with different reconstruction method.

We can clearly see from Figure

Output SINR versus the input SNR with different

It can be seen from Figure

Output SINR versus the input SNR with different

Output SINR versus the input SNR with different

Figure

It is worth noting that

Values of

We can observe from Figure

Effectiveness of our designed method is verified on the basis of the same simulation experiments. Figure

Output SINR versus the input SNR (INR = 30 dB).

Output SINR versus the input SNR (INR = 40 dB).

We can see from Figures

In order to further improve the robustness of adaptive beamformer, a high performance robust adaptive beamformer in the presence of various kinds of array imperfections was proposed and its performance was verified in detail. The proposed beamformer was realized via a modified method to reconstruct the IPN covariance matrix. The IPN covariance matrix comes from a weighted summation of two estimated covariance matrices and the proportion of the two estimated covariance matrices can be adjusted adaptively according to the input SNR and interference power. The simulation results demonstrate that the proposed beamformer can provide a superior performance against unknown arbitrary-type mismatches in a very large range of SNR.

The authors declare that there are no competing interests regarding the publication of this paper.

This paper was supported by National Defense “973” Basic Research Development Program of China (no. 6131380101). This paper is also supported by Preresearch Fund of the 12th Five-Year Plan (no. 4010403020102 and no. 4010103020103) and the Fundamental Research Funds for the Central Universities (nos. HEUCFD1433 and HEUCF1508).