International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 3743509, 12 pages

http://dx.doi.org/10.1155/2016/3743509

## High Performance Robust Adaptive Beamforming in the Presence of Array Imperfections

^{1}College of Information and Communications Engineering, Harbin Engineering University, Harbin 150001, China^{2}Naval Academy of Armament, Beijing 100161, China

Received 18 April 2016; Accepted 8 June 2016

Academic Editor: Shih Yuan Chen

Copyright © 2016 Wenxing Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–3]. Beamformers can be regarded as spatial filters, which can enhance the desired signal and suppress the interference effectively. The standard Capon beamformer (SCB), as one of the well-known adaptive beamformers, has excellent resolution and interference rejection capability. However, the SCB is sensitive to the model mismatches, especially, when the signal of interest (SOI) is presented in the training data [4–7]. In some applications like passive sonar and wireless communications, the training data usually contains the SOI. Thus, the performance of the adaptive beamformers may degrade significantly in the presence of array imperfections.

To improve the robustness of adaptive beamformers, many robust adaptive beamforming methods have been developed over the past several decades [8–12]. In [8], a diagonal loading method is proposed to improve the robustness of array against array steering vector (ASV) and covariance matrix mismatches. However, it does not provide any guidance to select the optimal diagonal loading factor. In [9], a robust adaptive beamforming scheme is obtained which aims to cope with the worst-case performance optimization, where the array steering vector is assumed to lie in an uncertain ellipsoidal set. It has been shown that this beamformer coping with the worst-case belongs to a kind of diagonal loading techniques, where the optimal diagonal loading factor can be adjusted according to the ellipsoidal uncertainty set. The robustness of the worst-case beamformer has been greatly improved compared with the simplest diagonal loading method. However, the performance of the worst-case beamformer is mainly determined by the uncertain parameter set, and the uncertainty of the ASV mismatch is difficult to be known accurately in practice.

Robust adaptive beamforming based on steering vector estimation has been proposed in [10]. To estimate the steering vector, one needs to maximize the beamformer output power and guarantee the ASV does not converge to any interference steering vectors or their linear combinations, which is a quadratically constrained quadratic programming (QCQP) problem and can be converted to semidefinite programming (SDP). Certainly, the global optimal solution can be found efficiently.

Recently, robust adaptive beamforming based on interference-plus-noise (IPN) covariance matrix reconstruction and ASV estimation has been proposed in [11]; the IPN covariance matrix was reconstructed by utilizing the Capon spectrum to integrate over a region separated from the SOI direction. This method can achieve good performance in the case of ASV direction error. However, this method is ineffective in the presence of array calibration errors, especially in low signal-to-noise ratios (SNRs) [12, 13]. In [14], a modified method to reconstruct the IPN covariance matrix has been given, where an annulus uncertainty set is used to constrain the steering vectors of the interference. Then integrate the Capon spectrum over the surface of the annulus, by which the reconstructed IPN matrix without containing the SOI can be obtained. Compared with method in [11], this method can reduce the sensitivity of beamformers to array calibration errors, but the performance improvement is limited, especially at low SNRs.

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 [11]. The second part is obtained by removing the SOI component from the sample covariance matrix, where the SOI component is estimated through eigendecomposition of the sample covariance matrix. Then a weighted summation of two parts is used to reconstruct the IPN matrix, and the weighting parameter is related to the desired signal energy compared with the interference energy. The detailed investigation of the parameters is also provided. To further improve the performance of the proposed method in low SNR, a hybrid method is proposed by using the diagonal loading method in the IPN matrix reconstruction to overcome overestimation of the signal subspace. In order to estimate the actual ASV of the desired signal, the presumed steering vector of SOI is subsequently corrected by using the orthogonal constraint. The estimated ASV is enforced to keep orthogonal to the noise subspace, while avoiding convergence to any of the interference steering vectors. That means the estimated ASV can only converge to the actual ASV of the desired signal.

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.

#### 2. The Signal Model

We consider a uniform linear array (ULA) with unidirectional antennas with spacing . We assume that there are signals arriving from the directions . The received data of the array can be expressed aswhere is array observations data vector. is the time index. , and denotes the complex waveform of the th signal. Here, is considered as the SOI, while , are the interference. is a vector of the additive white sensor noise, , where represents a steering vector in the direction, and is the wave number that can be represented as .

We assume that the signal and noise are statistically independent. The output of the beamformer is given bywhere is the optimal weight vector.

The minimum variance distortionless response (MVDR) beamformer is formulated as the following linearly constrained quadratic optimization problem: where is the presumed ASV of the SOI, is the optimal weight vector, and is the IPN covariance matrix. In practice, is commonly replaced by the sample covariance matrixwhere is the number of snapshots. Thus, the optimal solution to (3) is

The solution (5) is commonly referred to as the sample matrix inverse (SMI). The output SINR of the beamformer is defined as where is the power of the desired signal and is the actual ASV of the desired signal. The standard MVDR beamformer can produce sharp nulls at the direction of interference with a good interference rejection performance and high output SINR in the ideal case. However, in practice, the knowledge of ASVs can be imprecise, which means the mismatch may exist between the presumed and actual ASVs. In such a case, the standard MVDR beamformer may attempt to suppress the desired signal as it was interference. The performance of the beamformer will degrade seriously.

Recently, in [11], the authors proposed a robust beamforming technique by reconstructing the IPN covariance matrix, which uses the following Capon spectrum as an estimate of the spatial power spectrum over all possible directions [15]. It is well known that the Capon spatial spectrum estimator iswhere is the ASV associated with direction that has the structure defined by the antenna array geometry. There are also other candidate spatial spectrum estimators except (7). Using the Capon spatial spectrum, the IPN covariance matrix can be reconstructed aswhere is the complement sector of in the whole spatial domain and is an angular sector in which the desired signal may be located. It can be observed from (8) that is obtained by collecting the information of interference and noise in the sector , and the desired signal was not included in it as long as the direction of SOI is located in . That means that the effect of the desired signal has been removed from the reconstructed covariance matrix, and thus superior performance can be provided compared with the existing robust beamforming method in the case of ASV errors. However, this method is based on the premise that the precise information about the array structure is known exactly in advance, which is almost impossible in practice. As a consequence, the method in [11] will be ineffective in the presence of array calibration errors, such as the gain and phase perturbations and antenna location error. The performance of this method degraded seriously in the case of array calibration error, especially in low input SNRs [12, 13].

#### 3. The Proposed Algorithm

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 which is expressed as in (8). As for the second part, the desired signal component is estimated from the eigenvectors of the sample covariance matrix , and the desired signal can be removed from . Thus, the rest of the covariance matrix can be regarded as the second part of the estimated IPN matrix. A weighted summation of two parts is used to reconstruct the IPN matrix, and the weighting parameter is used to reflect the desired signal energy compared with the interference energy. With the reconstructed IPN matrix, the presumed ASV of the desired signal is subsequently corrected by solving the QCQP problem.

##### 3.1. The Basic Proposed Beamformer

The sample covariance matrix defined by (4) can be decomposed aswhere , are the eigenvalues of and , are the corresponding eigenvectors. represents the signal-plus-interference (SPI) subspace, which is formed by the interference plus a SOI. represents the noise subspace, are the eigenvalues of the SPI, and are the eigenvalues of noise. As we know, the eigenvectors of and the ASVs of the SPI lie in the same subspace. What is more, the mismatch between and is not too large in fact. We project the presumed ASV of the SOI onto the eigenvectors to get . can be expressed as

The projections can be arranged in descending order, as . Meanwhile, the corresponding eigenvectors can be arranged as , and the corresponding eigenvectors can be arranged as . It is important to note that the eigenvectors and the corresponding eigenvalues are the same as in (9), but they have been reordered according to .

As well known, the maximum of the projections is obtained when is the eigenvector corresponding to the SOI [12]. That means the eigenvector corresponding to the SOI is determined according to the projections . It is easy to see that is the eigenvector of SOI and is the corresponding eigenvalue. We remove and from and obtain . Then, we have

Since the SOI component has been removed from , can be regarded as a IPN matrix. We now propose a new method to estimate the IPN matrix by using the weighted combination as follows: where and . We can see that the parameter , which can reflect the SOI energy compared with the interference energy. The parameter is used to adjust the proportion of and according to the input SNR. When the input SNR is high, is close to 1, and is mainly composed of . When the input SNR is low, is close to 0, and is mainly composed of .

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:where is the corrected ASV, , , denotes the Euclidean norm, and is the norm bound of the mismatches. The objective function (13) keeps the estimated within the signal or interference limits, while the constraints (14) and (15) can guarantee that does not converge to any of the interference steering vectors. Thus, the accurate ASV of the desired signal can be obtained.

This optimization problem can be efficiently solved by convex optimization toolbox [16]. Finally, with the corrected ASV and estimated IPN matrix , the proposed beamformer weighting vector can be calculated as

##### 3.2. The Improvement of Basic Beamformer

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 dB due to the overestimation of signal subspace, which is an inherent shortcoming of the subspace decomposition. However, the LSMI, worst-case, and SDP-RAB perform almost equivalently when dB. That is to say, we prefer to use LSMI method in low SNR due to its low computational complexity and good performance.

We can use the parameter to reflect the input SNR directly, which can be expressed as

As we all know, if the input SNR is very small, , whereas, in high SNR situations, the large value of can be achieved. Here, we give an example to discuss the relationship between and the input SNR.

We consider a ULA of antennas spaced at a half wavelength distance. Additive noise is modeled as independent complex Gaussian noise with zero mean and unit variance. Two independent interference vectors are from the directions of and , respectively. The interference-to-noise ratios (INRs) of the interference are 20 dB unless otherwise is specified. The sample covariance matrix is collected based on data snapshots. The SOI is assumed from the direction of . The possible angular sector of the SOI is set to , so the complement sector is and the parameter . The difference between the presumed and actual positions of each antenna element is modeled as a uniform random variable distributed in the interval , where represents the wavelength. The actual DOA of SOI is , which means the DOA mismatch is . Figure 1 shows the values of versus input SNRs.