Elsevier

Signal Processing

Volume 131, February 2017, Pages 92-98
Signal Processing

Robust adaptive beamforming with minimum sensitivity to correlated random errors

https://doi.org/10.1016/j.sigpro.2016.08.004Get rights and content

Highlights

  • We propose a robust adaptive beamformer from the perspective of the general form of the beamformer sensitivity, in which we consider the correlated random errors.

  • We find that the inverse of the sample covariance matrix is a reasonable choice for the random error covariance.

  • Numerical results demonstrate the superior performance of the proposed beamformer in the presence of large mismatch relative to other existing approaches.

Abstract

The standard Capon beamformer is subject to substantial performance degradation in the presence of estimation errors of the signal steering vector and the array covariance matrix. In order to address this problem, robust adaptive beamformers (RABs) have been designed. In this study, we propose a novel RAB from the perspective of the beamformer sensitivity. In particular, we consider the general form of the beamformer sensitivity, implying that the random errors may be not white noise but correlated. Then we suggest to use the inverse of the array sample covariance matrix as the random error covariance. Using this, we propose to compute the Capon beamformer with minimum sensitivity to correlated random errors, considering a Euclidean ball as the uncertainty set for the signal steering vector. Moreover, the Lagrange multiplier methodology can be employed to solve the proposed optimization problem. Numerical results demonstrate the superior performance of the proposed beamformer in the presence of large mismatch relative to other existing approaches such as ‘diagonal loading’, ‘robust Capon’ and ‘maximally robust Capon’ beamformers.

Introduction

Capon beamformer is a representative example of adaptive array beamformer which intends to allow the signal of interest (SOI) to pass through without any distortion while the interference signals and noise are suppressed as much as possible, thereby maximizing the output signal-to-interference-plus-noise ratio (SINR). The standard Capon beamformer (SCB) can be formulated asminwwHR^ws.t.wHa¯=1with the solutionwc=βcR^1a¯where a¯ denotes the nominal SOI steering vector. The immaterial scalar βc=1a¯HR^1a¯ does not affect the array output SINR. The estimated covariance matrix R^ can be formed byR^=1Kk=1Kx(k)xH(k)where {x(k)}k=1K denote the array observations or snapshots. Performing eigen-decomposition on R^ yieldsR^=U^Γ^U^H=i=1Nγ^iu^iu^iHwhere N is the array sensor number. The matrix U^=[u^1,,u^N] collects all the eigenvectors, and Γ^=diag{γ^1,,γ^N} is a diagonal matrix with the eigenvalues γ^1γ^N being nonincreasingly ordered. By incorporating knowledge of the white noise variance σ2n, we can obtain the maximum likelihood estimation with a noise floor constraint [1]R^ML=i=1Nmax{γ^i,σn2}u^iu^iH.

It has been pointed out that the SCB may suffer from substantial performance degradations even for small mismatch between the presumed a¯ (or R^) and its actual value a0 (or R) [2], [3], [4], [5]. This is because in such situation the SOI may be treated as an interference signal and therefore be suppressed erroneously, which is commonly referred to as signal self-nulling [10]. In order to address this problem, some robust adaptive beamformers (RABs) have been designed which aim to provide acceptable performance even when the nominal steering vector and covariance matrix depart from their actual values. An excellent review and comparison of the existing robust techniques have been provided in [11], [12]; see also the references contained therein.

The most popular RAB technique is the diagonal loading (DL) method [2] along with its generalized versions [3], [4], [5]. The conventional DL beamformer can be formulated asminwwHR^w+ξwHws.t.wHa¯=1.The optimum solution of (6) is given byw^DL=βDL(R^+ξI)1a¯where I denotes the identity matrix with appropriate size and the scaling factor βDL=1a¯H(R^+ξI)1a¯ is also immaterial. We can see that the DL method in (6) differs from the SCB of (1) in that an additional term ξwHw is used, which can be explained by the following fact. When the signal self-nulling occurs, we have wHa00 (where a0 stands for the true steering vector of the SOI) and simultaneously we also have wHa¯=1 due to the distortionless constraint on the nominal SOI. Consequently, we can obtain the approximation expression wH(a¯a0)1. However, the nominal and actual steering vectors are often close and hence a¯a0 is relatively small, where the notation · represents the Euclidean norm. This implies that the relation wH(a¯a0)1 holds only if w is large [13]. Therefore, we use the term ξwHw in (6) to prevent the norm of the weight vector to become large and in turn avoid signal self-cancellation. In the traditional DL method [2], the loading factor ξ is set in an ad hoc way, typically ξ=10σn2 where σn denotes the noise power.

The generalized versions of DL [3], [4], [5] specifically attempt to establish the relationship between the loading factor and the steering vector uncertainty level. For example, in the RAB presented in [3] it is assumed that the true SOI steering vector belongs to the uncertainty setA(ε){a|aa¯ε}where ε is the pre-selected upper bound on the norm of the steering vector mismatch. Then the RAB in [3] maintains a gain no less than unity within the uncertainty set, while minimizing the output power. That is minwwHR^ws.t.|wHa|1,aA(ε).Moreover, it is found that the least gain within the uncertainty set, which corresponds to the worst-case steering vector, has the following form [3]:minaA(ε)|wHa|=|wHa¯|εw1.As a consequence, the problem in (9) can be rewritten in the convex second order cone programming form and be solved efficiently using the interior point method. It is also shown that this RAB technique based on worst-case performance optimization belongs to the diagonal loading approaches. Also, in [4] it is shown that the RABs proposed in [3], [4], [5] are equivalent and the essence of them is to replace the nominal SOI steering vector by the vector from the presumed uncertainty set, which results in the maximum output power.

Recently, a novel RAB has been considered in [10] from the perspective of the beamformer sensitivity which is defined asTsew2|wHa¯|2.From (10), we observe that the largest deviation of the array gain within the uncertainty set is εw. Therefore, the beamformer sensitivity measures the relative deviation in array response (which is normalized by the uncertainty level ε). Then, the beamforming problem in [10] is formulated as mina,ww2|wHa¯|2s.t.w=R^1aaHR^1aaA(ε).The basic idea of (12) is to find the vector within the uncertainty set which achieves the minimum beamformer sensitivity. As will be shown subsequently, the assumption of uncorrelated random errors is used implicitly in [10]. In other words, the signals are assumed to be perturbed by the white noise. However, this white noise assumption is not always satisfied [6].

In this paper, we suggest a robust adaptive beamforming which is also based on the beamformer sensitivity. Here we treat a more general case in which the signals are perturbed by correlated random errors. Then we find that it is reasonable to use the inversion of the sample covariance matrix as the random error covariance. We consider a beamformer optimization problem which intends to obtain the minimum beamformer sensitivity to the correlated random errors. Such optimization problem can be solved by the Lagrange multiplier methodology.

Section snippets

Proposed robust beamformer

In [2], a general definition of the beamformer sensitivity is given byTg-sewHEw|wHa¯|2where the matrix E denotes the covariance of the random errors. It is pointed out in [2] that Tg-se is a classic measure of sensitivity to tolerance errors. When the errors are uncorrelated, the covariance becomes the identity matrix and thus the above definition reduces to that in (11). However, this white noise assumption is not always valid. If we take the general form into account, the problem that may

Simulation results

Assume that three signals (one SOI plus two interferers) are incident on a uniform linear array with N=10 isotropic sensors and half-wavelength sensor spacing. The direction-of-arrivals (DOAs) of these two interfering signals are fixed at [θ1,θ2]=[30°,50°]. For each interfering signal, the interference-to-noise ratio (INR) in a single sensor is equal to 30 dB. Unless stated otherwise, we use K=50 snapshots to estimate the covariance matrix, the input signal-to-noise ratio (SNR) is 10dB, the

Conclusions

The beamformer sensitivity is a classic measure of sensitivity to tolerance errors. In many practical applications, however, the random errors may be not white noise but correlated. Despite the lack of the accurate knowledge of the random error covariance matrix, we find that the inverse of the sample covariance matrix can be a good substitute of the random error covariance. Then we propose to compute the Capon beamformer with minimum sensitivity to correlated random errors. Furthermore, the

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61571090, China Postdoctoral Science Foundation under Grant 2015M580785 and the Fundamental Research Funds for the Central Universities under Grant ZYGX2014J007.

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