Robust adaptive beamforming with minimum sensitivity to correlated random errors
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 aswith the solutionwhere denotes the nominal SOI steering vector. The immaterial scalar does not affect the array output SINR. The estimated covariance matrix can be formed bywhere denote the array observations or snapshots. Performing eigen-decomposition on yieldswhere N is the array sensor number. The matrix collects all the eigenvectors, and is a diagonal matrix with the eigenvalues 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]
It has been pointed out that the SCB may suffer from substantial performance degradations even for small mismatch between the presumed (or ) and its actual value (or ) [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 asThe optimum solution of (6) is given bywhere denotes the identity matrix with appropriate size and the scaling factor is also immaterial. We can see that the DL method in (6) differs from the SCB of (1) in that an additional term is used, which can be explained by the following fact. When the signal self-nulling occurs, we have (where stands for the true steering vector of the SOI) and simultaneously we also have due to the distortionless constraint on the nominal SOI. Consequently, we can obtain the approximation expression . However, the nominal and actual steering vectors are often close and hence is relatively small, where the notation represents the Euclidean norm. This implies that the relation holds only if is large [13]. Therefore, we use the term 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 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 setwhere 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 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]: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 asFrom (10), we observe that the largest deviation of the array gain within the uncertainty set is . 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 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 bywhere the matrix denotes the covariance of the random errors. It is pointed out in [2] that 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 . 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 , 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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2020, Signal ProcessingCitation Excerpt :Vorobyov has provided an excellent review and comparison of the existing robust techniques in [2]; see also the references contained therein. Roughly speaking, these robust methods can be categorized into two main groups [3]: methods based on previous mismatch assumptions (such as [4–8]) and techniques that estimate the mismatch or equivalently the actual steering vector (such as [9–15]). Among these approaches, the diagonal loading (DL) beamformer and its extension versions may be the most common, in which the input data consist of the raw data collected by array sensors and an artificial white Gaussian noise [16,17], thereby reducing the input signal-to-noise ratio (SNR) and detuning the beamformer response at the main lobe to gain the robustness.
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