Abstract—This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon’s method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples. Index Terms—Ellipsoidal calculus, Hadamard product, robust beamforming, second-order cone programming.

Abstract—In this letter, we show that worst-case robust beamforming, with uncertain weights subject to multiplicative variations, can be cast as a convex optimization problem. We interpret this problem as a weighted complex I-regularization of the nominal beamforming problem, and show that it can be solved with the same computational complexity as nominal beamforming, ignoring the variations. We derive a simple lower bound on how much worse the robust beamformer will be compared to the nominal beamformer solution with no weight uncertainty. We demonstrate the robust approach with a simple narrowband beamformer. Index Terms—Regularization, robust beamforming, robust optimization, robust sensor array signal processing. I. BEAMFORMING WE consider an array of sensor elements. Let be the array response to a wave of unit amplitude

It is well known that the performance of the minimum variance distortionless response (MVDR) beamformer is very sensitive to steering vector mismatch. Such mismatches can occur as a result of direction-of-arrival (DOA) errors, local scattering, near-far spatial signature mismatch, waveform distortion, source spreading, imperfectly calibrated arrays and distorted antenna shape. In this paper, an adaptive beamformer that is robust against the DOA mismatch is proposed. This method imposes two quadratic constraints such that the magnitude responses of two steering vectors exceed unity. Then, a diagonal loading method is used to force the magnitude responses at the arrival angles between these two steering vectors to exceed unity. Therefore, this method can always force the gains at a desired range of angles to exceed a constant level while suppressing the interferences and noise. A closed-form solution to the proposed minimization problem is introduced, and the diagonal loading factor can be computed systematically by a proposed algorithm. Numerical examples show that this method has excellent signal-to-interference-plus-noise ratio performance and a complexity comparable to the standard MVDR beamformer.

Abstract—We address the matched detector problem in the case the signal to be detected is imperfectly known. While in the standard detector the signal is known to lie along a particular direction, we consider the case where this direction is known up to additive white Gaussian noise. This somehow amounts to assuming that the signal lies in a cone the aperture of which depends upon the level of uncertainty. We build the associated generalized likelihood ratio (GLR), analyze its statistical properties, indicate how to set the threshold to achieve a given false alarm rate, and how to predict the associated probability of detection. The so-obtained detector reduces to the conventional one when the uncertainty vanishes and we analyze its behavior when the level of uncertainty, which has to be known a priori, is mis-evaluated. It appears that the sensitivity of the detector is quite low with respect to this kind of errors. More importantly several realistic examples are presented that indicate

Multiantenna MIMO channels have recently become a popular means to increase the spectral efficiency and quality of wireless communications by the use of spatial diversity at both sides of the link [1–4]. In fact, the MIMO concept is much more general and embraces many other scenarios such as wireline digital subscriber line (DSL) systems [5] and singleantenna

We consider the array signal processing problem of choosing the weight vector to minimize noise power, subject to a unit array gain for the desired wave, and subject to rejection constraints on interferences. We model the variations in the array response with ellipsoidal uncertainty, and take the worst-case robust optimization approach, i.e., we require the constraints to hold for all possible data in the uncertainty ellipsoid. We show that this robust array signal processing problem can be formulated as a second-order cone program, which interior-point algorithms can solve efficiently. The robust solution is demonstrated with an example.

An order recursive algorithm for minimum mean square error (MMSE) estimation of signals under a Bayesian model defined on the steering vector is introduced. The MMSE estimate can be viewed as a mixture of conditional MMSE estimates weighted by the posterior probability density function (PDF) of the random steering vector given the observed data. This paper derives an adaptive closed form Kalman-filter implementation that updates the weight vector by successive incorporations of data collected from additional array elements in the steering vector. The performance of the Bayesian beamformer is compared against several robust beamformers in terms of mean square error (MSE) and output signal-to-interference-plus-noise ratio (SINR). 1. BACKGROUND The received data vector of an N-element sensor array at sample time k has the form x(k) =a s ∗ (k)+i(k)+n(k), (1) where s(k) is the desired signal with known power σ 2 s, a ∈ C N is the steering vector, i(k) is the interence and n(k) is the noise. Let Ri+n � E[(i(k) +n(k))(i(k) +n(k)) H] be the interference-plus-noise covariance. Let (·) ∗ , (·) T and (·) H be the complex conjugate, transpose and Hermitian transpose, respectively. Assume that s(k), i(k) and n(k) are zero mean, temporally white, complex Gaussian processes that are mutually independent to each other. In practice, the true steering vector often deviates from its presumed value for various reasons such as improper array modeling, asynchronous sampling, pointing error, miscalibration, or source motion. It is often reasonable to model these

Abstract—A Bayesian approach to adaptive narrowband beamforming for uncertain source direction-of-arrival (DOA) is presented. The DOA is modeled as a random variable with prior statistics that describe its level of uncertainty. The Bayesian beamformer is the corresponding minimum mean-square error (MMSE) estimator, which can be viewed as a mixture of directional beamformers combined according to the posterior distribution of the DOA given the data. Under a deterministic DOA, the mean-square error (MSE) of the Bayesian beamformer becomes as low as that of the directional beamformer equipped with the DOA candidate in the prior set that is the closest to the true DOA at exponential rate, where closeness is defined in the Kullback–Leibler sense. Two efficient algorithms using a uniform linear array (ULA) are presented. The first method utilizes the efficiency of the fast Fourier transform (FFT) to compute the posterior distribution on a large number of DOA candidates. The second method approximates the posterior distribution by a Gaussian distribution, which leads to a directional beamformer incorporated with a particular spreading matrix and an adjusted DOA. Numerical simulations show that the proposed beamformer outperforms other related blind or robust beamforming algorithms over a wide range of signal-to-noise ratios (SNRs). Index Terms—Adaptive beamforming, Bayesian model, direction-of-arrival (DOA) uncertainty, minimum mean-square error (MMSE) estimation. I.

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Abstract—In downlink beamforming in a multiple-input multiple-output (MIMO) wireless communication system, we design beamformers that minimize the power subject to guaranteeing given signal-to-interference noise ratio (SINR) threshold levels for the users, assuming that the channel responses between the base station and the users are known exactly. In robust downlink beamforming, we take into account uncertainties in the channel vectors, by designing beamformers that minimize the power subject to guaranteeing given SINR threshold levels over the given set of possible channel vectors. When the uncertainties in channel vectors are described by complex uncertainty ellipsoids, we show that the associated worst-case robust beamforming problem can be solved efficiently using an iterative method. The method uses an alternating sequence of optimization and worstcase analysis steps, where at each step we solve a convex optimization problem using efficient interior-point methods. Typically, the method provides a fairly robust beamformer design within 5–10 iterations. The robust downlink beamforming method is demonstrated with a numerical example. I.