Robust Beamforming Research Articles

Robust beamforming with imprecise array geometry using steering vector estimation and interference covariance matrix reconstruction

Compared with state-of-the-art robust adaptive beamforming methods, the recently developed steering vector estimation-based beamformer and interference-plus-noise covariance matrix reconstruction-based beamformer are known to provide better robustness against model mismatch. However, both methods may suffer from performance degradation in presence of antenna array geometry error. To alleviate this problem, a novel algorithm is proposed in this paper. In contrast to previous works, the true desired signal steering vector is estimated by solving a new optimization problem, the objective function of which is minimizing the beamformer sensitivity, while the constraint of which is obtained by optimizing the worst-case performance of the constraint adopted in conventional steering vector estimation-based method. We show that the solution of the newly constructed optimization problem can be obtained in closed form by using the Lagrange multiplier methodology, which has a comparable computational complexity with that of the standard Capon beamformer. Subsequently, the precise interference-plus-noise covariance matrix is reconstructed by eliminating the estimated desired signal component from the sample covariance matrix, which mitigates computational burden and accumulated errors induced by the integration process employed in the conventional covariance matrix reconstruction-based method. Owing to the high accuracy of estimated and reconstructed results, the proposed robust adaptive beamforming algorithm can maintain satisfactory performance in cases with imprecise array geometry. Simulation results verify that the proposed method outperforms the existing ones in many scenarios.

Robust beamformer based on magnitude response constraint and steering vector correction

To form a beam toward the direction of a desired signal against steering vector mismatch, proposed is a robust beamformer based on magnitude response constraint and steering vector correction. First, the direction of the desired signal is restricted in a possible angular sector. An initial antenna array weight is obtained by adding a magnitude response constraint. Then, the presumed steering vector can be corrected iteratively based on a new objective function and constraint condition. Finally, an optimal weight and corrected steering vector can be obtained at the same time when the iterative process is accomplished. It is shown that both the mean square error of the corrected steering vector and the output signal-to-interference-plus-noise ratio of the proposed method outperform traditional beamformers.

Exploiting Known Interference as Green Signal Power for Downlink Beamforming Optimization

We propose a data-aided transmit beamforming scheme for the multi-user multiple-input-single-output (MISO) downlink channel. While conventional beamforming schemes aim at the minimization of the transmit power subject to suppressing interference to guarantee quality of service (QoS) constraints, here we use the knowledge of both data and channel state information (CSI) at the transmitter to exploit, rather than suppress, constructive interference. More specifically, we design a new precoding scheme for the MISO downlink that minimizes the transmit power for generic phase shift keying (PSK) modulated signals. The proposed precoder reduces the transmit power compared to conventional schemes, by adapting the QoS constraints to accommodate constructive interference as a source of useful signal power. By exploiting the power of constructively interfering symbols, the proposed scheme achieves the required QoS at lower transmit power. We extend this concept to the signal to interference plus noise ratio (SINR) balancing problem, where higher SINR values compared to the conventional SINR balancing optimization are achieved for given transmit power budgets. In addition, we derive equivalent virtual multicast formulations for both optimizations, both of which provide insights of the optimal solution and facilitate the design of a more efficient solver. Finally, we propose a robust beamforming technique to deal with imperfect CSI, that also reduces the transmit power over conventional techniques, while guaranteeing the required QoS. Our simulation and analysis show significant power savings for small scale MISO downlink channels with the proposed data-aided optimization compared to conventional beamforming optimization.

Robust Precoding and Beamforming Design of OSTBC Based CR AF-MIMO Relay System with Energy Harvesting Receiver

In this paper, we investigate the robust precoding and beamforming design problem for orthogonal space---time block code (OSTBC) based cognitive radio amplify-and-forward multiple-input---multiple-output relay systems with simultaneous wireless information and power transfer which consists of a primary receiver, a secondary source, a secondary relay, a secondary destination and an energy harvesting receiver. The secondary source and the secondary relay employ OSTBC based precoding and beamforming. The secondary relay and the secondary destination employ maximum-ratio combining for signal detecting and decoding. Our objective is to maximize the worst-case achievable rate under the sum transmit power constraint, the worst-case sum harvested power constraint and the worst-case interference power constraint. We transform the objective function into a convex function and convert the semi-infinite constraints into linear matrix inequalities by using $${\mathcal {S}}$$S-Lemma and the extension of $${\mathcal {S}}$$S-Lemma. Simulation results demonstrate that our proposed robust precoding and beamforming design has significant performance gain over the non-robust one.

Robust secrecy beamforming for wireless information and power transfer in multiuser MISO communication system

In this paper, we study the max-min fairness for robust secrecy beamforming design in a multiuser MISO communication system with simultaneous wireless information and power transfer (SWIPT). In particular, the imperfect channel state information (CSI) and power splitting information receiver (IR) are taken into account. We model the design as an optimization problem which maximizes the minimum harvested energy among the multi-antenna energy receivers (ERs). Besides, both the secure communication requirement and the lowest electrical energy storage threshold at IR must be guaranteed in our formulated optimization problem. The considered max-min problem is non-convex and hard to tackle. By introducing the technique of semidefinite relaxation (SDR), we prove the tightness of this relaxation and obtain the global optimal solution of our original optimization problem. Moreover, a suboptimal beamforming design scheme is proposed with low computational complexity. Simulation results show that our proposed robust schemes are more efficient than other schemes in terms of energy harvesting and transmit power saving.

Joint Probabilistic Constrained Robust Beamforming and Antenna Selection

A novel method is proposed for joint probabilistic constrained robust beamforming and antenna selection used in cognitive radio networks. Assuming complex Gaussian distributed channel state information errors, the Bernstein-type inequalities are introduced to transform no closed-form probabilistic constrained forms into the deterministic forms. Moreover, the l1-norm is used as the closest convex approximation of l0-norm. Thus the original NP-hard optimal problem can be relaxed as a tractable convex optimization problem. A computationally efficient and near-optimal solution is obtained by an iteratively re-weighted algorithm. Simulations show that the proposed algorithm satisfies the predetermined service levels at relatively small excess transmission power in a number of transmitter reduction scenarios.

A Variable Diagonal Loading Beamformer with Joint Uncertainties of Steering Vector and Covariance Matrix

In this paper, a novel robust adaptive beamforming is proposed in which both the uncertainties of steering vector and covariance matrix are taken into account. First we develop a min---max optimiza...

Robust amplify‐and‐forward relay beamforming for security with mean square error constraint

In this study, security schemes at the physical layer are proposed for an amplify-and-forward relay system in the presence of a passive eavesdropper. We focus on maximising the transmit power of the artificial noise (AN) to degrade the minimum mean square error (MMSE) level at the eavesdropper, while achieving MMSE constraint for the desired receiver. Assuming that the desired receiver and the passive eavesdropper possess perfect channel state information (CSI), AN is designed to lie in the null space of legitimate channels. However, the available CSI is impossible to be perfect in practice. Therefore the authors present a robust beamforming scheme which maximises the worst-case transmit power of the AN under the condition of imperfect CSI and recovers a large fraction of the performance in the perfect CSI case. A classical alternating optimisation algorithm is proposed to obtain the relay beamforming weights, and simulation results show the benefit of proposed schemes.

Beamforming schemes for hybrid satellite-terrestrial cooperative networks

This paper studies the relay beamforming (BF) schemes for a hybrid satellite-terrestrial cooperative network (HSTCN), where a cognitive base station (CBS) with multi-antenna is used not only as a relay to assist the signal transmission between a satellite and a terrestrial primary user (PU), but also as a base station to transmit signals to multiple secondary users (SUs). Our goal is to minimize the total transmit power, while guaranteeing the quality-of-service (QoS) of all users. Specifically, by supposing that the channel state information (CSI) of both the uplink and downlink are perfectly known at the relay, we first present a non-robust BF method. Then, note that downlink CSI often contains errors, we propose two approaches based on worst-case criterion, namely, lowest bound robust BF scheme (LB-RBF) and tight bound robust BF scheme (TB-RBF), each of which relaxes the constraints and converts the complex problem to a convex optimization problem so that it can be solved by using standard numerical optimization packages. Finally, computer simulations are provided to demonstrate the performance of the three BF algorithms.

Doubly constrained robust beamforming method using subspace-associated power components

The Capon beamformer can reject interfering signals by minimizing the array output power subject to the linear constraint exploiting a steering vector. When there exist steering vector errors, however, its performance can be severely degraded as the desired signal is misunderstood as interference. A robust beamforming method is presented which is based on part of the array output power associated with a subspace spanned by eigenvectors corresponding to some largest eigenvalues of the received signal covariance matrix. The use of the subspace-associated power component only, rather than all power components, allows us to find the minimum norm solution, which can result in a reduction in the output noise power since it is proportional to the squared norm of the weight vector. The proposed beamformer solves a minimization problem under the norm constraint and the set constraint on the steering vector. Though the constraint functions of the minimization problem are similar to those in the original doubly constrained beamformer (DCB), the cost function is differently defined based on the subspace power such that better robustness can be attained. Simulation shows that the proposed beamformer is superior in signal-to-interference-plus-noise ratio (SINR) to existing ones such as the conventional DCB.

Joint Beamforming and Power Optimization with Per-Antenna Power Constraints in Cognitive MISO System

Joint Beamforming and power allocation schemes are considered for cognitive radio (CR) multiple input single output (MISO) system in this paper. We considered the robust downlink beamforming and power allocation problem with per-antenna power constraints which is more realistic in practice, instead of sum-power constraint, under the imperfect channel state information (CSI). Since the given optimization problem is a non-convex problem which is difficult to handle, we transformed it into second order cone program (SOCP) problem, which can be solved efficiently. Simulation shows that the proposed scheme indeed has a better performance in CR system.

A Robust Adaptive Beamformer Based on Semidefinite Programming with Quadratic Constraints

A robust beamforming with quadratic constraints, formulated as a semidefinite programming (SDP) problem, is proposed in this paper. With this formulation, the constraints on magnitude response can be easily imposed on the adaptive beamformer. And the non-convex quadratic constraints can be transformed into linear constraints. Therefore, the proposed method can be robust against the steering direction error. In practice, there are many array imperfections except steering direction error. In order to resist all kinds of array imperfections, the adaptive beamformer based on worst-case optimization technique is proposed by minimizing the array output power with respect to the worst-case array imperfections. Simulation results demonstrate that the proposed method is effective and can achieve a better performance.

Robust Beamforming Design in MISO Interference Channels With RVQ Limited Feedback

This paper proposes robust beamforming designs for a multiple-input-single-output (MISO) interference channel (IC). Considering the random vector quantization (RVQ) feedback mechanism, we first derive the closed-form expression for the mean-squared-error (MSE) metric by averaging over the noise, the quantization error, and the channel amplitude. With the derived MSE result, we devise the robust beamformer by minimizing total MSEs and by minimizing the maximum per-user MSE. Since the optimizations of both designs are not convex, we first propose an iterative method to find out the solutions for the minimum total MSE criterion, where the transmitter beamforming vectors and the receiver decoding scalars are iteratively obtained. For the other design criterion, we propose an exhaustive search that can transfer the design problem into a feasible search problem. With the same design criterion, we propose another iterative method to find out the solutions, where the decoding scalars and the beamforming vectors are iteratively derived by minimizing MSE and second-order cone programming (SOCP), respectively. The simulation results verify the robustness of both designs when the quantization error exists.

Robust secrecy rate optimisations for multiuser multiple‐input‐single‐output channel with device‐to‐device communications

In the present study, the authors investigate robust secrecy rate optimisation problems for a multiple-input-single-output secrecy channel with multiple device-to-device (D2D) communications. The D2D communication nodes access this secrecy channel by sharing the same spectrum, and help to improve the secrecy communications by confusing the eavesdroppers. In return, the legitimate transmitter ensures that the D2D communication nodes achieve their required rates. In addition, it is assumed that the legitimate transmitter has imperfect channel state information of different nodes. For this secrecy network, the authors solve two robust secrecy rate optimisation problems: (a) robust power minimisation problem, subject to the probability based secrecy rate and the D2D transmission rate constraints; (b) robust secrecy rate maximisation problem with the transmit power, the probabilistic based secrecy rate and the D2D transmission rate constraints. Owing to the non-convexity of robust beamforming design based on two statistical channel uncertainty models, the authors present two conservative approximation approaches based on ‘Bernstein-type’ inequality and ‘S-Procedure’ to solve these robust optimisation problems. Simulation results are provided to validate the performance of these two conservative approximation methods, where it is shown that ‘Bernstein-type’ inequality based approach outperforms the ‘S-Procedure’ approach in terms of achievable secrecy rates.

Optimal Probabilistic Robust Beamforming for MISO SWIPT Systems with Gaussian Channel Uncertainties

For simultaneous wireless information and power transfer in the multi-input single-output broadcast channel, we propose a probabilistic robust beamforming scheme which minimizes the outage probability of the information receiver subject to the probabilistic energy harvesting constraint where the channel uncertainties are modeled as Gaussian random variables. Expressing the optimization problem by Marcum's Q-functions, we obtain the optimal closed-form beamforming solution. Simulation results show that the proposed scheme outperforms the worst-case robust beamforming scheme.

Low-Complexity Robust Capon Beamforming Based on Reduced-Rank Technique

Existing robust Capon beamformers achieve robustness against steering vector errors at a high cost in terms of computational complexity. Computationally efficient robust Capon beamforming approach based on the reduced-rank technique is proposed in this paper. The proposed method projects the received data snapshots onto a lower dimensional subspace consisting of the matched filters of the multistage Wiener filter (MSWF). The subsequent adaptive beamforming will then be performed within this subspace. The combination of the benefit of the robust adaptive beamforming and the reduced-rank technique improves the performance on combating steering vector errors and lowering the computational complexity.

Robust Adaptive Beamforming Based on Worst-Case and Norm Constraint

A novel robust adaptive beamforming based on worst-case and norm constraint (RAB-WC-NC) is presented. The proposed beamforming possesses superior robustness against array steering vector (ASV) error with finite snapshots by using the norm constraint and worst-case performance optimization (WCPO) techniques. Simulation results demonstrate the validity and superiority of the proposed algorithm.

Fast and robust adaptive beamforming method based on l 1‐norm constraint for large array

A robust adaptive beamformer (BF) with low computational complexity is proposed, where the adaptive weight is formulated as a linear combination of the training samples vectors and the target steering vector in the high interferences-to-noise ratio (INR) case. When the number of samples is greater than that of the interferences, an l1-norm constraint is imposed on the combinational vector to force its sparsity. Simulation results indicate that the proposed algorithm outperforms some classical robust BFs.

Robust sub-regional Frost beamforming based on the signal subspace reconstruction

For the problem of broadband beamforming in a subregion of interest, a robust Frost beamforming algorithm is derived by reconstructing the signal subspace. The basic idea of the proposed algorithm is to extract the characteristic components of the signal of interest (SOI) from the estimated signal-plus-interference subspace by a matrix filter first, then employ these characteristic components to reconstruct the signal subspace, and finally construct a set of linearly constrained minimum variance (LCMV) constraints to protect the SOI components. Compared with some other robust Frost beamformers, the proposed algorithm has a significant advantage, i.e., its steering-angle and band are effective to match the SOI without prior information. Hence, the performance of the proposed algorithm is almost always close to the optimal value across the whole region of interest. Theoretical analysis and simulation results validate the effectiveness of the proposed algorithm.

Quadratically Constrained Minimum Dispersion Beamforming via Gradient Projection

A quadratically constrained minimum dispersion (QCMD) beamformer that is robust against model uncertainties is devised for non-Gaussian signals. Different from the minimum variance based beamformers, the QCMD beamformer minimizes the $\ell_{p}$ -norm $(p\geq 1)$ of the output while constraining the magnitude response of any steering vector within a spherical uncertainty set to exceed unity. A gradient projection algorithmic framework is proposed to efficiently solve the resulting convex optimization problem instead of directly applying the standard optimization algorithm which has a high computational complexity. In each iteration, the gradient projection updates the solution along the gradient direction and projects it back to the constraint set. Importantly, a closed-form expression of the projection onto the constraint set is derived, which only needs a low complexity of ${\cal O}(M)$ with $M$ being the sensor number. Therefore, the proposed algorithm is much faster and simpler to implement compared with the standard method. In addition, the robust constant modulus beamformer (RCMB) is also discussed as a special case of the QCMD beamformer. Simulation results demonstrate the efficiency of the gradient projection algorithm and superiority of the QCMD beamformer over several representative robust beamformers, indicating that it can approach the optimal performance bound.