Consider a joint robust design problem for the transmit and receive beamvectors for a dual-functional radar and communication (DFR-C) system in the secondary communication of a cognitive radio (CR) network. The base station (BS) sends signals to detect a MIMO radar target while serving the secondary downlink users. Then a maximization problem of the worst-case radar output signal-to-interference-plus-noise ratio is formulated, subject to the total power constraint for the BS, the robust signal-to-noise constraints for the secondary users and the robust interference constraints for the primary users in the CR network, under the assumptions of imperfect CSI and uncertainty of the transmit and receive steering vectors for the radar sensing. To tackle the nonconvex problem, we derive the closed-form optimal values for two specific quadratic problems with a spherical constraint, and reexpress the robust constraints into quadratic constraints. Then an alternating optimization strategy is adopted to solve the problem. Specifically, when optimizing the transmit beamvector, a second-order cone programming (SOCP) approximation algorithm is proposed. On the other hand, when optimizing the receive beamvector, an SOCP problem is reformulated and solved. Then, simulation results demonstrate the improved performance of the DFRC system by the proposed algorithm, comparing with existing schemes.

The robust adaptive beamforming (RAB) problem is considered via the worst-case signal-to-interference-plus-noise ratio (SINR) maximization over distributional uncertainty sets for the random interference-plus-noise covariance (INC) matrix and desired signal steering vector. The distributional uncertainty set of the INC matrix accounts for the support and the positive semidefinite (PSD) mean of the distribution, and a similarity constraint on the mean. The distributional uncertainty set for the steering vector consists of the constraints on the known first- and second-order moments. The RAB problem is formulated as a minimization of the worst-case expected value of the SINR denominator achieved by any distribution, subject to the expected value of the numerator being greater than or equal to one for each distribution. Resorting to the strong duality of linear conic programming, such a RAB problem is rewritten as a quadratic matrix inequality problem. It is then tackled by iteratively solving a sequence of linear matrix inequality relaxation problems with the penalty term on the rank-one PSD matrix constraint. To validate the results, simulation examples are presented, and they demonstrate the improved performance of the proposed robust beamformer in terms of the array output SINR.

Sharing a multiple-input multiple-output (MIMO) radar, the single platform can achieve dual functions of radar and communication (DFRC) within the same frequency spectrum, via the same transmit waveforms. In this paper, a space-time coding scheme is developed for transmit beamforming of DFRC and embedding communication information, without their cross-interference. For transmit beampattern design of DFRC, the shape approximation and integrated power approximation criteria are adopted respectively for waveforms optimization with the constant-envelope constraints of transmit waveforms and the equivalent signal in the communication direction. Based on the space-time coding scheme, the direct constellation mapping (DCM) and phase-rotation constellation mapping (PRCM) methods are proposed to embed information symbols. It turns out that the proposed space-time coding scheme for information constellation mapping can prevent missing information symbols and have better performance in bit-error rate (BER), compared to the existing information-embedding techniques. Moreover, the scheme can reduce the dependence of the communication data rate on radar pulse repetition frequency (PRF). Simulation results are presented to demonstrate the effectiveness of the proposed methods.

Consider a robust multiple-input single-output downlink beamforming optimization problem in a frequency division duplexing system. The base station (BS) sends training signals to the users, and every user estimates the channel coefficients, quantizes the gain and the direction of the estimated channel and sends them back to the BS. Suppose that the channel state information at the transmitter is imperfectly known mainly due to the channel direction quantization errors, channel estimation errors and outdated channel effects. The actual channel is modeled as in an uncertainty set composed of two inequality homogeneous and one equality inhomogeneous quadratic constraints, in order to account for the aforementioned errors and effects. Then the transmit power minimization problem is formulated subject to robust signal-to-noise-plus-interference ratio constraints. Each robust constraint is transformed equivalently into a quadratic matrix inequality (QMI) constraint with respect to the beamforming vectors. The transformation is accomplished by an equivalent phase rotation process and the strong duality result for a quadratically constrained quadratic program. The minimization problem is accordingly turned into a QMI problem, and the problem is solved by a restricted linear matrix inequality relaxation with additional valid convex constraints. Simulation results are presented to demonstrate the performance of the proposed method, and show the efficiency of the restricted relaxation.

The worst-case robust adaptive beamforming problem for generalrank signal model is considered. This is a nonconvex problem, and an approximate version of it (by introducing a matrix decomposition on the presumed covariance matrix of the desired signal) has been studied in the literature. Herein the original robust adaptive beamforming problem is tackled. Resorting to the strong duality of a linear conic program, the robust beamforming problem is reformulated into a quadratic matrix inequality (QMI) problem. There is no general method for solving a QMI problem in the literature. Here- in, employing a linear matrix inequality (LMI) relaxation technique, the QMI problem is turned into a convex semidefinite programming problem. Due to the fact that there often is a positive gap between the QMI problem and its LMI relaxation, a deterministic approximate algorithm is proposed to solve the robust adaptive beamforming in the QMI form. Last but not the least, a sufficient optimality condition for the existence of an optimal solution for the QMI problem is derived. To validate our theoretical results, simulation examples are presented, which also demonstrate the improved performance of the new robust beamformer in terms of the output signal-to-interference- plus-noise ratio.

The MVDR robust adaptive beamforming design problem based on estimation of the signal-of-interest (SOI) steering vector is considered. In this case, the optimal beamformer is obtained by computing the sample matrix inverse and an optimal estimate of the SOI steering vector. In order to find the optimal steering vector estimate of the SOI, a new beamformer output power maximization problem is formulated subject to a double-sided norm perturbation constraint, a generalized similarity constraint, and a direction-of-arrival (DOA) constraint that guarantees that the DOA of the SOI is away from the DOA region of all linear combinations of the interference steering vectors. It turns out that the power maximization problem is a nonconvex quadratically constrained quadratic program (QCQP) with two homogenous and one inhomogeneous constraints. In general, a globally optimal solution for the QCQP is not guaranteed; however, we herein derive sufficient optimality conditions to ensure the existence of an optimal solution, and develop an efficient algorithm to find the solution. To validate our results, simulation examples are presented, and they demonstrate the improved performance of the new robust adaptive beamformer in terms of the output SINR.

The worst-case robust adaptive beamforming problem for general-rank signal model is considered. Its formulation is to maximize the worst-case signal-to-interference-plus-noise ratio, incorporating a positive semidefinite constraint on the actual covariance matrix of the desired signal. In the literature, semidefinite program (SDP) techniques, together with others, have been applied to approximately solve this problem. Herein, an inner second-order cone program (SOCP) approximate algorithm is proposed to solve it. In particular, a sequence of SOCPs are constructed and solved, while the SOCPs have the nonincreasing optimal values and converge to a locally optimal value (it is in fact a globally optimal value through our extensive simulations). As a result, our algorithm does not use computationally heavy SDP relaxation technique. To validate our inner approximation results, simulation examples are presented, and they demonstrate the improved performance of the new robust beamformer in terms of the averaged cpu-time (indicating how fast the algorithms converge) in a high signal-to-noise region.

This paper considers the beamforming design for a multiuser multiple-input single-output (MU-MISO) downlink with an arbitrary number of (context-specific) shaping constraints. In this setup, the state-of-the- art beamforming schemes cannot attain the well-known performance bound promised by the semidefinite program (SDP) relaxation technique. To close the gap, we propose a linear space-time beamforming (LiST-BF) scheme, consisting of a circulant space-time symbol mapper followed by the beamforming design with orthogonality constraints. It is shown that the proposed LiST-BF scheme can perform general rank-$K$ beamforming for user symbols in a low-complexity and structured manner. Sufficient conditions are derived to guarantee that the LiST-BF scheme always achieves the SDP bound for linear beamforming schemes. Based on such conditions, an efficient algorithm is then developed to obtain the optimal LiST-BF solution in polynomial time. Numerical results demonstrate that the proposed scheme enjoys substantial performance gains over the existing alternatives.

In this paper, we develop the robust transceiver optimization for the multiple-input single-output (MISO) interference channels where each transmitter (Tx) is equipped with multiple antennas and each single-antenna receiver performs simultaneous wireless information and power transfer (SWIPT) based on a power-splitting architecture. Assuming imperfect channel state information (CSI) at the Txs, we design jointly optimal transmit beamforming and receive power-splitting scheme that minimizes the total transmission power under the worst-case signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints. When the channel uncertainties are bounded by ellipsoidal regions, we show that the worst-case SINR and EH constraints can be recast into quadratic matrix inequality forms, and the intended problem can be relaxed as a tractable semi-definite program. Furthermore, relying on the alternating direction method of multipliers (ADMM), we propose a decentralized algorithm capable of computing the optimal beamforming and power- splitting schemes with local CSI and limited information exchange among the Txs.

This letter considers a downlink multi-input single-out (MISO) system where each user performs simultaneous wireless information and power transfer (SWIPT) based on a power splitting receiver architecture. Assuming imperfect channel state information (CSI) at the base station, we develop two robust joint beamforming and power splitting (BFPS) designs that minimize the transmission power under both the signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints per user. In the first design, we consider the worst-case (WC) SINR and EH constraints, and show that the WC-BFPS problem can be relaxed as a semidefinite program (SDP) through a linear matrix inequality representation for (infinitely many) robust quadratic matrix inequality constraints. In the second design, we consider the chance constraints (CCs) for SINR and EH, and resort to both semidefinite relaxation and Bernstein-type inequality restriction to transform the CC-BFPS problem into another convex SDP. Based on these convex reformulations, the (near-)optimal robust BFPS designs can be efficiently solved. Numerical results are provided to demonstrate the merit of the proposed robust designs.

This letter considers constrained steering direction estimation in the presence of additive Gaussian disturbance. The uncertainty region is modeled through double-sided quadratic constraints (up to three) and the Maximum Likelihood (ML) criterion is adopted to get the direction estimator. It is shown that the considered formulation leads to a fractional Quadratically Constrained Quadratic Program (QCQP) whose solution can be computed in polynomial time via semidefinite programming relaxation, Charnes-Cooper transformation, and suitable rank-one decomposition tools. At the analysis stage, with reference to a specific constraint set, the performance of the devised estimator is compared with the constrained Cramer Rao lower Bound (CRB).

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for non-perfect channel state information at the base station. The convexity of the robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP), and more importantly, it possesses an equivalent and explicit convex reformulation, due to an linear matrix inequality description of the cone of Lorentz-positive maps. Hence the robust beamforming problem can be efficiently solved by an optimization solver. The simulation results show that the conservativeness of the robust form of semi-infinite SOCP is appropriate in terms of problem feasibility rate and the average transmission power.

Two-way relaying can improve spectral efficiency in two-user cooperative communications. It also has great potential in multiuser systems. A major problem of designing a multiuser two-way relay system (MU-TWRS) is transceiver or precoding design to suppress co-channel interference. This paper aims to study linear precoding designs for a cellular MU-TWRS where a multi-antenna base station (BS) conducts bi-directional communications with multiple mobile stations (MSs) via a multi-antenna relay station (RS) with amplify-and-forward relay strategy. The design goal is to optimize uplink performance, including total mean-square error (Total-MSE) and sum rate, while maintaining individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. We show that the BS precoding design with the RS precoder fixed can be converted to a standard second order cone programming (SOCP) and the optimal solution is obtained efficiently. The RS precoding design with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Then, the joint BS-RS precoding is obtained by solving the BS precoding and the RS precoding alternately. Comprehensive simulation is conducted to demonstrate the effectiveness of the proposed precoding designs.

Consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) network. The minimization problem of transmit power is formulated subject to both the quality-of-service (QoS) constraints on the secondary receivers and the interference temperature constraints on the primary users, under the assumption of imperfect channel state information (CSI). The problem is non-convex quadratically constrained quadratic program (QCQP), and it is hard to achieve the global optimality. As a compromise, we present a randomized approximation algorithm for the problem via convex optimization techniques. In particular, we point out that the robust beamforming problem is efficiently solvable when the number of primary and secondary links in the CR network is not larger than three. Simulation results are presented to demonstrate the performance gains of the proposed algorithm over an existing robust design.

Consider the downlink beamforming optimization problem with signal-to-interference-plus-noise ratio constraints, null-shaping interference constraints and multiple groups of individual shaping constraints. We propose an efficient algorithm for the problem, which consists of firstly solving the dual of the semidefinite programm (SDP) relaxation, secondly formulating a linear program (LP) and solving it to find a rank-one solution of the SDP relaxation. In contrast to the existing algorithms, the analysis of the proposed algorithm includes neither the rank reduction steps (purification process) nor the Perron-Frobenius theorem.

Consider a downlink communication system where multi-antenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors so as to minimize the transmission powers while ensuring the signal-to-interference-noise ratio (SINR) requirement of individual users within the system, and keeping lower interference level to other systems which operate in the same frequency band and in the same region. This optimal beamforming problem is a separable homogeneous quadratically constrained quadratical programming (QCQP), and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds, and propose an efficient algorithm for the optimal beamforming problem. First, we study rank-constrained solutions of a general separable semidefinite programming (SDP), and propose a rank reduction procedure to achieve a lower rank solution. Then we show that the SDP relaxation of a class of the optimal beamforming problem has a rank-one solution, which can be obtained by invoking the rank reduction procedure.