Co-Ordinated Secure Beamforming in -User Interference Channel With Multiple Eavesdroppers

This paper studies cooperative secure transmission in wireless relay networks, where a source sends a confidential message to a legitimate destination through a set of amplify-and-forward (AF) relays in the presence of multiple eavesdroppers. To prevent the eavesdroppers from retrieving the message, we assume that the relays employ a joint cooperative beamforming (CB) and artificial noise (AN) strategy. Under both the total and individual relay power constraints, we propose an optimal joint CB-AN design based on secrecy rate maximization (SRM). We treat the SRM problem by a combination of two-level optimization and semidefinite relaxation (SDR). For the latter, we prove that SDR always has a rank-one solution for the SRM problem - which identifies that SDR is optimal. Simulation results are presented to show the efficacy of the proposed joint CB-AN design.

In this paper, considering a mobile wireless sensor network, we study the problem of exploiting sensor mobility information in the process of sensor localization under two range measurement models, namely the time-of-arrival (TOA) model and the received signal strength (RSS) model. To do so, for each model, we first derive the maximum likelihood (ML) location estimator for the case of error-free velocity measurements. As the corresponding optimization problems are non-convex, we resort to semi-definite relaxation (SDR) techniques to find approximate solutions to each problem using semi-definite programming (SDP). We then extend our results to the cases where the velocity measurements are subject to measurement errors. Our simulation results show that exploiting the mobility information in the localization process can significantly improve the performance of the sensor localization. Moreover, mobility-aided localization has the potential to address some of typical positioning problems, such as sensitivity to the ranging measurement errors and the requirement on the number of the anchors needed to uniquely localize the sensor nodes.

We consider physical-layer security in a novel MISO cooperative overlay cognitive radio network (CRN) with a single eavesdropper. We aim to design an artificial noise (AN) aided secondary transmit strategy to maximize the joint achievable secrecy rate of both primary and secondary links, subject to a global secondary transmit power constraint and guaranteeing any transmission of secondary should at least not degrade the receive quality of primary network, under the assumption that global CSI is available. The resulting optimization problem is challenging to solve due to its non-convexity in general. A computationally efficient approximation methodology is proposed based on the semidefinite relaxation (SDR) technique and followed by a two-step alternating optimization algorithm for obtaining a local optimum for the corresponding SDR problem. This optimization algorithm consists of a one-dimensional line search and a non-convex optimization problem, which, however, through a novel reformulation, can be approximated as a convex semidefinite program (SDP). Analysis on the extension to multiple eavesdroppers scenario is also provided. Simulation results show that the proposed AN-aided joint secrecy rate maximization design (JSRMD) can significantly boost the secrecy performance over JSRMD without AN.

For wireless multicast channels, it is prospective to provide quality of service (QoS) differentiation at physical layer. In this paper, we propose a rank-two transmit beamforming mechanism to support differentiated services (Diffserv) in multi-group multicasting scenarios based on Alamouti space-time coding (ASTC). In QoS-Diffserv multi-group multicast systems, two kinds of users exist, named high-priority users with guaranteed QoS and low-priority users with limited QoS. First the original QoS-Diffserv issue is formulated as a non-convex quadratically constrained quadratic programming (QCQP) program. With the aid of semi-definite relaxation (SDR) technique, we then relax the QCQP problem to a convex semi-definite programming (SDP) problem, which can be easily handled by basic convex optimization tools. To obtain approximate solutions to the original problem which should meet the rank-two constraints, a Gaussian randomization based process and a power control problem are employed together as the last step. Relationship between the QCQP and SDP problems and computational complexity of the proposed algorithm are analysed. Simulation results demonstrate that the proposed rank-two beamforming algorithm could provide near-optimal transmission power under different QoS constraints.

This paper studies the coordinated beamforming design problem for the multiple-input single-output (MISO) interference channel, assuming only channel distribution information (CDI) at the transmitters. Under a given requirement on the rate outage probability for receivers, we aim to maximize the system utility (e.g., the weighted sum rate, weighted geometric mean rate, and the weighed harmonic mean rate) subject to the rate outage constraints and individual power constraints. The outage constraints, however, lead to a complicated, nonconvex structure for the considered beamforming design problem and make the optimization problem difficult to handle. {Although} this nonconvex optimization problem can be solved in an exhaustive search manner, this brute-force approach is only feasible when the number of transmitter-receiver pairs is small. For a system with a large number of transmitter-receiver pairs, computationally efficient alternatives are necessary. The focus of this paper is hence on the design of such efficient approximation methods. In particular, by employing semidefinite relaxation (SDR) and first-order approximation techniques, we propose an efficient successive convex approximation (SCA) algorithm that provides high-quality approximate beamforming solutions via solving a sequence of convex approximation problems. The solution thus obtained is further shown to be a stationary point for the SDR of the original outage constrained beamforming design problem. {Furthermore}, we propose a distributed SCA algorithm where each transmitter optimizes its own beamformer using local CDI and information obtained from limited message exchange with the other transmitters. Our simulation results demonstrate that the proposed SCA algorithm and its distributed counterpart indeed converge, and near-optimal performance can be achieved for all the considered system utilities.

A new blind channel estimation approach for orthogonally coded multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) systems is proposed. The main idea of our technique is, using specific properties of the orthogonal space-time block codes (OSTBCs), to estimate the finite impulse response (FIR) channel parameters in the time domain instead of doing this in the frequency domain independently for each subcarrier. This results in a substantially improved parsimony of the channel parametric model as compared to the direct per subcarrier channel estimation methods and allows coherent processing across the subcarriers. It is shown that using the semidefinite relaxation (SDR) technique, our channel estimation problem can be approximated as a convex semidefinite programming (SDP) problem and, hence, can be solved efficiently using modern convex optimization methods. Our simulations validate the performance and/or computational complexity advantages of the proposed method as compared to some current state-of-the-art blind MIMO-OFDM channel estimation methods.

In this paper, we investigate the problem of robust beamforming for multiple-input multiple-output (MIMO) dual-function radar-communication (DFRC) system, when there exist radar look direction mismatch and communication channel state information (CSI) error. To address these system imperfections, a region-of-interest (ROI) rather than a specific angle is considered for radar sensing, and the communication CSI is assumed to be available with a bounded unknown error. With these considerations, we design a robust dual-function beamformer through optimizing the radar sensing performance in terms of the energy radiated towards the ROI, while guaranteeing the communication quality-of-service (QoS) measured by the per-user signal-to-interference-plus-noise ratio (SINR) as required. For this robust design, a max-min optimization problem is formulated and a semidefinite programming (SDP) reformulation based on the S-procedure is presented. The popular semidefinite relaxation (SDR) technique is employed to solve the resulting non-convex problem, and the tightness of SDR is discussed. Further, we propose to incorporate an extra constraint of main beam control into the problem to achieve a flat magnitude response of the beampattern over the ROI. In addition, it is shown that the proposed approach is straightforwardly applicable to the scenario of multiple targets. Numerical examples are carried out to show the tightness, and demonstrate the effectiveness and robustness of the proposed MIMO-DFRC beamforming approach.

The secrecy performance of wireless networks powered up by a dedicated power beacon (PB) is addressed in this work. Particularly, the closed-form expression of the secrecy outage probability (SOP) under the presence of multiple eavesdroppers is given. The considered networks are asymmetric since there are multiple eavesdroppers but only a single legitimate receiver. As a consequence, two wiretap schemes are employed, namely, non-colluding and colluding schemes, to take advantage of multiple eavesdroppers. To draw insights from the derived framework, the asymptotic framework of the SOP under the high transmit power regime is also provided. Next, Monte Carlo simulations are provided to validate the theoretical foundations of the proposed approach. Our findings show that by increasing the transmit power from −10 dB to 30 dB, the SOP ameliorates almost a thousand times. Additionally, the SOP is a monotonic decreasing function with a time-switching ratio. We also make comparisons with work in the literature in which the full-duplex relay is employed to help the legitimate link. Numerical results illustrate that the proposed network outperforms those described in the literature. Particularly, through reliance on the setup, the SOP under the proposed system is better than those of the literature by about twofold.

This paper investigates the application of simultaneous wireless information and power transfer (SWIPT) to cooperative non-orthogonal multiple access (NOMA). A new cooperative multiple-input single-output (MISO) SWIPT NOMA protocol is proposed, where a user with a strong channel condition acts as an energy-harvesting (EH) relay to help a user with a poor channel condition. The power splitting (PS) scheme is adopted at the EH relay. By jointly optimizing the PS ratio and the beamforming vectors, the design objective is to maximize the data rate of the "strong user" while satisfying the QoS requirement of the "weak user". It boils down to a challenging nonconvex problem. To resolve this issue, the semidefinite relaxation (SDR) technique is applied to relax the quadratic terms related with the beamformers, and then it is solved to its global optimality by two-dimensional exhaustive search. We prove the rank-one optimality, which establishes the equivalence between the relaxed problem and the original one. To further reduce the high complexity due to the exhaustive search, an iterative algorithm based on successive convex approximation (SCA) is proposed, which can at least attain its stationary point efficiently. In view of the potential application scenarios, e.g., IoT, the single-input single-output (SISO) case of the cooperative SWIPT NOMA system is also studied. The formulated problem is proved to be strictly unimodal with respect to the PS ratio. Hence, a golden section search (GSS) based algorithm with closed-form solution at each step is proposed to find the unique global optimal solution. It is worth pointing out that the SCA method can also converge to the optimal solution in SISO cases. In the numerical simulation, the proposed algorithm is numerically shown to converge within a few iterations, and the SWIPT-aided NOMA protocol outperforms the existing transmission protocols.

Computation of the phase and gain margins of MIMO control systems

Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

The physical layer beamforming design of downlink wireless energy transfer (WET) is investigated. To minimize the power of WET with respect to the terminals’ quality of service requirements, the physical beamforming design is modeled as a non-convex quadratically constrained quadratic program (QCQP) problem. A two-step method to solve the QCQP is proposed. First, the QCQP is tightened into an ellipsoidal approximation problem with the aid of a Lowner-John ellipsoid, by solving a semidefinite program (SDP). Second, due to the approximation problem’s ellipsoidal nature, it is solved as another SDP. The proposed method is compared using numerical results to the conventional semidefinite relaxation (SDR) approach. It is demonstrated that for equivalent computing time cost, the proposed method usefully improves the energy efficiency of the downlink WET compared with SDR.

In this paper, we address the motion parameter estimation problem using time difference of arrival (TDOA) measurements, where a mobile source starts from an initial position with constant velocity. The problem is formulated as three non-convex constrained weighted least squares (CWLS) problems. Owing to their non-convex nature, local convergence may occur when solving them by an iterative algorithm, implying good estimates are not guaranteed. We propose to solve these CWLS problems by applying semidefinite relaxation (SDR) to generate three different semidefinite programming (SDP) problems, namely, the motion unconstrained semidefinite programming (MU-SDP), motion constrained semidefinite programming (MC-SDP), and motion parameter direct semidefinite programming (MPD-SDP). The MU-SDP, MC-SDP, and MPD-SDP methods are then extended to the scenario of unknown propagation speed (PS), which is common in underwater and underground localizations. Specifically, for this scenario, the two-step MU-SDP, two-step MC-SDP, and two-step MPD-SDP methods are designed to keep the relaxed SDP problems tight by introducing the penalty function. Simulation results show that MU-SDP performs worse than MC-SDP and MPD-SDP owing to the ignorance of the motion equation, and MC-SDP and MPD-SDP are able to reach the Cramér-Rao lower bound (CRLB) accuracy.

We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.

Many important NP-hard combinatorial problems can be efficiently approximated using semidefinite programming relaxations. We propose a new hierarchy of semidefinite relaxations for classes of such problems that are based on graphs and for which the projection of the problem onto a subgraph shares the same structure as the original problem. This includes the well-studied max-cut and stable-set problems. Each level k of the proposed hierarchy consists of the basic semidefinite relaxation of the problem augmented by the constraints enforcing the structural projection condition on every k-node subgraph of the problem. This hierarchy has the distinguishing feature that all the relaxations are formulated in the space of the original semidefinite relaxation. Because the size of the relaxations increases rapidly with the number of subgraphs, we explore the possibility of adding the projection constraints only for selected subgraphs. Preliminary computational results show that the proposed hierarchy yields improved bounds when compared to the initial relaxation for benchmark instances of the max-cut and stable-set problems, and that the improved bounds result in significantly smaller enumeration trees when the relaxation is used in a branch-and-bound scheme.