Individual Power Constraints Research Articles

Water-Filling: A Geometric Approach and its Application to Solve Generalized Radio Resource Allocation Problems

In this paper, a simple and elegant geometric water-filling (GWF) approach is proposed to solve the unweighted and weighted radio resource allocation problems. Unlike the conventional water-filling (CWF) algorithm, we eliminate the step to find the water level through solving a non-linear system from the Karush-Kuhn-Tucker conditions of the target problem. The proposed GWF requires less computation than the CWF algorithm, under the same memory requirement and sorted parameters. Furthermore, the proposed GWF avoids complicated derivation, such as derivative or gradient operations in conventional optimization methods, while provides insights to the problems and the exact solutions to the target problems. Most importantly, the GWF can be extended to solve a generalized form of radio resource allocation problem with more stringent constraints: (weighted) optimization problem with individual peak power constraints (GWFPP), and to include (weighted) group bounded power constraints (GWFGBP). On the other side, the CWF cannot solve these two general forms of the RRA problems, due to the difficulty to solve the non-linear system with multiple non-linear equations and inequalities in multiple dual variables. Optimality of the proposed water-filling solution is strictly proved for each of the proposed algorithms. Furthermore, numerical results show that the proposed approach is effective, efficient, easy to follow and insight-seeing.

Linear Coherent Estimation With Spatial Collaboration

A power-constrained sensor network that consists of multiple sensor nodes and a fusion center (FC) is considered, where the goal is to estimate a random parameter of interest. In contrast to the distributed framework, the sensor nodes may be partially connected, where individual nodes can update their observations by (linearly) combining observations from other adjacent nodes. The updated observations are communicated to the FC by transmitting through a coherent multiple access channel. The optimal collaborative strategy is obtained by minimizing the expected mean-square error subject to power constraints at the sensor nodes. Each sensor can utilize its available power for both collaboration with other nodes and transmission to the FC. Two kinds of constraints, namely the cumulative and individual power constraints, are considered. The effects due to imperfect information about observation and channel gains are also investigated. The resulting performance improvement is illustrated analytically through the example of a homogeneous network with equicorrelated parameters. Assuming random geometric graph topology for collaboration, numerical results demonstrate a significant reduction in distortion even for a moderately connected network, particularly in the low local signal-to-noise ratio regime.

Impact of Channel Estimation Error on Bidirectional MABC-AF Relaying With Asymmetric Traffic Requirements

Channel estimation error results in severe performance deterioration in wireless networks. In this paper, we study the impact of imperfect channel estimation (ICE) on the outage performance of bidirectional relaying where the two sources have asymmetric traffic requirements (ATRs). In particular, we focus on the amplify-and-forward (AF) relay-assisted multiple-access broadcast (MABC) protocol, i.e., MABC-AF, which has received much attention due to its high spectrum efficiency and low complexity. In the single-relay scenario, we derive exact and generalized closed-form expressions for system outage probability, which indicate that the system outage is determined by the unidirectional outage in the case of highly asymmetric traffic patterns. For more insights into our approach, the closed-form asymptotic expressions are also evaluated, manifesting the interesting error floor (EF) phenomenon due to ICE. Using these analytical results, we further develop a robust and practical optimum power-allocation (OPA) algorithm that minimizes the system outage probability under aggregate and individual node power constraints. In the multirelay scenario, by taking into account the traffic knowledge, a novel relay selection criterion is proposed for asymmetric MABC-AF, followed by its impact on the system outage probability in the presence of ICE. Numerical results validate the accuracy of our analytical results and highlight the effect of the proposed OPA algorithm under various traffic requirements and channel estimation errors. Furthermore, the results also show that the proposed relay selection criterion is superior to the classical max-min criterion with ATRs, and the performance improvement becomes remarkable as channel estimation error increases.

Max-Min Fairness Linear Transceiver Design for a Multi-User MIMO Interference Channel

Consider the max-min fairness linear transceiver design problem for a multi-user multi-input multi-output (MIMO) interference channel. When the channel knowledge is perfectly known, this problem can be formulated as the maximization of the minimum signal-to-interference-plus-noise ratio (SINR) utility, subject to individual power constraints at each transmitter. We prove in this paper that, if the number of antennas is at least two at each transmitter (receiver) and is at least three at each receiver (transmitter), the max-min fairness linear transceiver design problem is computationally intractable as the number of users becomes large. In fact, even the problem of checking the feasibility of a given set of target SINR levels is strongly NP-hard. We then propose two iterative algorithms to solve the max-min fairness linear transceiver design problem. The transceivers generated by these algorithms monotonically improve the min-rate utility and are guaranteed to converge to a stationary solution. The efficiency and performance of the proposed algorithms compare favorably with solutions obtained from the channel matched beamforming or the leakage interference minimization.

The Effect of Correlated Observations on the Performance of Distributed Estimation

Estimating unknown signal in Wireless Sensor Networks (WSNs) requires sensor nodes to transmit their observations of the signal over a multiple access channel to a Fusion Center (FC). The FC uses the received observations, which is corrupted by observation noise and both channel fading and noise, to find the minimum Mean Square Error (MSE) estimate of the signal. In this paper, we investigate the effect of the source-node correlation (the correlation between sensor node observations and the source signal) and the inter-node correlation (the correlation between sensor node observations) on the performance of the Linear Minimum Mean Square Error (LMMSE) estimator for three correlation models in the presence of channel fading. First, we investigate the asymptotic behavior of the achieved distortion (i.e., MSE) resulting from both the observation and channel noise in a non-fading channel. Then, the effect of channel fading is considered and the corresponding distortion outage probability, the probability that the distortion exceeds a certain value, is found. By representing the distortion as a ratio of indefinite quadratic forms, a closed-form expression is derived for the outage probability that shows its dependency on the correlation. Finally, the new representation of the outage probability allows us to propose an iterative solution for the power allocation problem to minimize the outage probability under total and individual power constraints. Numerical simulations are provided to verify our analytic results.

Optimal Power Allocation for Wideband Cognitive Radio Networks Employing SC-FDMA

In this paper, we investigate the optimal power allocation (OPA) strategy of secondary user (SU) transmitters in wideband cognitive radio networks employing single-carrier frequency-division multiple access (SC-FDMA). To protect the link of the primary user, we study two types of constraints on the SUs: Individual per-tone interference power constraints (IPCs) for each SU and a joint aggregate IPC across all frequency tones and users. Using tools from convex optimization, we derive the sum rate optimal OPA across frequency tones for the SU transmitters for both types of constraints. Simulation results show that the proposed OPA schemes can considerably improve the achievable sum rate of SC-FDMA based SU systems compared to equal power allocation and conventional power allocation schemes.

Source/relays/destination combined MMSE-based optimization for AF-MIMO multiple-relay systems with correlated channel uncertainties

In this paper, source/relays-precoders and destination-equalizer combined optimization are proposed as a dual-hop amplify-and-forward (AF) multiple-input multiple-output (MIMO) multiple-relay system with Gaussian random and correlated channel uncertainties in both hops. Taking correlated channel uncertainties into account, a robust transceiver joint optimization design is developed based on the minimum mean-squared error (MMSE) criterion under individual power constraints at the source and the relays. Simulation results illustrate that the robust multiple relays/transceiver joint design architecture for an AF-MIMO system equipped with multiple relays substantially outperforms a nonrobust transceiver design that assumes estimated channels as actual channels.

Joint Cooperative Beamforming and Jamming to Secure AF Relay Systems With Individual Power Constraint and No Eavesdropper's CSI

Cooperative beamforming and jamming are two efficient schemes to improve the physical-layer security of a wireless relay system in the presence of passive eavesdroppers. However, in most works these two techniques are adopted separately. In this letter, we propose a joint cooperative beamforming and jamming scheme to enhance the security of a cooperative relay network, where a part of intermediate nodes adopt distributed beamforming while others jam the eavesdropper, simultaneously. Since the instantaneous channel state information (CSI) of the eavesdropper may not be known, we propose a cooperative artificial noise transmission based secrecy strategy, subjected to the individual power constraint of each node. The beamformer weights and power allocation can be obtained by solving a second-order convex cone programming (SOCP) together with a linear programming problem. Simulations show the joint scheme greatly improves the security.

Cooperative Secure Beamforming for AF Relay Networks With Multiple Eavesdroppers

This letter studies cooperative secure beamforming for amplify-and-forward (AF) relay networks in the presence of multiple eavesdroppers. Under both total and individual relay power constraints, we propose two schemes, namely secrecy rate maximization (SRM) beamforming and null-space beamforming. In the first scheme, our design problem is based on SRM. Using a suboptimal, but convex, technique-semidefinite relaxation (SDR), we show that this problem can be handled by performing a one-dimensional search which involves solving a sequence of semidefinite programs (SDPs). To reduce the complexity, in the second scheme, we instead maximize the information rate at the destination while completely eliminating the information leakage to all eavesdroppers. We prove that this problem can be exactly solved by SDR with one SDP only. Simulation results demonstrate the performance gains of the two proposed designs.

Optimization of Cooperative Beamforming for SC-FDMA Multi-User Multi-Relay Networks by Tractable D.C. Programming

This paper addresses the optimal cooperative beamforming design for multi-user multi-relay wireless networks in which the single-carrier frequency division multiple access (SC-FDMA) technique is employed at the terminals. The problem of interest is to find the beamforming weights across relays to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among source users subject to individual power constraints at each relay. Such a beamforming design is shown to be a hard nonconvex optimization problem and therefore it is mathematically challenging to find the optimal solution. By exploring its partial convex structures, we recast the design problem as minimization of a d.c. (difference of two convex) objective function subject to convex constraints and develop an effective iterative algorithm of low complexity to solve it. Simulation results show that our optimal cooperative beamforming scheme realizes the inherent diversity order of the relay network and it performs significantly better than the equal-power beamforming weights.

Max-Min Fairness Linear Transceiver Design Problem for a Multi-User SIMO Interference Channel is Polynomial Time Solvable

Consider the linear transceiver design problem for a multi-user single-input multi-output (SIMO) interference channel. Assuming perfect channel knowledge, we formulate this problem as one of maximizing the minimum signal to interference plus noise ratio (SINR) among all the users, subject to individual power constraints at each transmitter. We prove in this letter that the max-min fairness linear transceiver design problem for the SIMO interference channel can be solved to global optimality in polynomial time. We further propose a low-complexity inexact cyclic coordinate ascent algorithm (ICCAA) to solve this problem. Numerical simulations show the proposed algorithm can efficiently find the global optimal solution of the considered problem.

Secrecy Capacity Enhancement With Distributed Precoding in Multirelay Wiretap Systems

The secrecy capacity of relay communications is investigated in this paper, where an eavesdropper is present. Distributed precoding through multiple relay nodes can be used to enhance the received signal power at the destination node and to mitigate the signal leakage to the eavesdropper, simultaneously. Due to individual power constraints at relay nodes, the distributed precoding scalar at each relay node is equivalent to a distributed precoding angle. An iterative algorithm is proposed to find the suboptimum distributed precoding angles at all the relay nodes, where the channel state information (CSI) sharing overhead among the relay nodes can also be substantially reduced. Each relay node receives the equivalent CSI from its preceding relay node, computes its distributed precoding angle, and updates the equivalent CSI for the next relay node. Compared with the simple decode-and-forward relaying protocol with random distributed precoding, the proposed iterative distributed precoding algorithm is able to further improve on the secrecy capacity of the multirelay wiretap system with an acceptable CSI sharing overhead.

Energy-Efficient Power Allocation for Fixed-Gain Amplify-and-Forward Relay Networks with Partial Channel State Information

In this letter, energy-efficient transmission and power allocation for fixed-gain amplify-and-forward relay networks with partial channel state information (CSI) are studied. In the energy-efficiency problem, the total power consumed is minimized while keeping the signal-to-noise-ratio (SNR) above a certain threshold. In the dual problem of power allocation, the end-to-end SNR is maximized under individual and global power constraints. Closed-form expressions for the optimal source and relay powers and the Lagrangian multiplier are obtained. Numerical results show that the optimal power allocation with partial CSI provides comparable performance as optimal power allocation with full CSI at low SNR.

Coordinated Beamforming for Multiuser MISO Interference Channel Under Rate Outage 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.

Joint Linear Receiver Design and Power Allocation Using Alternating Optimization Algorithms for Wireless Sensor Networks

In this paper, we consider a two-hop wireless sensor network (WSN) with multiple relay nodes where the amplify-and-forward (AF) scheme is employed. We present strategies to design jointly linear receivers and the power-allocation parameters via an alternating optimization approach subject to global, individual, and neighbor-based power constraints. Two design criteria are considered: The first criterion minimizes the mean-square error (MSE), and the second criterion maximizes the sum-rate (SR) of the WSN. We derive constrained minimum mean-square error (MMSE) and constrained maximum sum-rate (MSR) expressions for the linear receivers and the power-allocation parameters that contain the optimal complex amplification coefficients for each relay node. Computer simulations show good performance of our proposed methods in terms of bit error rate (BER) or SR compared with the method with equal power allocation and to a two-stage power-allocation technique. Furthermore, the methods with neighbor-based constraints bring flexibility to balance the performance against the computational complexity and the need for feedback information, which is desirable for WSNs to extend their lifetime.

Joint Scheduling and Power Allocation in OFDMA Network with Decode-and-Forward Relaying

This paper investigates the resource allocation of Orthogonal Frequency Division Multiplexing Access (OFDMA) cellular network with DF relaying in the downlink. A joint scheme of relay selection, subcarrier assignment and power allocation is proposed to maximize the system sum rate subject to individual power constraints and heterogeneous users' rate requirements. Since the original problem is a mixed integer programming (MIP), we transform it to a standard convex optimization by continuous relaxation and solve it via iterative water-filling (IWF) and subgradient methods in the dual domain. Beside the joint scheme, the upper bound is studied through an exhaustive bisection search. Simulations results show that the proposed scheme outperforms the traditional methods in terms of both system throughput and users' satisfactions.

Optimal resource allocation for multiple network-coded two-way relay in orthogonal frequency division multiplexing systems

This paper studies optimal resource allocation for multiple network-coded two-way relay in orthogonal frequency division multiplexing systems. All the two-way relay nodes adopt amplify-and-forward and operate with analog network coding protocol. A joint optimization problem considering power allocation, relay selection, and subcarrier pairing to maximize the sum capacity under individual power constraints at each transmitter or total network power constraint is first formulated. By applying dual method, we provide a unified optimization framework to solve this problem. With this framework, we further propose three low-complexity suboptimal algorithms. The complexity of the proposed optimal resource allocation ORA algorithm and three suboptimal algorithms are analyzed, and it is shown that the complexity of ORA is only a polynomial function of the number of subcarriers and relay nodes under both individual and total power constraints. Simulation results demonstrate that the proposed ORA scheme yields substantial performance improvement over a baseline scheme, and suboptimal algorithms can achieve a trade-off between performance and complexity. The results also indicate that with the same total network transmit power, the performance of ORA under total power constraint can outperform that under individual power constraints. Copyright © 2012 John Wiley & Sons, Ltd.

Optimality of Received Energy in Decision Fusion Over Rayleigh Fading Diversity MAC With Non-Identical Sensors

Received-energy test for non-coherent decision fusion over a Rayleigh fading multiple access channel (MAC) without diversity was recently shown to be optimum in the case of conditionally mutually independent and identically distributed (i.i.d.) sensor decisions under specific conditions [1], [2]. Here, we provide a twofold generalization, allowing sensors to be non identical on one hand and introducing diversity on the other hand. Along with the derivation, we provide also a general tool to verify optimality of the the received energy test in scenarios with correlated sensor decisions. Finally, we derive an analytical expression of the effect of the diversity on the large-system performances, under both individual and total power constraints.

Weighted-Sum-Rate-Maximizing Linear Transceiver Filters for the K-User MIMO Interference Channel

This letter is concerned with transmit and receive filter optimization for the K-user MIMO interference channel. Specifically, linear transmit and receive filter sets are designed which maximize the weighted sum rate while allowing each transmitter to utilize only the local channel state information. Our approach is based on extending the existing method of minimizing the weighted mean squared error (MSE) for the MIMO broadcast channel to the K-user interference channel at hand. For the case of the individual transmitter power constraint, however, a straightforward generalization of the existing method does not reveal a viable solution. It is in fact shown that there exists no closed-form solution for the transmit filter but simple one-dimensional parameter search yields the desired solution. Compared to the direct filter optimization using gradient-based search, our solution requires considerably less computational complexity and a smaller amount of feedback resources while achieving essentially the same level of weighted sum rate. A modified filter design is also presented which provides desired robustness in the presence of channel uncertainty.

Joint iterative power allocation and linear interference suppression algorithms for cooperative DS-CDMA networks

This work presents joint iterative power allocation and interference suppression algorithms for spread spectrum networks, which employ multiple hops and the amplify-and-forward cooperation strategy for both the uplink and the downlink. The authors propose a joint constrained optimisation framework that considers the allocation of power levels across the relays subject to individual and global power constraints and the design of linear receivers for interference suppression. The authors derive constrained linear minimum mean-squared error (MMSE) expressions for the parameter vectors that determine the optimal power levels across the relays and the linear receivers. In order to solve the proposed optimisation problems, the authors develop cost-effective algorithms for adaptive joint power allocation, and estimation of the parameters of the receiver and the channels. An analysis of the optimisation problem is carried out and shows that the problem can have its convexity enforced by an appropriate choice of the power constraint parameter, which allows the algorithms to avoid problems with local minima. A study of the complexity and the requirements for feedback channels of the proposed algorithms is also included for completeness. Simulation results show that the proposed algorithms obtain significant gains in performance and capacity over existing non-cooperative and cooperative schemes.