Joint Beamforming and Power Allocation for Cognitive MIMO Systems Under Imperfect CSI Based on Game Theory

The problem of joint beamforming and power allocation for cognitive multi-input multi-output systems is studied via game theory. The objective is to maximize the sum utility of secondary users (SUs) subject to the primary user (PU) interference constraint, the transmission power constraint of SUs, and the signal-to-interference-plus-noise ratio (SINR) constraint of each SU. In our earlier work, the problem was formulated as a non-cooperative game under the assumption of perfect channel state information (CSI). Nash equilibrium (NE) is considered as the solution of this game. A distributed algorithm is proposed which can converge to the NE. Due to the limited cooperation between the secondary base station (SBS) and the PU, imperfect CSI between the SBS and the PU is further considered in this work. The problem is formulated as a robust game. As it is difficult to solve the optimization problem in this case, existence of the NE cannot be analyzed. Therefore, convergence property of the sum utility of SUs will be illustrated numerically. Simulation results show that under perfect CSI the proposed algorithm can converge to a locally optimal pair of transmission power vector and beamforming vector, while under imperfect CSI the sum utility of SUs converges with the increase of the transmission power constraint of SUs.     

认知MIMO系统中波束成形和功率的联合控制博弈算法

针对认知MIMO系统中波束成形和功率的联合优化问题,建立了非合作可分离对策联合博弈模型,提出了新的代价函数,根据KT条件证明了该模型中的各个子博弈均存在最优纳什均衡,从而得到联合博弈存在最优纳什均衡的结论,并提出了新的交替迭代算法。仿真结果表明,该算法有较快的收敛速度;同时,在动态环境下,算法也能达到最优纳什均衡,具有较好的稳定性。     

Multiradio Channel Allocation in Multihop Wireless Networks

Channel allocation was extensively investigated in the framework of cellular networks, but it was rarely studied in the wireless ad hoc networks, especially in the multihop networks. In this paper, we study the competitive multiradio multichannel allocation problem in multihop wireless networks in detail. We first analyze that the static noncooperative game and Nash equilibrium (NE) channel allocation scheme are not suitable for the multihop wireless networks. Thus, we model the channel allocation problem as a hybrid game involving both cooperative game and noncooperative game. Within a communication session, it is cooperative; and among sessions, it is noncooperative. We propose the min-max coalition-proof Nash equilibrium (MMCPNE) channel allocation scheme in the game, which aims to maximize the achieved data rates of communication sessions. We analyze the existence of MMCPNE and prove the necessary conditions for MMCPNE. Furthermore, we propose several algorithms that enable the selfish players to converge to MMCPNE. Simulation results show that MMCPNE outperforms NE and coalition-proof Nash equilibrium (CPNE) schemes in terms of the achieved data rates of multihop sessions and the throughput of whole networks due to cooperation gain.     

Competitive Pricing for Spectrum Sharing in Cognitive Radio Networks: Dynamic Game, Inefficiency of Nash Equilibrium, and Collusion

We address the problem of spectrum pricing in a cognitive radio network where multiple primary service providers compete with each other to offer spectrum access opportunities to the secondary users. By using an equilibrium pricing scheme, each of the primary service providers aims to maximize its profit under quality of service (QoS) constraint for primary users. We formulate this situation as an oligopoly market consisting of a few firms and a consumer. The QoS degradation of the primary services is considered as the cost in offering spectrum access to the secondary users. For the secondary users, we adopt a utility function to obtain the demand function. With a Bertrand game model, we analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium (i.e., equilibrium pricing adopted by the primary services). We present distributed algorithms to obtain the solution for this dynamic game. The stability of the proposed dynamic game algorithms in terms of convergence to the Nash equilibrium is studied. However, the Nash equilibrium is not efficient in the sense that the total profit of the primary service providers is not maximized. An optimal solution to gain the highest total profit can be obtained. A collusion can be established among the primary services so that they gain higher profit than that for the Nash equilibrium. However, since one or more of the primary service providers may deviate from the optimal solution, a punishment mechanism may be applied to the deviating primary service provider. A repeated game among primary service providers is formulated to show that the collusion can be maintained if all of the primary service providers are aware of this punishment mechanism, and therefore, properly weight their profits to be obtained in the future.     

Non-Cooperative Multicast and Facility Location Games

We consider a multicast game with selfish non- cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with in players, we establish an upper bound of O(radicnlog² n) on the price of anarchy, and a lower bound of Omega(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.     

Optimal Precoder Design for Non-Regenerative MIMO Cognitive Two-Way Relay Systems with Underlay Spectrum Sharing

We study the optimal precoder design for a MIMO cognitive two-way relay system with underlay spectrum sharing. The system consists of two secondary users (SUs) and one relay station (RS). We jointly optimize the precoders for SUs and RS with perfect and imperfect channel state information (CSI) between SUs/RS and the primary user, where our design approach is based on the alternate optimization method. For the perfect CSI case, we derive the optimal structure of the RS precoding matrix, which generalizes the result for single-antenna SUs and helps to reduce the search complexity. We develop gradient projection (GP) algorithm to calculate the optimal RS precoder numerically. When the RS precoder is given, we propose a fast algorithm based on generalized water-filling theorem to compute the optimal SU precoders. For the imperfect CSI case, we derive equivalent conditions for the interference power constraints and convert the robust SU precoder optimization into the form of semi-definite programming. As for the robust RS precoder optimization, we relax the interference power constraint related with the RS precoder to be convex and then the GP algorithm can be applied. Finally, simulation results demonstrate the effectiveness of the proposed schemes.     

不完全信道状态信息下非线性波形非正交多址接入系统吞吐量分析

针对目前关于非正交多址接入(NOMA)的研究均局限于线性调制的问题,且考虑到实际系统中不完全的信道状态信息(CSI),基于最小频移键控类(MSK-type)调制,该文提出一种不完全CSI下的非线性波形NOMA系统。首先,分别采用过采样和匹配滤波技术对两用户的上行异步和同步系统进行了建模。进一步,基于矩阵分解理论和连续干扰消除(SIC)机制,分别推导出完全或不完全CSI下系统吞吐量达到最大时的发射功率和归一化时延。最后,数值仿真结果表明了完全与不完全CSI下系统吞吐量与帧长、响应长度、波形以及频率脉冲等系统参数的关系。     

Performance analysis of maximum ratio transmission with imperfect channel estimation

Maximal ratio transmission (MRT) is designed assuming the availability of perfect channel state information (CSI) at both the transmitter and the receiver. However, perfect CSI is not available in practice. This paper investigates the impact of Gaussian estimation errors on the MRT performance in independently and identically distributed (i.i.d.) Rayleigh fading channels. We derive the cumulative distribution function (cdf), the probability density function (pdf) and the moment generating function (mgf) of the MRT output signal-to-noise ratio (SNR) with imperfect CSI, enabling the evaluation of some useful performance metrics such as the average error rate and the outage performance. Numerical and simulation results are provided to show the impact of imperfect CSI on the MRT performance.     

MIMO认知无线电分布式波形自适应算法

该文研究在主用户干扰温度约束下多个竞争的MIMO认知无线电最大化自身信息速率的波形自适应问题。该文使用非协作博弈论将其表述为Nash均衡问题,给出了Nash均衡解存在唯一的条件,并提出了一种求解Nash均衡的带惩罚价格的分布式迭代注水算法,通过对干扰进行价格惩罚使得MIMO认知无线电在达到Nash均衡时满足干扰温度约束。仿真结果表明相对于不考虑干扰温度约束的经典迭代注水算法,该文算法能够满足干扰温度约束,适用于认知无线电网络。     

A Nash game approach for OSNR optimization with capacity constraint in optical links

This paper develops a Nash game towards optimizing optical signal-to-noise ratio (OSNR) in the presence of link capacity constraint. In optical wavelength-division multiplexing (WDM) networks, all wavelength-multiplexed channels in a link share the optical fiber. In order to limit nonlinear effects, the total power launched into a fiber has to be below a nonlinearity threshold. This can be regarded as the optical link capacity constraint. We formulate an extended OSNR Nash game that incorporates this underlying system constraint. The status of an optical link is considered directly in the cost function. Sufficient conditions for existence and uniqueness of the Nash equilibrium (NE) solution are given. Two iterative algorithms for channel power control are proposed to compute the NE solution: a parallel update algorithm (PUA) and a relaxed PUA (r-PUA). Their convergence is studied under different conditions, both theoretically and numerically.     

Robust AM-MIMO based on minimized transmission power

We propose a design of minimizing transmission power adaptive modulation (AM) that utilizes imperfect channel state information (I-CSI) in multi-input and multi-output (MIMO) systems. By taking channel estimation errors into account, the proposed algorithm provides the quality of services (QoS) closed to the expected values when larger channel estimation error occurs. Furthermore, we, make comparison between the proposed algorithm and the ideal one based on perfect CSI assumption, and present the simulations results to illustrate the performance advantages at last.     

Asynchronous Iterative Water-Filling for Gaussian Frequency-Selective Interference Channels

This paper considers the maximization of information rates for the Gaussian frequency-selective interference channel, subject to power and spectral mask constraints on each link. To derive decentralized solutions that do not require any cooperation among the users, the optimization problem is formulated as a static noncooperative game of complete information. To achieve the so-called Nash equilibria of the game, we propose a new distributed algorithm called asynchronous iterative water-filling algorithm. In this algorithm, the users update their power spectral density (PSD) in a completely distributed and asynchronous way: some users may update their power allocation more frequently than others and they may even use outdated measurements of the received interference. The proposed algorithm represents a unified framework that encompasses and generalizes all known iterative water-filling algorithms, e.g., sequential and simultaneous versions. The main result of the paper consists of a unified set of conditions that guarantee the global converge of the proposed algorithm to the (unique) Nash equilibrium of the game.     

Dynamic resource allocation with beamforming for MIMO OFDM systems: performance and effects of imperfect CSI

We propose three different dynamic resource allocation algorithms using adaptive beamforming for multiple-input multiple-output (MIMO) OFDM systems, and investigate their performance over multipath fading channels under perfect and imperfect channel state information (CSI). These approaches involve the use of adaptive modulation, adaptive frequency-domain power allocation, and/or adaptive sub-channel allocation. By employing the proposed approaches in MIMO/OFDM systems, significant performance improvement can be achieved compared to the conventional adaptive antenna array based OFDM. The investigation of the effects of imperfect CSI reveals that the adaptive-modulation based approach is too sensitive to channel estimation errors, and that its performance is worse than the adaptive frequency-domain power allocation and/or adaptive sub-channel allocation approaches. The performance analysis also shows that combining adaptive power allocation with sub-channel allocation yields the best performance under imperfect CSI while being robust to channel estimation errors.     

自适应调制MIMO系统的最小化发射功率准则

该文提出了一种新的适用于多输入多输出系统的最小化发射总功率自适应调制准则,给出了其在完美信道反馈条件下最佳功率/比特分配方案的理论闭式解及非完美反馈时实现算法。最小化发射功率准则的显著优点是能够在满足用户需要的传输速率和误比特率的前提下,以尽可能小的发射功率实现数据传输,使无线网络中的跨层优化成为可能。     

Radio Resource Management in MIMO-OFDM- Mesh Networks: Issues and Approaches

We present a survey on the radio resource management issues in MIMO-OFDM-based infrastructure wireless mesh networks. The major components in radio resource management (i.e., scheduling and admission control) and related research issues are discussed. We review related work in the literature. We propose a game-theoretic model for admission control in IEEE 802.11n-based WMNs using the MIMO-OFDM technology. The proposed scheme uses Q-learning, which is a reinforcement learning algorithm, to gain knowledge on system performance. Then this knowledge is used to determine payoff for the game formulation to obtain the Nash equilibrium for the decision on admitting or rejecting a new connection at a mesh router.     

Performance of selective space‐time coding and selection diversity under perfect and imperfect CSI

Selective space‐time coding and selection diversity can be viewed as practical means to reduce the implementation complexity of multiple‐input multiple‐output (MIMO) systems while still taking benefit of the use of multiple antennas. In this paper, we evaluate the performance of selective space‐time block coding (selective‐STBC) and antenna selection diversity, and analyze the performance of both techniques under perfect and imperfect channel state information (CSI) available at both ends of the transmission link. Our performance analysis reveals that, under perfect or imperfect CSI and ideal feedback channel, selective‐STBC yields a loss in selection diversity gains and that selecting just a single antenna at the transmitter side is the best transmission strategy. We also show that selective‐STBC and antenna selection diversity have different behaviors when the feedback channel is imperfect. Indeed, it is shown that selection diversity outperforms selective‐STBC when the feedback channel is of high quality, while selective‐STBC yields better performance when the feedback channel is of low quality. Copyright © 2008 John Wiley & Sons, Ltd.     

Game‐theoretic modeling and analysis of relay selection in cooperative wireless networks

In this paper, distributed single relay selection in cooperative wireless networks is modeled as a Chinese restaurant game (CRG). Specifically, the CRG is used to model strategic relay selection decisions of source nodes, taking into account negative network externality due to the potential sharing of relay nodes among source nodes. Two cases are studied as follows: (i) perfect relay transmit power (RTP) knowledge and (ii) imperfect RTP knowledge. Under the first case, a distributed relay selection algorithm is proposed and shown to converge to a Nash equilibrium grouping. Under the second case, a reinforcement learning algorithm is proposed and combined with the distributed relay selection algorithm to allow network nodes to select rate‐maximizing relays. Simulation results verify the efficiency of the proposed distributed relay selection algorithm when compared with other relay selection schemes and demonstrate that it yields a network sum‐rate that is comparable with that of centralized relay selection. Copyright © 2014 John Wiley & Sons, Ltd.     

多用户下行链路收发机鲁棒性优化算法

该文提出了一种多用户MIMO系统中预矩阵及其匹配接收矩阵的鲁棒性优化算法。针对信道估计误差构造均方误差代价函数,在发射功率受限的约束条件下联合设计发射和接收矩阵,推导出了采用线性多用户MMSE接收时的最优收发机迭代算法。仿真表明,系统具有良好的鲁棒性。     

The MIMO Iterative Waterfilling Algorithm

This paper considers the noncooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general multiple-input multiple-output (MIMO) case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness/convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game.     

Joint user pairing and power allocation in uplink virtual MIMO with imperfect CSI

User pairing strategy for virtual multi-input multi-output （VMIMO） has been widely studied to improve system throughput, but most studies are based on perfect channel state information （CSI） and uniform power allocation. However, perfect CSI is very difficult or even impossible to obtain in practical system. Moreover power allocation has significant impact on algorithm performance. Therefore, in this paper, a low-complex joint user pairing and power allocation algorithm based on aggressive discrete stochastic optimization and Lagrangian dual solution is proposed for uplink VMIMO with imperfect CSI. Simulation results show that the proposed algorithm can achieve desirable throughput performance, and restrict inter-user interference （IUI） efficiently.