Joint beamforming and power allocation between a multistatic MIMO radar network and multiple targets using game theoretic analysis

https://doi.org/10.1016/j.dsp.2021.103085Get rights and content

Highlights

  • A basic game framework of joint beamforming and power allocation is constructed.

  • The existence and uniqueness of the Nash equilibrium are strictly proved in this game.

  • Two game algorithms of joint beamforming and power allocation are proposed.

Abstract

This paper investigates a countermeasure model between a multistatic multiple-input multiple-output (MIMO) radar network and multiple targets in the presence of continuous surface clutter. As a member of the multistatic MIMO radar network, the main purpose of each radar is to minimize its transmit power under a certain target detection criterion. Based on the selfishness of each radar, a strategic non-cooperative game (SNG) framework between MIMO radars is constructed. For the SNG between MIMO radars, the existence and uniqueness of the Nash equilibrium (NE) solution are proved strictly. And then, an iterative power allocation strategy for each MIMO radar is developed using optimization theory. The receive beamformer weight vectors of the multistatic MIMO radars are obtained by minimum variance distortionless response (MVDR) and linearly constrained minimum variance (LCMV) to suppress cross-channel interferences, respectively. Furthermore, two joint beamforming and power allocation game algorithms are proposed, which converge to the NE of the game. Finally, in order to illustrate the superiority of the proposed algorithms, we compare them with the relevant game algorithm. Numerical results are provided to show the advantages of the proposed algorithms in terms of power allocation and interference suppression.

Introduction

With the development of modern war, multiple-input multiple-output (MIMO) radar, as a new radar system, has been proposed and studied for decades [1], [2], [3]. It has shown superior performances in target detection, location and tracking, anti-jamming, and other aspects, and has been widely investigated. In order to improve the location accuracy and coverage of monostatic MIMO radar, multistatic MIMO radar system has also been studied and shown better performance in various combat scenarios [4]. Due to the normal operation of multistatic radar system, there will be cooperation and conflict between the radars. Therefore, it is necessary to further study the coordination relationship between radars and reasonably allocate their resources so that radars can work better together to enhance radars' abilities to detect targets and suppress interferences. Game theory can be applied to the resource allocation of multistatic MIMO radar as a research theory for cooperative and non-cooperative abilities [5].

Recently, game theory has been applied to the resource allocation of the communication networks, which mainly includes automatic, anonymous and feasible mobile network resource allocation. In practice, network devices can use independent and reasonable decisions to deal with the cooperation and competition of network members [6]. In addition, for the optimization problems of resource allocation in communication system, such as power control, beamforming, interference suppression, spectrum allocation, and antenna layout, game theory can be used to solve them. Then, decision makers can make correct decisions and improve the working efficiency of the communication system [7], [8], [9], [10], [11], [12], [13], [14], [15]. For the uniform power control problem, a kind of uncertain MIMO channel information is considered in [7]. Meanwhile, a robust solution is obtained by using game theory framework, which further satisfies the general situation of multiple access channels. Aiming at the problem of downlink power allocation in two-layer cellular networks, the authors of [8] use a Stackelberg Bayesian game to simulate and analyze the maximization of transmission energy without exceeding the interference threshold. Similarly, the authors in [9] also consider two-layer cellular networks, but they propose a Stackelberg game method for resource allocation management strategy based on consistent and inconsistent price mechanism. Literature [10] studies the energy consumption of node localization in wireless sensor networks and constructs a non-cooperative game model, which the game belongs to the supermodular game and has the pure strategic Nash equilibrium (NE). Meanwhile, the uniqueness of the NE is also confirmed. Besides, the problems of joint power allocation and beamforming are also studied by using game theory, and different game strategies are proposed to analyze these problems [11], [12], [13]. In order to solve the problem of communication interference suppression, a Stackelberg game model is directly constructed base on the power allocation strategy in [14]. The existence and uniqueness of the Stackelberg equilibrium are proved to ensure the convergence of the proposed Stackelberg game method. In [15], the interference avoidance game of code division multiple access (CDMA) wireless system is studied. The game is formulated by joint codewords and power allocation, and two non-cooperative sub games are proposed.

In the field of radar research, many scholars also introduce game theoretic analysis into cooperation and conflict of radar and jammer, radar and target, radar and radar, including power allocation [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], target tracking and positioning [27], [28], [29], [30], [31], target detection [32], [33], time scheduling [34], coding and waveform design [35], [36], [37], radar communication integration [38], [39]. For the power allocation problem in [16], the authors study two power allocation games between an intelligent jammer and an intelligent MIMO radar, including a two person zero sum (TPZS) game and a Stackelberg game model. Then, according to the theoretic analysis, both games can converge to the equilibrium state. In addition, for the non-cooperative game of power allocation between radars, the authors of [20], [21] construct different non-cooperative game models to study conflict relationship and propose corresponding game strategies, so that the radars can well meet the demand of utility function. In [27], [28], the cooperative game is used to study the power allocation of multistatic MIMO radar in target tracking and positioning. In [30], the authors propose a novel particle filter technology, which combines concurrent particle filter and data association with game theory to track multiple maneuvering targets in the presence of clutter. Additionally, in the aspect of target detection, a utility function of joint detection probability and false alarm probability is constructed in [32], which satisfies supermodular game and has pure strategic NE. The work in [33] designs a target detection method for polarimetric MIMO radar based on game theory, which has superior detection performance than that of the single horizontal polarization or vertical polarization. The studies in [34] analyze the time scheduling game of radar and jammer in peacetime and wartime. In order to better suppress the interference and sidelobe effects, game theory is applied as an effective strategy analysis theory to the radar waveform design [35], [36], [37], which improves the operational and survival performance of radar. In recent years, radar communication integration as a research hotspot has been studied by many researchers. Because radar and communication as two different systems have both cooperative and conflict relationships. Therefore, in order to promote their work efficiency in practical application, game theory can be used to improve their cooperative abilities between radar and communication [39].

According to the above researches, this paper investigates a countermeasure model between a multistatic MIMO radar network and multiple targets in the presence of continuous surface clutter. In general, the clutter should be continuous and widely distributed [40]. The strong local point clutter model in [21] is only a special case, which can be suppressed in the spatial domain. However, for the suppression problem of the continuous surface clutter, it is difficult to use the algorithm in [21] to suppress it in the spatial domain. If the continuous surface clutter is suppressed by the algorithm in [21], it requires not only a large number of degrees of freedom (DoFs) of array elements, but also a lot of training sampling data. Furthermore, it will result in a lot of wasted resources and affect the radar system design. In order to suppress the continuous surface clutter effectively and easily, we can use moving target indication (MTI) or moving target detection (MTD) to suppress it in the temporal domain. After the clutter processing, the radars are affected by the cross-channel interference. Then, we use the proposed algorithms to suppress the cross-channel interference between the radars. For clarification, we list the main contribution of this paper as follows:

  • A game framework of joint beamforming and power allocation between a multistatic MIMO radar network and multiple targets is constructed. The existence and uniqueness of the NE are strictly proved in this game.

  • The transmit beamformer and receive beamformer optimization models of the multistatic MIMO radars with common access of transmit and receive antennas are built, respectively. The receive beamforming weight vectors are obtained by the minimum variance distortionless response (MVDR) and linearly constrained minimum variance (LCMV), and the corresponding transmit beamforming weight vectors are also obtained, respectively.

  • The game theory and the dual optimization theory are developed to put the continuous clutter and noise into the dual function and consider them as a bounded energy in the transmitter.

  • Two game algorithms of joint beamforming and power allocation are proposed, where the LCMV game algorithm can suppress cross-channel interferences and produce deeper nulls than the algorithm in [21]. Numerical results verify the convergence and superiority of the proposed algorithms.

The rest of this paper is organized as follows. The system model is presented in section 2. Section 3 constructs the basic game framework and proves the existence and uniqueness of the NE. Section 4 proposes two game algorithms of joint beamforming and power allocation. Section 5 uses numerical results to verify the convergence and superiority of the proposed algorithms. Finally, section 6 is to conclude this paper.

Notation: ()T, () and ()H represent the transpose, conjugate and conjugate transpose operation, respectively. F defines the Frobenius norm. defines the Euclidean norm. IL is the L×L identity matrix. 1L is all one vector whose length is L. n is independent and identically distributed Gaussian white noise, nCN(0,σn2I).

Section snippets

System model

As shown in Fig. 1, this paper considers a countermeasure model between a multistatic radar network and multiple targets. The multistatic radar network consists of K separate radars each consisting of M transmit and M receive antennas. The main objective of each radar is to meet a desired detection performance for the targets using a minimum possible power allocation, which is constraint by a desired signal-to-interference noise ratio (SINR) requirement. Besides, it is assumed that multiple

Basic game theoretic formulation

There is a countermeasure relationship between radars and targets. Besides, the radars of the same side not only cooperate with each other, but also interfere with each other. Thus, a strategic non-cooperative game (SNG) framework of joint beamforming and power allocation between multistatic radars is established. The radars are regarded as players, and the power allocation is regarded as the corresponding game strategy. The utility function is the sum of the radar power of each station. Thus,

Power allocation

In order to gain more advantages in the game process, each player expects to obtain channel information with other players. Because the multistatic radars belong to the same organization, all the interradar channel information of participants know each other, and each radar knows the exact location of other radars. The optimization model (21a), (21b), (21c) can be converted into:minWt(k)l=1Lwt(kl)2s.t.jlL|wt(kl)Hhr(kkj)|2+ηkl1γkl|wt(kl)Hhr(kkl)|20,lwt(kl)Hgr(kil)gr(kil)Hwt(kl)=0,ik,

Numerical results

In this section, numerical results are presented to show the convergence of the power allocation and beamformer design game algorithms between multistatic MIMO radars, and the adaptive beamforming can be used to suppress cross-channel interferences. Meanwhile, compared with the recent related work, it shows the superiority of the proposed algorithm. It is assumed that the radars and the targets are in a far-field environment, and the targets enter the radar power radiation range. Clutter can

Conclusion

In this paper, we have used convex optimization and game theory to study a countermeasure model between a multistatic MIMO radar network and multiple targets in the presence of continuous surface clutter. For the strategic non-cooperative game between radars, the existence and uniqueness of the NE are proved. According to introducing the utility function of the game and its corresponding best response strategy, we have proposed two joint beamforming and power allocation game algorithms to

CRediT authorship contribution statement

Bin He: Conceptualization, Methodology, Software, Investigation, Writing – original draft. Hongtao Su: Validation, Formal analysis, Visualization. Junsheng Huang: Validation, Formal analysis, Visualization, Software.

Declaration of Competing Interest

We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable and insightful comments and suggestions which helped improve the paper. Thank my good friends Dr. Xu Liu and Dr. Ning Yang for their warm help.

Bin He received the B.E. and M.S. degrees in department of mathematics from North University of China, Taiyuan, Shanxi, China, in 2013 and 2017, respectively. He is currently pursuing the Ph.D. degree with the National Laboratory of Radar Signal Processing, Xidian University, Xi'an, China. His research interests include signal processing, resource allocation, game theory and optimization theory.

References (47)

  • W. Saad et al.

    Coalitional game theory for communication networks

    IEEE Commun. Mag.

    (2009)
  • D.P. Palomar et al.

    Uniform power allocation in mimo channels: a game-theoretic approach

    IEEE Trans. Inf. Theory

    (2003)
  • N.D. Duong et al.

    Stackelberg bayesian game for power allocation in two-tier networks

    IEEE Trans. Veh. Technol.

    (2016)
  • X. Kang et al.

    Price-based resource allocation for spectrum-sharing femtocell networks: a Stackelberg game approach

    IEEE J. Sel. Areas Commun.

    (2012)
  • A. Moragrega et al.

    Supermodular game for power control in TOA-based positioning

    IEEE Trans. Signal Process.

    (2013)
  • D.N. Nguyen et al.

    Power minimization in mimo cognitive networks using beamforming games

    IEEE J. Sel. Areas Commun.

    (2013)
  • D.H.N. Nguyen et al.

    Multiuser downlink beamforming in multicell wireless systems: a game theoretical approach

    IEEE Trans. Signal Process.

    (2011)
  • H. Dahrouj et al.

    Coordinated beamforming for the multicell multi-antenna wireless system

    IEEE Trans. Wirel. Commun.

    (2010)
  • D. Yang et al.

    Coping with a smart jammer in wireless networks: a Stackelberg game approach

    IEEE Trans. Wirel. Commun.

    (2013)
  • C. Lacatus et al.

    Adaptive interference avoidance for dynamic wireless systems: a game theoretic approach

    IEEE J. Sel. Top. Signal Process.

    (2007)
  • X. Song et al.

    The mimo radar and jammer games

    IEEE Trans. Signal Process.

    (2012)
  • A. Deligiannis et al.

    Power allocation game between a radar network and multiple jammers

  • H. Godrich et al.

    Power allocation strategies for target localization in distributed multiple-radar architectures

    IEEE Trans. Signal Process.

    (2011)
  • Cited by (0)

    Bin He received the B.E. and M.S. degrees in department of mathematics from North University of China, Taiyuan, Shanxi, China, in 2013 and 2017, respectively. He is currently pursuing the Ph.D. degree with the National Laboratory of Radar Signal Processing, Xidian University, Xi'an, China. His research interests include signal processing, resource allocation, game theory and optimization theory.

    Hongtao Su received the B.S., M.S., and Ph.D. degrees in electronics engineering from Xidian University, Xi'an, China, in 1997, 2000, and 2005, respectively. He is currently a Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include high-frequency over-the-horizon radar signal processing, adaptive array signal processing, and statistical signal processing.

    Junsheng Huang received the B.S. degree in electronic engineering from Lanzhou Jiaotong University, Lanzhou, China, in 2014. Since 2015, he has been working toward the Ph.D. degree in electronic engineering from Xidian University, Xi'an, China, where he is currently a graduate student at the National Laboratory of Radar Signal Processing. His research interests include array signal processing, adaptive anti-jamming techniques, and radar waveform design.

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