IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO.2, FEBRUARY 2012 687 The MIMO Radar and Jammer Games Xiufeng Song, Student Member, IEEE, Peter Willett, Fellow, IEEE, Shengli Zhou, Senior Member, IEEE, and Peter B, Luh, Fellow, IEEE Abstract-The interaction between a smart target and a smart MIMO radar is investigated from a game theory perspective. Since the target and the radar form an adversarial system, their interac- tion is modeled as a two-person zero-sum game. The mutual infor- mation criterion is used in formulating the utility functions. The unilateral, hierarchical, and symmetric games are studied, and the equilibria solutions are derived. Index Terms-Game theory, hierarchical game, jamming, MIMO radar, Nash equilibrium, Stackelberg equilibrium, wave- form. 1. INTRODUCTION T HE success of the multiple-input multiple-output (MIMO) structure in communications has inspired investigation of MIMO radar. MIMO radars do not have a standard definition, and current literature divides them into statistical [1] and co-located [2], based on the antenna con- figuration. Generally, a statistical MIMO radar leverages the diversity of propagation path with sufficiently dissimilar trans- mitter-receiver geometry to improve detection, estimation, and information extraction [1], [3]-[8]; while a co-located one implies spatially coherent processing such as beamforming and direction-of-arrival estimation [2], [9]-[11]. The eventual acceptance of MIMO radar still remains unclear [12]. Waveform diversity is a key feature of a MIMO radar system [2]-[16]. It emphasizes illumination cooperation, and may provide an opportunity to upgrade radar performance. The specification of a waveform set largely depends on the system task. For propagation path separation, waveforms are required to be (near) orthogonal in order to avoid cross interference [3], [13]-[15]. In beampattem design, waveforms are corre- lated, so maximal transmission power can be focused in a certain direction [9]-[11]. In information extraction, the mutual information (MI) between the target response and collected echoes is maximized [4]-[8]. In target detection, optimized waveforms are designed to assure least likely missed detection Manuscript received January 28, 2011; revised June 26, 2011; accepted Au- gust 27, 2011. Date of publication September 22,2011; date of current version January 13, 2012. The associate editor coordinating the review of this manu- script and approving it for publication was Prof. Stefano Marano. This work was supported by the U.S. Office of Naval Research under Grants N00014-07-10429 and NOOO 14-09-1 0613. The authors are with the Department of Electrical and Computer En- gineering, University of Connecticut, Storrs, CT 06269 USA (e-mail: xi- ufeng.song@ gmail.com; willett@engr.uconn.edu; shengli@engr.uconn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109fTSP.20l1.216925l for a given false alarm rate [8] or to maximize the signal-to-in- terference-plus-noise ratio [16]. And in target scatterer matrix estimation, waveforms are optimized for minimum mean square error [4]-[6]. Among those waveform design criteria, MI has acquired ex- tensive attention. In the pioneering work [17], Woodward first suggested the application of information theory to radar receiver design. Later, Bell showed that maximizing the MI between target impulse response and measurement may enable the radar system a better capacity in characterizing the target in a contam- inated environment [18]. Some interesting extensions including MI based waveform design in the presence of multiple targets [19], MI based MIMO radar space time code optimization [8] and waveform design [4]-[7] emerge thereafter. In this paper, we will concentrate on the application of the MI criterion to sta- tistical MIMO radar. Current literature on MIMO radar waveform design prefers to investigate the interaction between a smart radar and a dumb target, where the former has some knowledge of the latter such as radar cross section (RCS) distribution, while the latter is in- capable of interfering with the former. Actually, with the de- velopment of electronic warfare, many noncooperative targets such as fighters are equipped with countermeasure systems to prevent a radar from operating as well as it might [20]. In this paper, the interaction involves a smart target, which carries jam- ming equipment that could intelligently confuse the radar. If the target always tries to prevent a radar from fulfilling its task, the interaction between them can be modeled as a two-person zero-sum (TPZS) game [21]. As in [4]-[8], the MI criterion is utilized to formulate the utility functions. The radar controls the waveform matrix to maximize the MI, while the latter has some access to its jam- ming matrix to minimize it. The contributions of this paper are as follows. • We suggest the use of a MI based TPZS game to model the interaction between a target and MIMO radar, and cat- egorize the game into one of three-unilateral, hierarchial, and symmetric-based on the information set available for each player. • In the unilateral case, where one player can intercept the other's strategy while the latter does not notice that this is happening, the TPZS games are simplified as single person optimizations. For this case, the optimal (water- filling) strategies are derived. • In the hierarchial case, where one player can intercept the other's strategy while the latter does notice that, the TPZS game is recast as a conservative minmax or maxmin two- stage optimization. The Stackelberg equilibria-optimiza- tion solutions-are derived. 1053-587Xf$26.00 © 2011 IEEE