Beam-Domain Anti-Jamming Transmission for Downlink Massive MIMO Systems: A Stackelberg Game Perspective

In this paper, beam-domain (BD) anti-jamming transmission in a downlink massive multiple-input multiple-output (MIMO) system is investigated. A smart jammer with multiple antennas attempts to interfere with the signal reception of users with the desired energy efficiency (EE), whereas a base station (BS) tries to minimize the transmission cost while ensuring uninterrupted communication. A Bayesian Stackelberg game between the BS and jammer, where the jammer is the follower and the BS acts as the leader, is modeled. In the follower subgame, the optimal jamming precoding with a closed-form power solution is introduced. The optimal jamming power is proportional to the transmission power in the downlink, and thus, for the BS, the strategy of suppressing malicious attacks by increasing the transmission power fails. In the leader subgame, generalized zero-forcing (ZF), whose closed-form power solution constitutes the unique Stackelberg equilibrium (SE) with that of the jammer, is found to be the optimal anti-jamming precoding for robust transmission. The results show that there always exists a precoding solution for the BS that ensures reliable transmission when the SE is obtained. A proper increase in the minimum signal-to-interference-and-noise ratio (SINR) threshold or the BD channel approximation error helps the BS save power during the resistance against the jammer. Then, a simplified power solution without the instantaneous channel state information (CSI) of jamming channels is further introduced for practical implementation. Numerical results are provided to verify the proposed solutions.