Noncooperative iterative MMSE beamforming algorithms for ad hoc networks

An asynchronous unicast ad hoc network is considered, where each node i is equipped with a receive/transmit beam-former pair (W_i, g_i) designed under a quality-of-service (QoS) SNR constraint. It is first shown that the minimum sum-power beamformers for the network satisfy a weak duality condition, in which the pairs ((g_i^opt)*, (W_i^opt)*) achieve the same sum power as the primal network. However, the optimum receive beamformer w_i^opt is not in general equal to (g_i^opt)*, in contrast to the case of cellular and time-division duplexing networks. Iterative minimum mean-square error (IMMSE) beamforming algorithms are then proposed in which w_i = g_i* is enforced. These algorithms are shown to be instances of the Power Algorithm in which gi is the maximizing eigenvector of an SNR-related objective matrix. The IMMSE algorithm can also be viewed as a noncooperative beamforming game, in which the payoff includes normalized SNR, and the tax is related to interference caused at other nodes. The existence of fixed points (Nash equilibria) is proved for IMMSE. Furthermore, fixed points of IMMSE are shown to satisfy the first-order necessary conditions for optimization using a network Lagrangian. The IMMSE game is modified to yield a sequential distortionless-response beamforming algorithm, which is shown to be convergent using a Total Interference Function. Extensive simulation results illustrate that IMMSE yields better power efficiency than a greedy noncooperative SNR-maximizing game.