Distributed spatial multiplexing with 1-bit feedback

We analyze the feasibility of distributed spatial multiplexing with limited feedback in a slow-fading interference network with MN non-cooperating single-antenna sources and M non-cooperating single-antenna destinations. In particular, we assume that the sources are divided into M mutually exclusive groups of N sources each, every group is dedicated to transmit a common message to a unique destination, all transmissions occur concurrently and in the same frequency band and a dedicated 1-bit broadcast feedback channel from each destination to its corresponding group of sources exists. We provide a feedbackbased iterative distributed (multi-user) beamforming algorithm, which “learns” the channels between each group of sources and its assigned destination. This algorithm is a straightforward generalization, to the multi-user case, of the feedback-based iterative distributed beamforming algorithm proposed recently by Mudumbai, Hespanha, Madhow and Barriac in [1] for networks with a single group of sources and a single destination. Putting the algorithm into a Markov chain context, we provide a simple convergence proof. We then show that, for M finite and N → ∞, spatial multiplexing based on the beamforming weights produced by the algorithm achieves full spatial multiplexing gain of M and full per-stream array gain of N , provided the time spent “learning” the channels scales linearly in N . The network is furthermore shown to “crystallize” in the sense that, in the largeN limit, the M individual fading links not only decouple (as reflected by full spatial multiplexing gain) but also converge to non-fading links. Finally, we quantify the impact of the performance of the iterative distributed beamforming algorithm on the crystallization rate, and we show that the multi-user nature of the network leads to a significant reduction in the crystallization rate, when compared to the M = 1 case.