Distributed spatial multiplexing with 1-bit feedback

We analyze 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 feedback-based 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 et al., in IEEE Trans. Inf. Th. (submitted) 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 approaching infinity, 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''. Finally, we characterize the corresponding crystallization rate.