Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design

In this paper, we consider single-source, single-sink (ss-ss) multi-hop relay networks, with slow-fading Rayleigh links and single-antenna relay nodes operating under the half-duplex constraint. We present protocols and codes to achieve the optimal diversity-multiplexing tradeoff (DMT) of two classes of networks. Networks belonging to the first class can be viewed as the union of K node-disjoint parallel paths, each of length > 1, labeled here as KPP networks. The results are extended to variants including KPP(I) networks which permit causal interference between paths and KPP(D) networks which posses a direct link from source to sink. The second class is comprised of layered networks in which each layer is fully connected. We also draw some results for more general networks. For an arbitrary network with multiple flows, we show that the maximum achievable diversity gain for each flow is equal to the corresponding min-cut and present a simple amplify-andforward (AF) scheme for achieving the same. For arbitrary ss-ss directed networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable using an AF protocol. Explicit codes with short block-lengths based on cyclic division algebras are given for all the proposed protocols. Two key implications of the results in the paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two as previously believed and that simple AF protocols are often sufficient to attain the best possible DMT.