DMT of Multi-hop Cooperative Networks - Part I: Basic Results

In this two-part paper, the DMT of cooperative multi-hop networks is examined. The focus is on single-source single-sink (ss-ss) multi-hop relay networks having slow-fading links and relays that potentially possess multiple antennas. In this first part, some basic results that help in determining the DMT of cooperative networks as well as in characterizing the two end-points of the DMT for arbitrary full-duplex networks is established. In the companion paper, two families of half-duplex networks are studied. The present paper examines the two end-points of the DMT of ss-ss networks. In particular, the maximum achievable diversity of arbitrary multi-terminal wireless networks is shown to be equal to the min-cut between the corresponding source and the sink. The maximum multiplexing gain (MMG) of arbitrary fullduplex ss-ss networks is shown to be equal to the min-cut rank, using a new connection to a deterministic network for which the capacity was recently found. This connection is operational in the sense that a capacity-achieving scheme for the deterministic network can be converted into a MMG-achieving scheme for the original network. We also prove some basic results including a proof that the colored noise encountered in AF protocols for cooperative networks can be treated as white noise for DMT computations. We derive lower bounds for the DMT of triangular channel matrices, which are subsequently utilized to derive alternative, and often simpler proofs of several existing results. The DMT of a parallel channel with independent MIMO links is also computed here. As an application of these basic results, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable for arbitrary, ss-ss single-antenna, directed-acyclic networks equipped with full-duplex relays. All protocols in this paper are explicit and rely only upon amplify-and-forward (AF) relaying. Explicit codes for all protocols introduced here are included in the companion paper.