On the achievable degrees of freedom in a class of multi-user half-duplex relay networks

Abstract

We study the achievable sum degrees of freedom (DoF) in a class of wireless single-antenna multi-hop half-duplex relay networks. The networks contain M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> information sources, Md information destinations, and arbitrary layers of relays, each with 2K (K ≥ max{M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ,M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> }) half-duplex relays, in between. A cluster successive relaying transmission scheme is applied to conduct the communication: We divide the relays in each layer into two equal-size clusters and activate them successively to efficiently use the available channel. It is shown that in a time-varying fading environment this scheme asymptotically achieves the sum DoF min { M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> K/M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> +K, M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> K/M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> +K-1}, which is irrelevant to the number of hops the source messages have to pass through. This result also implies that, when the number of relays in each layer is infinitely large, the available DoF (i.e. the optimally achievable sum DoF) of the considered networks is min{M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ,M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> }. Neither distributed signal processing nor multiple layers of half-duplex relay operation negatively affects the system DoF performance.