We study a layered $K$-user $M$-hop Gaussian relay network consisting of $K_m$ nodes in the $m^{ \operatorname{ th}}$ layer, where $M\geq2$ and $K=K_1=K_{ M+1}$. We observe that the time-varying nature of wireless channels or fading can be exploited to mitigate the inter-user interference. The proposed amplify-and-forward relaying scheme exploits such channel variations and works for a wide class of channel distributions including Rayleigh fading. We show a general achievable degrees of freedom (DoF) region for this class of Gaussian relay networks. Specifically, the set of all $(d_1,..., d_K)$ such that $d_i\leq 1$ for all $i$ and $\sum_{ i=1}^K d_i\leq K_{ \Sigma}$ is achievable, where $d_i$ is the DoF of the $i^{ \operatorname{ th}}$ source--destination pair and $K_{ \Sigma}$ is the maximum integer such that $K_{ \Sigma}\leq \min_m\{ K_m\}$ and $M/K_{ \Sigma}$ is an integer. We show that surprisingly the achievable DoF region coincides with the cut-set outer bound if $M/\min_m\{ K_m\}$ is an integer, thus interference-free communication is possible in terms of DoF. We further characterize an achievable DoF region assuming multi-antenna nodes and general message set, which again coincides with the cut-set outer bound for a certain class of networks.
Categories: cs.IT math.IT