Multi-hop Analog Network Coding: An Amplify-and-Forward Approach
  In this paper, we study the performance of an amplify-and-forward (AF) based analog network coding (ANC) relay scheme in a multi-hop wireless network under individual power constraints. In the first part, a unicast scenario is considered. The problem of finding the maximum achievable rate is formulated as an optimization problem. Rather than solving this non-concave maximization problem, we derive upper and lower bounds for the optimal rate. A cut-set like upper bound is obtained in a closed form for a layered relay network. A pseudo-optimal AF scheme is developed for a two-hop parallel network, which is different from the conventional scheme with all amplification gains chosen as the maximum possible values. The conditions under which either the novel scheme or the conventional one achieves a rate within half a bit of the upper bound are found. Then we provide an AF-based multi-hop ANC scheme with the two schemes for a layered relay network. It is demonstrated that the lower bound of the optimal rate can asymptotically achieve the upper bound when the network is in the generalized high-SNR regime. In the second part, the optimal rate region for a two-hop multiple access channel (MAC) via AF relays is investigated. In a similar manner, we first derive an outer bound for it and then focus on designing low complexity AF-based ANC schemes for different scenarios. Several examples are given and the numerical results indicate that the achievable rate region of the ANC schemes can perform close to the outer bound. 
Binyue Liu and Ning Cai
cs.IT math.IT