Full diversity achieving analog network coding for asynchronous two-way relay networks with linear receivers

Time asynchronism is a practical issue needs to be addressed for a general distributed two-way relay network, where two terminal nodes exchange information through multiple spatial-separated relay nodes. In this paper, we propose an analog network coding (ANC) scheme for a time asynchronous two-way relay network. In the proposed scheme, each relay node linearly transforms the received mixed asynchronous signals in the first time-slot by a Toeplitz matrix, and then broadcasts them back to the terminals in the second time-slot. A sufficient condition is derived for the proposed ANC to achieve full cooperative diversity using only linear receivers at the terminal nodes, such as zero-forcing (ZF), or minimum mean square error (MMSE) receivers, with any delay profiles of the timing errors. The design of the coefficients in the Toeplitz matrix to satisfy the sufficient condition is also proposed, which is shown coincides with the design we proposed in the previous work. Simulation results verify our analysis of the diversity order.