A linear analog network coding for asynchronous two-way relay networks

Time asynchronism is a practical issue need 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 letter, 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 decoding of the proposed ANC scheme is computationally efficient and the symbol rate can approach 1, when the coding block length is sufficiently large compared to the number of relay nodes R and the timing errors.