In this paper, we propose two schemes for asynchronous multi-relay two-way relay (MR-TWR) systems in which neither the users nor the relays know the channel state information (CSI). In an MR-TWR system, two users exchange their messages with the help of $N_R$ relays. Most of the existing works on MR-TWR systems based on differential modulation assume perfect symbol-level synchronization between all communicating nodes. However, this assumption is not valid in many practical systems, which makes the design of differentially modulated schemes more challenging. Therefore, we design differential modulation schemes that can tolerate timing misalignment under frequency-selective fading. We investigate the performance of the proposed schemes in terms of either probability of bit error or pairwise error probability. Through numerical examples, we show that the proposed schemes outperform existing competing solutions in the literature, especially for high signal-to-noise ratio (SNR) values.
Download