  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.
