Differential Modulation and Demodulation for Decode-and-Forward Multiple Relay Systems

We consider differential modulation and demodulation in a decode-and-forward (DF) multiple relay system when channel state information (CSI) is unknown to any node. This paper first derives the maximum likelihood (ML) detector and proposes a low complexity pairwise-linear (PL) detector to well approximate the nonlinear ML detector. In both the ML and PL detectors, the receive signals at the destination from the source and all the relays are combined with different weights, since the transmission links in the relay system have different statistics. Then, the bit error rate (BER) for the PL detector is analyzed. For single relay systems, the exact BER is obtained as a simple function of signal-to-noise ratios (SNRs), the variances, and the fading rates of the transmission links. Based on the exact BER, a BER approximation at high SNR is derived, and explicitly shows the diversity order and the asymmetry between the source-relay link and the relay-destination link. For multiple relay systems, a Cherenoff upper bound on the BER is obtained. In addition, our results are applicable to the cases where all the transmission links have arbitrary fading rates.