Maximum-Likelihood Detector for Differential Amplify-and-Forward Cooperative Networks

The exact maximum-likelihood (ML) detector for amplify-and-forward (AF) cooperative networks employing M-ary differential phase-shift keying (DPSK) in Rayleigh fading is derived in a single-integral form, which serves as a benchmark for differential AF networks. Two algorithms are then developed to reduce the complexity of the ML detector. Specifically, the first algorithm can eliminate a number of candidates in the ML search, while causing no loss of optimality of ML detection. In high signal-to-noise ratios (SNRs), this algorithm almost surely identifies a single candidate that amounts to the ML estimate of the signal. For low to medium SNRs with multiple candidates determined, we then derive an accurate closed-form approximation for the integral involved in the likelihood function, which only requires a five-sample evaluation per symbol candidate. Finally, combining these algorithms, we propose a closed-form approximate ML detector, which achieves an almost identical bit-error-rate (BER) performance to the exact ML detector at practical complexity. In particular, it is shown that the proposed approximate ML detector is far less complex than the well-known diversity combiner in high SNRs, while achieving approximately 1.7-dB gain in the 10^-5 BER when the relay is closer to the destination.