One of the most important applications of network coding is the Multiple Access Relay Channel (MARC) technique in which source terminals transmit independent data blocks over one or more relay terminals to a common destination terminal. In recent years, MARC has become one of the vital network coding applications in the area of innovative wireless cooperative communications for enhancing spectral efficiency. In this paper, non-coherent binary frequency shift keying (BFSK) modulation and differential binary phase shift keying (DPSK) modulation techniques are applied to the decode-and-forward (DF)-based MARC model and the average bit error rate (BER) of the proposed system is derived theoretically by using the maximum likelihood (ML) decision rule. The simulation results show that the BER performance of DPSK modulation is better than the non-coherent BFSK and full diversity gain is achieved in both modulation techniques. We also propose a power allocation scheme for the proposed system model which optimizes the power allocated to the source and relay transmissions. Depending on the location of the relay, performance gains up to 1.25 dB are achieved with the proposed optimum power allocation (OPA) scheme. Also, the mathematical derivations are verified through Monte-Carlo simulations and the superior performance of the OPA over equal power allocation (EPA) is observed. (C) 2019 Elsevier B.V. All rights reserved.