We derive a decoder for differential data of two multiple access channel (MAC) based uplink users. Both users utilize differential modulation for transmitting their data. It is assumed that the users cannot exchange their information. In addition, it is assumed that the users transmit data over multiple access uplink channels simultaneously in the same frequency band and without any orthogonal signatures (no CDMA). Their transmissions are assumed to be perfectly synchronized in time, phase, and frequency. The decoder is obtained by maximizing the joint probability distribution function (p.d.f.) of consecutively received data samples. We first obtain a partially differential decoder which avoids the knowledge of channel of one user at the base-station. Then we derive a heuristic differential decoder which completely avoids the knowledge of the channels of both users. For the heuristic differential decoder to work properly, both users must use rotated constellations. Optimized values of rotation angles for different M-PSK constellations are numerically obtained. It is further shown by simulations that the proposed differential decoders perform better than a same rate existing TDMA based two-user transmission scheme.
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