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Separation Theorems for Phase-Incoherent Multiple-User Channels

We study the transmission of two correlated and memoryless sources $(U,V)$ over several multiple-user phase asynchronous channels. Namely, we consider a class of phase-incoherent multiple access relay channels (MARC) with both non-causal and causal unidirectional cooperation between encoders, referred to as phase-incoherent unidirectional non-causal cooperative MARC (PI-UNCC-MARC), and phase-incoherent unidirectional causal cooperative MARC (PI-UCC-MARC) respectively. We also consider phase-incoherent interference channels (PI-IC), and interference relay channel (PI-IRC) models in the same context. In all cases, the input signals are assumed to undergo non-ergodic phase shifts due to the channel. The shifts are assumed to be unknown to the transmitters and known to the receivers as a realistic assumption. Both necessary and sufficient conditions in order to reliably send the correlated sources to the destinations over the considered channels are derived. In particular, for all of the channel models, we first derive an outer bound for reliable communication that is defined with respect to the source entropy content (i.e., the triple $(H(U|V),H(V|U),H(U,V))$). Then, using {\em separate} source and channel coding, under specific gain conditions, we establish the same region as the inner bound and therefore obtain tight conditions for reliable communication for the specific channel under study. We thus establish a source-channel separation theorem for each channel and conclude that without the knowledge of the phase shifts at the transmitter sides, separation is optimal. It is further conjectured that separation in general is optimal for all channel coefficients.