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In the article, we present an ad hoc (AH) detector for two differentially encoded BPSK sources in the Hierarchical MAC (H-MAC), i.e., for the case when the receiver sees the superposition of non-orthogonal signals from individual sources. (Prefix “H-” means Hierarchical, it emphasizes that the entity is related to the many-to-one principle.) The AH detector decodes the XOR H-map of the two BPSK streams—in other words, it decides whether the transmitted symbols from the two sources are the same or opposite. The BER of the detection in H-MAC is denoted as H-BER. The H-BER is compared with the other two differential detectors, with the coherent (Coh) detector, and with an approximate coherent (ApC) detector. The exact analytical H-BER formula is derived for the ad hoc and coherent detectors. The proposed ad hoc detector is very simple for evaluation, does not require the estimation of subchannel phases, does not depend on noise variance, and it is uniformly only roughly 3.5 dB worse than the coherent one.

Exact Analytical H-BER for Ad Hoc XOR H-Map Detector for Two Differentially Modulated BPSK Sources in H-MAC Channel