The heuristic, GLRT and MAP detectors for double differential modulation are identical

We consider the relationship between several single-symbol detectors for double differential modulation. We have shown in a previous paper that a simple heuristic detector is identical to the generalized likelihood ratio test (GLRT) detector. We introduce a maximum a posteriori (MAP) detector and show that it is identical to the heuristic and GLRT detectors. Consequently, neither GLRT nor MAP can offer any gain over the simple heuristic detector, which means that the latter should be the detector of choice.