Maximum-likelihood double differential detection clarified

The maximum-likelihood detector (MLD), also called the generalized likelihood ratio test (GLRT) detector, for single differential modulation is easy to derive. On the other hand, the MLD problem associated with double differential modulation is much more complicated and solving it was deemed to require a computationally unattractive nonlinear search. Consequently, a simple heuristic detector was usually preferred to the MLD, on computational grounds. In this correspondence, we prove the somewhat unexpected result that the aforementioned heuristic detector coincides with the exact MLD.