A reflection on the conventional formulations of correlation lower bounds for M-PSK/CDMA sequences

We argue that due to the existence of the even-odd transforms, the smallest possible maximum periodic and odd correlation magnitudes are identical for complex-valued sequence sets under the equi-energy (or constant-envelope) constraint. This trivializes most of the conventional formulations of odd correlation lower bounds for BPSK/CDMA sequences. By generalizing the even-odd transforms, we also extend this insightful result for complex-valued sequence sets applied to asynchronous M-PSK/CDMA schemes with M ≥ 2, whose multiple access interferences are characterized by the M-phase correlation magnitudes. Next, a general transform that translates known inner-product lower bounds into M-phase correlation bounds is introduced. Finally, we conclude that in order to formulate nontrivial M-phase correlation lower bounds for equi-energy (or constant-envelope) sequences, additional phase alphabet constraints must be incorporated into the formulation.