When Is Differential Detection Optimum for Ideal and Partially Coherent Reception/Demodulation of M-PSK?

We investigate the low SNR behavior of the optimum MAP sequence (blockwise) detection metric for ideal and partially coherent detection of differentially encoded M- PSK modulation. In particular, we show that based on an observation of N ges 2 symbols in the presence of a phase ambiguity introduced by the carrier tracking loop and assumed to be constant over the observation, this optimum coherent metric approaches that of multiple symbol differential detection (MSDD) asymptotically as the SNR approaches zero. More importantly, we show that for practical low SNR applications (e.g., error-correction coded systems), the region of SNR where the optimum coherent metric becomes synonymous with MSDD grows with the modulation order M. Finally, we consider the MSDD metric itself modified for symbol-by-symbol detection and demonstrate that for sufficiently low SNR (asymptotically as the SNR approaches zero), its behavior is equivalent to conventional (N = 2) differential detection, i.e., no gain to be had be extending the observation interval beyond two symbols. Unfortunately, unlike the sequence detection case, here the SNR region of validity of the low SNR approximation does not increase with increasing modulation order M.