Optimum Noncoherent Detection of Constant-Envelope Signals using Received Signal Magnitudes -- Energy Detection and Amplitude Detection

Constant-envelope signals are widely employed in wireless communication systems due to their hardware-friendly design, energy efficiency, and enhanced reliability. However, detecting these signals reliably using simple, power-efficient receivers remains a critical challenge. While coherent detection methods generally offer superior performance, they require complex frequency synchronization, which increases receiver complexity and power consumption. In contrast, noncoherent detection is inherently simpler since it avoids frequency synchronization. However, traditional noncoherent detection approaches still rely on In-phase and Quadrature-phase (IQ) demodulators to extract the noisy received signal magnitudes, and assume the energy detector as the test statistic according to the IQ structure, without rigorous theoretical justification. Motivated by the practical need for robust and low-complexity detection, this paper proposes a comprehensive framework for optimal signal detection using a simple bandpass-filter envelope-detector (BFED) in conjunction with a Bayesian approach and the generalized likelihood ratio test (GLRT) under unknown amplitude conditions. By leveraging approximations of the modified Bessel functions, we derive two distinct regimes of the optimal detector. In the low SNR regime, we rigorously prove that the energy detector emerges as the Bayesian-optimal solution, thereby establishing its theoretical validity for the first time. In the high SNR regime, we derive a novel amplitude-based detector that directly compares the estimated amplitude against the noise standard deviation, leading to a simple yet optimal detection strategy. Numerical simulations validate the theoretical findings, confirming that both the energy and amplitude detectors achieve the minimum error probability in their respective SNR domains.