Optimum Detection of Random Phase Modulation with Phase‐Lock Loops

A study was conducted to determine the optimum design for a second‐order phase‐lock loop when used to detect a randomly phase‐modulated sine wave of unknown average frequency in a white‐noise background. The optimum physically realizable loop was also found for the case when the average frequency is known a priori. A quasilinear model was used to account for the nonlinear characteristic of the phaselock loop and to predict the threshold effect. The results of the study give formulas for calculating the root‐mean‐square loop‐phase error and detector‐output signal‐to‐noise ratio as a function of the input signal‐to‐noise ratio and the power and bandwidth of the assumed modulating function. The results show a strong dependence of the loop threshold on the characteristics of the modulating function. Careful consideration must be given these parameters in system designs employing phase‐lock loops for the processing of this class of signal.