An Analytic Approximation of Phase-Lock Receiver Threshold

The quasi-linearization procedure of Booton is applied to obtain an analytic approximation to phase-lock receiver threshold. Only the situation of an unmodulated sinusoid embedded in additive white Gaussian noise has been considered. The threshold signal-to-noise power ratio in the two-sided loop noise bandwidth of a phase-lock receiver of arbitrary transfer function was found to be 1.34 db. At threshold the rms loop error is 1.0 radian. The special situation of a high gain second-order receiver was also treated. In order to compare the analytical results with possible future measurements, the high S/N bandwidth was chosen as a reference point. Referred to this high S/N bandwidth, the threshold signal-to-noise power ratio of -0.2 db and a corresponding rms loop error of 1.16 radians were derived. The applicability of Booton's linearization procedure to nonlinear systems with statistical inputs has been experimentally verified in control system applications similar in nature to the phase-lock loop with excellent results. It is therefore anticipated that the application to phase-lock loop analysis should yield a mathematical model which describes the system more closely than strictly linear approximations.