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.