Transient analysis of phase-locked tracking systems in the presence of noise

This paper is concerned with the problem Of obtaining time-dependent solutions to a class of Fokker-Planck equations that arise in the analysis and synthesis of a variety of first-order synchronization systems employing the phase-lock principle. These include the classical sinusoidal phase-locked loop, squaring and Costas loops, data-aided loops, hybrid loops, various symbol synchronizer mechanizations, and tunnel-diode oscillators. By analyzing the spectral properties of the associated time-dependent Fokker-Planck boundary value problem, eigenfunction expansions of the reduced modulo- phase-error-transition probability-density function are developed for a class of first-order synchronization systems. Whereas previous work treated the steady-state probability density function, this approach yields a complete statistical description of the phase-error process reduced modulo- .