In solving the Fokker-Planck equation as applied the first-order phase-locked loop, a Fourier series, with time varying coefficients, is used to represent the modulo-2π phase error probability density function. Solutions for the Fourier coefficients are found analytically for some special cases. For these cases the modulo-2π phase error probability density functions are given analytically. However, for the general case, a new technique for computing the Fourier co-efficients is developed. The method is based upon the observation that the Fourier coefficients can be interpreted as the state variables in an appropriate RLC ladder network. The Fourier coefficients are computed efficiently by an RLC ladder network simulation on a digital computer. Results are shown for cases involving various signal-to-noise ratios and initial conditions. Also, with the aid of the RLC ladder approach, the steady state modulo-2π phase error probability density function and the variance are given in closed forms.

This thesis also presents linearization methods applicable to first and second-order phase-locked loops. These have resulted in a systematic method of obtaining the transient statistics of both loops without recourse to the Fokker-Planck technique. Results for different values of signal-to-noise ratios are shown and compared with those obtained by the RLC ladder approach.