Transient Statistics of the First-Order Phase-Locked Loop

The transient behavior of the phase-error probability density function of the first-order phase-locked loop (PLL) in the presence of additive white Gaussian noise is determined using numerical techniques. In addition, the modulo-2π probability density function and cycle-slipping density functions are found. The algorithm used in these calculations involves numerical integration of a modified form of the Fokker-Planck (F-P)equation. Results are shown for cases involving various signal-to-noise ratios (SNR's and initial conditions.