Skip to Main Content
The time-dependent probability-density function (pdf) of the state vector for the second-order phase-locked loop (PLL) is shown to be determined by an orthogonal function expansion. The expansion coefficients are the solutions to infinite dimensional ordinary differential equations, which are derived by Ito's stochastic calculus. Numerical solutions can be obtained by truncating them. It is also shown that certain parameters which characterize the distribution of a random variable on a circle, such as phase error, can be expressed by only one coefficient of the Fourier series of its pdf.