Transient analysis of the second-order phase-locked loop in the presence of noise (Corresp.)

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.