An analytical solution of the Fokker-Planck equation in the absence of noise

A Fourier series with time varying coefficients is used to represent the modulo-2π phase error probability density function (pdf) in order to solve the Fokker-Planck (FP) equation as applied to the first-order phase-locked loop (PLL). The Fourier coefficients and hence the pdf are obtained analytically for the case when the initial phase error is a random variable uniformly distributed and the additive noise is zero.