Performance of First-Order Digital Phase-Locked Loops in Mobile Radio Channels

The performance of a first-order digital phase-locked loop (DPLL) using nonuniform sampling is studied in the mobile radio environments. The mobile radio channel is characterized by introducing fast Rayleigh fading and random phase variation to the signal envelope and phase, respectively. The nonlinear stochastic difference equation describing the loop operation in fading environments is introduced. The joint probability density function (pdf) of the random variables of this equation is derived by transformation of random variables for fast Rayleigh fading channels. A closed-form expression for the transition probability of the ChapmanKolmogorov (C-K) equation is obtained for phase step plus noise input and for frequency step plus noise input. The probability density function of the steady-state phase error is obtained by solving the C-K equation numerically.