Statistical analysis of digital phase-locked loops in fading channels

Statistical analyses of zero-crossing sampling first- and second-order digital phase-locked loops in fading channels (DPLFC) are presented. A DPLFC is defined as one where the incoming signal is not only corrupted by additive Gaussian noise channel (AGN) but also by a fading communication channel. The three fading channels considered are the log-normal, Rayleigh, and Rician channels. The performance degradation of the DPLFC operation is characterized by the steady-state phase-error process probability density function and phase-error variance. Approximate analytic expressions for the phase-error statistics of a first-order DPLFC for both phase-step and frequency-step inputs are obtained by solving the Chapman-Kolmogorov equation associated with the phase-error process. The second-order DPLFC is analyzed for a frequency-step input. Numerical and simulation results are provided confirming the analyses presented for the three fading channels considered.