Timing Jitter Distribution and Power Spectral Density of a Second-Order Bang-Bang Digital PLL with Transport Delay Using Fokker-Planck Equations

In this paper, a second-order bang-bang digital phase-locked loop (BBPLL) with dominant random walk phase noise and transport delay is analyzed using Fokker-Plank equations. Explicit closed-form expressions are derived for the timing error probability distribution function, jitter variance, and power spectral density (psd). For the type-II BBPLL considered in this paper, the timing error distribution is shown to be Laplacian and not Gaussian distributed as previously assumed, while the derived psd is Lorentzian, which is consistent with earlier works. The analytical solutions are valid as long as the continuous-time approximation of the BBPLL dynamics is accurate as is the case for typical operating loop bandwidths. The accuracy of the derived expressions is validated via simulation.