Investigation of first-order digital bang-bang phase-locked loops with reference clock jitter

Bang-bang phase-locked loops (BBPLLs) are a class of PLLs with a binary-quantized phase detector (BPD). They are widely used in clock and data recovery circuits and have
recently been implemented as digital BBPLLs for high-bandwidth synthesis. This paper investigates a first-order digital BBPLL
with reference clock jitter. We derive the Chapman-Kolmogorov equation which statistically characterizes the timing jitter process. The numerical solution of this equation allows us to compute the timing jitter probability density function (PDF) in steadystate and to examine the effect of varying loop detuning and
RMS reference clock jitter on the timing offset, the RMS timing jitter and the mean number of steps to slip a cycle. The analysis
shows that the steady-state PDF is Gaussian-like only for a small range of RMS clock jitter values, which leads to a new curve for
the BPD gain as a function of jitter.