Binary Phase Detector Gain in Bang-Bang Phase-Locked Loops With DCO Jitter

Bang-bang phase-locked loops are hard nonlinear systems due to the nonlinearity introduced by the binary phase detector (BPD). In the presence of jitter, the nonlinear loop is typically analyzed by linearizing the BPD and applying linear transfer functions in the analysis. In contrast to a linear phase detector, the linearized gain of a BPD depends on the rms jitter and the type of jitter (either nonaccumulative or accumulative). Previous works considered the case of nonaccumulative reference clock jitter and showed that the BPD gain is inversely proportional to the rms jitter when the latter is small or large. In this brief, we consider the case of accumulative digitally-controlled oscillator (DCO) jitter and derive an asymptotic closed-form expression for the BPD gain, which becomes exact in the limit of small and large jitter. Contrary to the reference clock jitter case, the BPD gain is constant for small DCO jitter and is inversely proportional to the square of jitter for large DCO jitter; in the latter case, the timing jitter has a normal Laplace distribution.