Recently, several digital phase-locked loops (DPLLs) have been demonstrated to achieve the jitter performance of traditionalcharge-pump-based analog PLLs. This paper is concerned with a class of DPLLs employing a binary-quantized phase detector, referred to as bangbang PLLs (BBPLLs). They are widely used in clock and data recovery circuits and have recently been implemented as digital BBPLLs for high-bandwidth synthesis. Given that a DPLL implementation typically suffersfrom (excess) loop delay, this paper investigates the combined effect of loop delay and reference clock jitter in a first-order digital BBPLL. To statistically characterize the loop’s timing jitter we formulate it as a discrete-time vector Markov process and numerically solve the associated Chapman-Kolmogorov equation. This allows us to compute the timing jitter probability density function in steady-state and to evaluate the jitter performance (timing offset and RMS timing jitter) for varying loop detuning, RMS reference clock jitter and loop delay.
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