Bang–Bang phase detector (BBPD) as a bistable system has the metastability problem. In addition, BBPD is a non-linear block in the phase locked-loop (PLL) and clock and data recovery (CDR) that makes their analysis complicate. To simplify the analysis of the non-linear BBPD, the linear expression for the gain of multi-level BBPD (ML-BBPD) is derived in this paper by considering the metastability failure. The ML-BBPD gain is developed based on Markov chain to provide more realistic model. Simulation results show that the metastability failure not only attenuates the gain of ML-BBPD for the small input jitter, but also increases the output jitter and consequently the BER of the CDR. It is shown that by increasing the number of sampling clocks the output jitter and BER is reduced. Also, the gain of ML-BBPD is decreased for small values of the input jitter, but for the large values of input jitter ML-BBPD gain is the same for every over sampling ratio.

Bang-Bang 相位检测器 (BBPD) 作为双稳态系统存在亚稳态问题。此外，BBPD 是锁相环 (PLL) 和时钟和数据恢复 (CDR) 中的非线性模块，这使得它们的分析变得复杂。为了简化非线性BBPD的分析，本文考虑亚稳态失效，推导出多级BBPD增益的线性表达式（ML-BBPD）。ML-BBPD 增益是基于马尔可夫链开发的，以提供更真实的模型。仿真结果表明，亚稳态失效不仅会衰减小输入抖动的 ML-BBPD 增益，还会增加输出抖动，从而增加 CDR 的 BER。结果表明，通过增加采样时钟的数量，可以降低输出抖动和误码率。还，