Pseudo-Linear Analysis of Bang-Bang Controlled Timing Circuits

This paper describes an accurate, yet analytical method to predict the key characteristics of a bang-bang controlled timing loop: namely, the jitter transfer (JTRAN), jitter generation (JG), and jitter tolerance (JTOL). The analysis basically derives a linearized model of the system, where the bang-bang phase detector is modeled as a set of two linearized gain elements and an additive white noise source. This phase detector (PD) model is by far the most extensive one in literature, which can correctly estimate the effects of random jitter, transition density, and finite loop latency on the loop characteristics. The described pseudo-linear analysis assumes the presence of random jitter at the PD input and the minimum jitter necessary to keep the linear model valid is derived, based on a describing function analysis and Nyquist stability analysis. The presented analysis re-confirms the findings of prior theories and provides theoretical basis to the prior empirically-drawn equations, such as those for the quantization noise power and the gain reduction in presence of a finite loop delay. The predictions based on the presented analysis match well with the results from time-accurate behavioral simulations.