Phase Noise Analysis for Stochastically Injected Oscillators

Abstract

For energy-efficient low-noise clock generation, many multiplying delay-locked loops or injection-locked clock multipliers adopt probabilistic gating to calibrate circuit nonidealities. However, it has not yet been studied how this stochastic behavior affects the phase noise of the clock generator. To answer this question, this brief proposes a novel phase noise analysis methodology for stochastically injected oscillators. Specifically, the power spectral density of a non-wide-sense-stationary process, consisting of multiple cyclostationary processes occurring stochastically, is derived from time-averaged autocorrelation. In order to obtain the required expected value, a Markov-chain-based method to calculate the probability of each state is also presented. For validation, phase noise obtained from the proposed analysis is compared with time-domain simulation results, which demonstrates that the proposed analysis provides accurate estimation.