Stochastic steady-state and AC analyses of mixed-signal systems

This paper demonstrates that the steady-state and adjoint sensitivity analyses can be extended to stochastic mixed-signal systems based on Markov chain models. The examples of such systems include digital phase-locked loops and delta-sigma data converters, of which steady-state response is statistical in nature, consisting of an ensemble of waveforms with probability distribution. For efficient Markov-chain analysis, the paper describes three methods that can limit the number of states: a state discretization scheme based on Gaussian decomposition, a state exploration algorithm that discovers the recurrent states, and a state truncation algorithm that eliminates the states with negligible stationary probabilities. The stochastic AC analysis is performed by deriving a first-order ordinary differential equation governing the perturbations in the stationary probabilities and solving it via phasor analysis. In the digital PLL and first-order ΔΣ ADC examples, the number of states was reduced by a factor of 35 and the frequency-domain phase and noise transfer functions were simulated with a 57~22,000x speed-up compared to using transient, Monte-Carlo simulations.