Multiple-Population Moment Estimation: Exploiting Inter-Population Correlation for Efficient Moment Estimation in Analog/Mixed-Signal Validation

Moment estimation is an important problem during circuit validation, in both pre-Silicon and post-Silicon stages. From the estimated moments, the probability of failure and parametric yield can be estimated at each circuit configuration and corner, and these metrics are used for design optimization and making product qualification decisions. The problem is especially difficult if only a very small sample size is allowed for measurement or simulation, as is the case for complex analog/mixed-signal circuits. In this paper, we propose an efficient moment estimation method, called Multiple-Population Moment Estimation (MPME), that significantly improves estimation accuracy under small sample size. The key idea is to leverage the data collected under different corners/configurations to improve the accuracy of moment estimation at each individual corner/configuration. Mathematically, we employ the hierarchical Bayesian framework to exploit the underlying correlation in the data. We apply the proposed method to several datasets including post-silicon measurements of a commercial high-speed I/O link, and demonstrate an average error reduction of up to 2$\times$, which can be equivalently translated to significant reduction of validation time and cost.