Fast Statistical Analysis of Rare Circuit Failure Events via Scaled-Sigma Sampling for High-Dimensional Variation Space

Accurately estimating the rare failure rates for nanoscale circuit blocks (e.g., static random-access memory, D flip-flop, etc.) is a challenging task, especially when the variation space is high-dimensional. In this paper, we propose a novel scaled-sigma sampling (SSS) method to address this technical challenge. The key idea of SSS is to generate random samples from a distorted distribution for which the standard deviation (i.e., sigma) is scaled up. Next, the failure rate is accurately estimated from these scaled random samples by using an analytical model derived from the theorem of "soft maximum." Our proposed SSS method can simultaneously estimate the rare failure rates for multiple performances and/or specifications with only a single set of transistor-level simulations. To quantitatively assess the accuracy of SSS, we estimate the confidence interval of SSS based on bootstrap. Several circuit examples designed in nanoscale technologies demonstrate that the proposed SSS method achieves significantly better accuracy than the traditional importance sampling technique when the dimensionality of the variation space is more than a few hundred.