Accurate Multi-segment Probability Density Estimation Through Moment Matching

Abstract

The impact of process variations continues to grow as transistor feature size shrinks. Such variations in transistor parameters lead to variations and unpredictability in circuit output and may ultimately cause them to violate specifications leading to circuit failure. In fact, timely failures in critical circuits may lead to catastrophic failures in the entire chip. As such, statistical modeling of circuit behavior is becoming increasingly important. However, existing statistical circuit simulation approaches fail to accurately and efficiently analyze the high sigma behavior of probabilistic circuit output. To this end, we propose PDM (Piecewise Distribution Model) - a piecewise distribution modeling approach via moment matching using maximum entropy to model the high sigma behavior of analog/mixed-signal (AMS) circuit probability distributions. PDM is independent of the number of input dimensions and matches region specific probabilistic moments which allows for significantly greater accuracy compared to other moment matching approaches. PDM also utilizes Spearman’s rank correlation coefficient to select the optimal approximation for the tail of the distribution. Experiments on a known mathematical distribution and various circuits obtain accurate results up to 4.8 sigma with 2-3 orders of speedup relative to Monte Carlo. PDM also demonstrates better accuracy while compared against other state-of-the-art statistical modeling approaches, such as maximum entropy, importance sampling, and subset simulation.