為了達到低功率，低電壓的設計是必需的。再正常電壓下，電晶體裡面不同的變量影響著元件延遲使得元件延遲被表示成這些變量的一個非線性函數。因為這些電晶體裡面的變量再低電壓的情況下未必是常態分佈，因此元件延遲的分佈也未必是常態分佈。所以必須要有一個元件延遲特徵化的方法來處理低電壓的情況。在先前的方法裡面，NLOPALV是一個非常有效率的方法來處理再低電下的元件庫特徵化。此方法是使用一個線性函數來近似元件延遲與電晶體變量之間的關係。這個方法再元件延遲與電晶體變量之間的關係是近似一個線性函數的時候會有很好的結果，但是當此關係非常不像是一個線性函數此方法可能會造成很大的誤差。蒙地卡羅方法所得到元件延遲的分佈來當做實際上真正的結果，因此誤差是與蒙地卡羅方法的結果來比較。我們的方法是利用退化的二次多項式來近似元件延遲與電晶體變量之間的關係。在我們的實驗結果裡面，我們的方法會比NLOPALV的誤差更小並且誤差在5%以下。

[1] R. Rithe, S. Chou, J. Gu, A. Wang, S. Datla, G. Gammie, D. Buss, and A. Chandrakasan “Non-linear operating point statistical analysis for local variations in logic timing at low voltage,” in Proc. Design, Automation & Test in Europe Conference & Exhibition (DATE), pp. 965-968, 2010.

[2] R. Rithe, S. Chou, J. Gu, A. Wang, S. Datla, G. Gammie, D. Buss, and A. Chandrakasan “Cell library characterization at low voltage using non12 linear operating point analysis of local variations,” in Proc. International Conference on VLSI Design, pp. 112-117, 2011.

[3] R. Rithe, S. Chou, J. Gu, A. Wang, S. Datla, G. Gammie, D. Buss, and A. Chandrakasan “The effect of random dopant fluctuations on logic timing at low voltage,” IEEE Trans on Very Large Scale Integration
Systems, vol. 20, no. 5, pp. 911-924, 2012.

[4] J. Jaffari and M. Anis, “On Efficient Monte Carlo-Based Statistic Timing Analysis of Digital Circuits,” in proc. International Conference on Computer-Aided Design., pp. 196-203, 2008.

[5] Y. Kanoria, S. Mitra and A. Montanari “Statistical Static Timing Analysis using Markov Chain Monte Carlo,” in Proc. Design, Automation & Test in Europe Conference & Exhibition, pp. 813-818, 2010.

[6] A. Singhee, S. Singhal and Rob A. Rutenbar “Practical, fast monte carlo statistical static timing analysis: Why and how,” in Proc. International Conference on Computer-Aided Design., pp. 190-195, 2008.

[7] V. Veetil, D. Sylvester and D. Blaauw “Efficient monte carlo based incremental statistical timing analysis,” in Proc. Design Automation Conference., pp. 676-681, 2008.

[8] Lerong Cheng, Jinjun Xiong and Lei He, “Non-Gaussian Statistical Timing Analysis Using Second-Order Polynomial Fitting,” in Proc. Asia and South Pacific Design Automation Conference., pp. 298-303, 2008.

[9] V. Khandelwal and A. Srivastava, “A general framework for accurate statistical timing analysis considering correlations,” in Proc. Design Automation Conference., pp. 89-94, 2005.

[10] X. Du and W. Chen, “A Most Probable Point Based Method For Uncertainty Analysis,” Journal of Design and Manufacturing Automation, vol. 4, no. 1, pp. 47-66, 2001.

[11] K. Breitung, “Asymptotic Approximations for Multinormal Integrals,” Journal of Engineering Mechanics, vol. 110, no. 3, pp. 357-366, 1984.

[12] L. Tvedt, “Distribution of Quadratic Forms in Normal Space Application to Structural Reliability,” Journal of Engineering Mechanics, vol. 116, no. 6, pp. 1183-1197, 1990.