Process-Variation Statistical Modeling for VLSI Timing Analysis

隨著製程的技術越來越進步,特徵尺寸已經小於曝光微影術的光源波長,這時候製程所產生的偏差變得非常重要,而且必須在晶片設計的階段就考慮這個問題。傳統以 corner value 為基礎的時序分析將會導致預測的時序過份被低估,這個原因其實是因為所有的閘都要運作在各自 corner value 的機會是非常渺小的。統計上的靜態時序分析 ( Statistical Static Timing Analysis - SSTA) 就是利用統計的方式去描述這些製成偏差,把他們視為一些統計的隨機變數,然後利用他們去預測時序並且得到更準確而且更逼真的結果。 目前已經有很多有關統計上的靜態時序分析演算法被提出,而且可以說是發展的十分成熟了。然而為了便於計算,他們大多是假設這些製成偏差為高斯分佈。利用高斯分佈他所能夠模型化的能力十分有限,而且對於各種非高斯分佈的這些製程誤差參數更是一籌莫展。 在這篇論文,我們使用了高斯多項式去模型化這些非高斯分佈的製成偏差。然而,一些含有高度偏態的分佈沒有辦法直接使用高斯多項式去呈現,我們會得到一些複係數。複係數代表被模型化的資料也是一組複係數的資料,在傳統的統計學裡,我們沒有辦法去針對這些資料做任何的統計推論。因此,我們發展了一個以統計上的動差匹配為基礎的漸進式波形估計方法 (AWE-type statistical moment matching – SMM) 去復原經過統計上的靜態時序分析演算法結果的機率密度函數。As the technology feature sizes are getting smaller than the wave length of optical lithography light source, the process variation issues are also getting significant and must be taken into consideration during design. Classical corner-based timing analysis produces timing predictions that are often too pessimistic and grossly conservative because we have only few chances to get parameters of all gates working on their corner values. Statistical static timing analysis (SSTA) that characterizes time variables as statistical random variables offers a better approach for more accurate and realistic timing prediction. Many SSTA algorithms have been proposed and reach maturity. However, most of them were built upon Gaussian distributions due to its simplicity while dealing with maximum operation which is essential during timing analysis. The modeling capability of a signal Gaussian is quite limited and may not be able to deal with various non-Gaussian process distributions. We use Gaussian polynomial to model the non-Gaussian process variation. However, some high skewness distribution can not be modeled by Gaussian polynomial directly, we may get complex coefficients. Complex coefficients mean the modeling data to be complex number and can not use traditional statistical inference for these datas. Therefore, we develop a AWE-type statistical moment matching (SMM) method to recover the PDF.1 Introduction 1 2 Accurate Statistical Process Data Modeling using Gaussian Polynomial 4 2.1 Parameter Variation and Measurement . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Quadratic Timing Model . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Parameter Variation Components . . . . . . . . . . . . . . . . . . . 6 2.1.3 Measurement Data Matrix . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Quadratic Parameter Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Moments in the Statistics Domain . . . . . . . . . . . . . . . . . . . 10 2.2.2 Moment Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Quadratic Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Covariance Translation . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Least Square Fitting Based Quadratic Gaussian Polynomials . . . . . . . . 16 3 More Flexible Methods: Extending to Complex Number 19 3.1 Complex Gaussian Polynomial(CGP) Model . . . . . . . . . . . . . . . . . 19 3.1.1 AWE-type Statistical Moment Matching . . . . . . . . . . . . . . . 20 3.1.2 Complex Quadratic Gaussian Polynomials (CQGP) . . . . . . . . . 22 3.1.3 Complex Cubic Gaussian Polynomial (CCGP) . . . . . . . . . . . . 25 3.1.4 Complex High Order Gaussian Polynomial (CHGP) . . . . . . . . . 28 3.1.5 Recovering PDF of our models by Modified SMM . . . . . . . . . . 30 3.2 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Conclusion 36 Bibliography 3