計算統計方法在積體電路最佳化及敏感度分析之研究

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國立交通大學

統計學研究所

碩士論文

計算統計方法在積體電路設計最佳化及敏感度分析之研究

Computational Statistics Approach to Integrated Circuit

Design Optimization and Sensitivity Analysis

研究生：羅婉文

指導教授：洪慧念博士

李義明博士

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計算統計方法在積體電路設計最佳化及敏感度分析之研究

Computational Statistics Approach to Integrated Circuit Design Optimization

and Sensitivity Analysi

s

研究生：羅婉文 Student：Wan-Wen Lo

指導教授：洪慧念博士 Advisor：Dr. Hui-Nien Hung

李義明博士 Advisor：Dr. Yiming Li

國立交通大學

統計學研究所

碩士論文

A Thesis

Submitted to the Department of Statistics National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master in Statistics July 2006 Hsinchu, Taiwan

中華民國九十五年七月

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Copyright by Wan-Wen Lo 2006

All Rights Reserved

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計算統計方法在積體電路設計最佳化及敏感度分析之研究

學生：羅婉文

指導教授：洪慧念博士

李義明博士

國立交通大學統計研究所碩士班

摘

要

現今電子產品中，為了滿足民生或工業消費上的需求，設計上須達到特定之商業規格，其中積體電路(ICs)在電子產業中扮演著重要角色與地位，要如何將積體電路設計達到想要的規格，通常設計者必須調整其中的主、被動元件以及積體電路佈局等參數，使得電氣規格可以達到我們想要的設計目標。要如何掌握電路行為的趨勢來符合嚴格的需求是現今市場競爭上困難的一件事，傳統上為了滿足工程需求，工程師往往反覆不斷的手動調整係數與執行電路模擬器，才能找出一組可行的參數組合來達到想要的設計結果；或者用數值最佳化的方法、演化式生物計算工程的方法、蒙地卡羅的方法設計參數，這些方法各有其優缺點。本論文嘗試提出ㄧ個整合電路模擬器與實驗設計的計算統計方法應用在積體電路的設計最佳化與規格敏感度分析，我們利用此方法研究類比與數位電路之設計問題展現出不錯的結果。藉由此系統化的方法，首先應用在由 0.25 微米金屬氧化物半導體場效應電晶體所組成之低雜訊放大器射頻積體電路設計。例如，若我們所討論的電路特性希望規格為：一、輸入反射損失小於-10dB；二、輸出反射損失小於-10dB；三、輸入端與輸出端隔離度小於-25dB；四、輸出增益望大；五、穩定因子大於 1；六、雜訊指數小於 2；七、第三階截斷點(IIP3)大於-10。藉由呼叫電路模擬器取得電路特性，吾人首先透過篩選實驗，十個顯著的電路參數由 13 個參數中被挑選出來做進一步的中央合成設計，進而導出各電路特性相對應的二次反應曲面模型，同時使用望想函數(desirability function)，吾人可取得最佳解；若電路特性未達到希望的規格，可適當的調整參數範圍，最後使得

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vi 所研究的電路特性達到所預期的規格範圍內。同時由取得最佳解之敏感度分析，得知我們所估算的參數組合對電路特性是穩定的。另外，吾人也進一步將此方法應用在數位電路的性能敏感度分析上，例如由 65 奈米金屬氧化物半導體場效應電晶體所組成之靜態隨機讀取記憶體的靜態雜訊邊際(static noise margin)敏感度分析，將靜態隨機讀取記憶體分成六個電晶體組態與四個電晶體組態討論，我們希望靜態雜訊邊際的變異越小越好。將這兩種結構以 10%元件長度與偏壓當作三個標準差來探討其變異，分析出六個電晶體組態與四個電晶體組態的靜態雜訊邊際落在我們測試的條件準則下達 98%和 95.8%，相較之下六個電晶體組態的靜態雜訊邊際來的穩定。總之，藉由以上的例子，吾人歸結得知，此有系統的計算統計方法，初步研究結果顯示，它可以成功的應用在類比與數位積體電路的設計上，且都有不錯的設計穩定性。吾人深信此統計方法可進ㄧ步推廣，適當地用在積體電路設計最佳化，並量化分析電路的操作特性與可靠度之變化趨勢，進而有效解決不同電路的設計問題。

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Computational Statistics Approach to Integrated Circuit Design

Optimization and Sensitivity Analysis

Student：Wan-Wen Lo

Advisor：Dr. Hui-Nien Hung

Dr. Yiming Li

Institute of Statistics

National Chiao Tung University

Abstrac

t

It is known that integrated circuits (ICs) design nowadays plays a crucial role for microelectronics industry; in particular, for highly competitive consumer products. To meet specified electrical characteristics and performance of designed product, designer in general has to tune parameters of the passive and active devices ranging from resistors, capacitors, inductors, line width, line length, to transistor size, etc. Diverse approaches have been proposed to reduce products’ designing cycles and accelerate time to market. These methods include (1) directly empirical procedure, (2) numerical optimization technique, (3) evolutionary algorithm, and (4) Monte Carlo statistical method, and have demonstrated their merit and validity. We believe that a systematical integration of circuit simulation tool, design of experiment, and response surface model may provide an alternative way to advanced IC design optimization and sensitivity analysis of performance.

In this thesis, by verifying two different analog and digital circuits, a low noise amplifier and static random access memory, we develop a computational statistics approach, which is mainly based upon SPICE circuit simulator, a screen design, a central composite design (CCD), and a 2nd order response surface model (RSM). We firstly state the computational algorithm by taking a low noise amplifier circuit with 0.25 μm MOSFETs as an example. The circuit specification consists of (1) the input return loss < -10 dB, (2) the output return loss < -10 dB, (3) reverse isolation < -25 dB, (4) voltage gain which is as great as possible, (5) stability factor > 1, (6) noise figure < 2 dB, and (7) the third-order-intercept point > -10 dB. To achieve the aforementioned

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seven circuit specifications, calling circuit simulator to obtain circuit performances is performed and then ten significant results among thirteen parameters are selected from the screening design. By simultaneously running SPICE circuit simulator, a ten-parameter face centered cube design is then performed in the step of central composite design. We use the 149 simulation results in constructing the corresponding 2nd order response surface model (it is a 10-variable 2nd order polynomial) by using statistical software, Design Expert®. We note that, for validating the constructed model, the model adequacy checking and the accuracy verification are necessary. If the model adequacy checking fails, we transform the circuit performance by BOX-COX transformation. Furthermore, adjustment of parameters’ range corresponding to the circuit specification will be enabled for accuracy verification. With the 2nd order RSM, design optimization and sensitivity analysis of performance will be explored. For the design optimization, if one of the circuit performances does not meet its specification, we adjust the parameter range corresponding to the circuit specification, and return to the step of CCD. If the optimized results are eventually satisfied the aforementioned seven specifications, the first three optimal recipes will be provided. Performance sensitivity with respect to certain optimized parameter (or all parameters) is investigated by using RSM to an optimized recipe with 100 randomly generated normal samples. The optimized recipe is right the mean of the normal distribution; and one per centum of the optimized recipe is assumed to be the standard deviation. Our result shows that the optimized recipe is stable to the circuit performance. Similar methodology is further applied to explore the variation of static noise margin (SNM) of six- and four-transistors (6T and 4T) static random access memory (SRAM) cells with respect to channel length and supply voltage. For SRAM with 65 nm CMOS devices, our result shows that 98 % (theoretically it should be 100 %) variation of SNM is within 3-sigma for the 6T SRAM with 3-sigma variation of parameters. It is better than that of the result of 4T SRAM (95.8 %). Thus, it quantitatively confirms that SRAM with 6T configuration is more stable than it with 4T configuration.

In conclusion, we systematically implement a computational statistics approach to ICs’ design optimization and sensitivity analysis. Successful application of the method to study analog and digital circuits shows its computational efficiency and engineering accuracy, compared with large-scale SPICE circuit simulations. This approach is suitable for optimization problems and diagnosis of quantify trade-offs in IC industry.

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ix

誌

謝

這份論文能夠順利完成，首先感謝洪慧念教授給予學生最大自由度，讓學生可以完成感興趣的研究，感謝洪老師在課業以及生活上的鼓勵及支持。其次，學生感謝指導老師李義明副教授，感謝恩師兩年來指導學生論文方向脈絡，研究方法之傳授及論文撰寫之推敲斟酌，讓學生研究能力之激發有深厚的影響。恩師學術研究態度嚴謹，在半導體數學模式及電腦模擬計算之專業知識，足以為學生日後之表率。學生在此謹獻上最誠摯的感謝與敬意。另外感謝許文郁老師、彭松村老師來當學生的口試委員，讓學生的論文可以更加完備，更符合統計和工程雙方面的觀點。在平行與科學計算實驗室方面，我要感謝周宏穆學長、建松、孟家學弟撥空教我積體電路上的觀念，真的讓我獲益匪淺；感謝卓彥羽學長百忙之中在程式上的協助，真的解決了我很多困難；紹銘學長、陳璞學長、傳盛學長、正凱學長總是無條件的當我論文問題的求救對象。景嵐學長、煒昕、柏賢，宏榮加上東祐學弟在此一並感謝。統研所方面，感謝沛君、謝宛茹、孟樺、秀慧、鷰筑、張宛茹、耀文、泓毅；因為有你們的關心，帶給我歡樂，替我加油打氣，謝謝你們平常聽我發牢騷，有你們這一群朋友真的很榮幸。統研所是一個很溫馨的地方，讓我總是笑嘻嘻的與你們玩在一起。我還要感謝我的好朋友們，運璿、怡婷、勇志、千姿、其沛，瑾魚，穎劭學姐總是在我徬徨失措的時候給我許多建議，有時半夜接到我騷擾的電話還陪我講講話，真的很感謝你們，有你們的相伴覺得很幸福。最後最應該感謝的是我的爸爸，媽媽，和我的哥哥姐姐以及景偉，週末回家總是讓我感覺家的溫馨，不用煩惱任何家裡的事情，讓我得到最充裕的休息，再重新出發，也因為你們的關心與支持讓我勇敢的在外面闖蕩，現在終於完成論文畢業了，真的很感謝你們。感謝這段期間大家對我的包容、關懷與愛護。這篇論文，這個工作，以及我在交大的一切成長，沒有你們，是完全沒有辦法達成的，一切的功勞都歸因於全部的人。謝謝大家一直挺我鼓勵我，使我順順利利的度過這段非凡且精采的日子。本論文感謝行政院國家科學委員會 ( 計畫編號 NSC-93-2115-E-492-008 、 NSC-94-2115-E-009-084) 、卓越延續計畫 ( 計畫編號 NSC-94-2752-E-009-003-PAE 、 NSC-95-2752-E-009-003-PAE) 、五年五百億計畫、經濟部科專計劃 ( 計畫編號 93-EC-17-A-07-S1-0011)及台灣積體電路製造股份有限公司 2005~2006 年研究計畫之資助。在此將這篇論文獻給所有關心我以及我所愛的人，謝謝你們。羅婉文謹誌中華民國九十五年七月三十一日于風城交大

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List of Tables

2.1 Difference between Taguchi approach and classical DOE. . . 33

4.1 The levels of screening design for the 13 factors. . . 48

4.2 A list of the results for the screening design, where ”1” means the most important term with respect to the corresponding response. . . 49

4.3 The minimum and maximum of seven responses in the six settings of Load. The Load factor is 4.5 which makes the voltage gain close to our target. . . 50

4.4 A list of the results of the predicted values after optimization in the first three experiments using CCF design. . . 59

4.5 Experiment levels for 10 factors after optimization of the 3th experiment . . 60

4.6 The information of 7 response surface models for the LNA circuit using CCF design. . . 62

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LIST OF TABLES xvii

4.7 The information of the 5 response surface models and 2 transformed re-sponse surface models for the LNA circuit using CCF design with stepwise regression method. . . 62 4.8 The coefficients of 1/S11 with coded factors in a significance order of the

stepwise regression. . . 63 4.9 The coefficients of S12 with coded factors in a significance order of the

stepwise regression. . . 64 4.12 The coefficients of K with coded factors in a significance order of the

step-wise regression. . . 65 4.13 The coefficients of NF with coded factors in a significance order of the

stepwise regression. . . 65 4.14 The coefficients of IIP3 with coded factors in a significance order of the

stepwise regression. . . 66 4.15 Accuracy verification of the results calculated from the constructed

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xviii LIST OF TABLES

4.16 Accuracy verification of the results obtained from circuit simulator. . . 79 5.1 The constraints of LNA circuit parameters. . . 85 5.2 The targets of responses. . . 85 5.3 The constraint for the case of satisfied all specifications. We modify the

specifications within 2ˆσ. . . 87 5.4 The constraint for the case of minimized noise figure. We modify the

spec-ifications within 2ˆσ. . . 88 5.5 The constraint for the case of maximized voltage gain. We modify the

specifications within 2ˆσ. . . 88 5.6 Optimal recipes for the case of satisfied all specifications calculated by the

2nd_{order response surface model. ”1” means the highest priority. . . 89}

5.7 Optimal recipes for the case of minimized noise figure calculated by the 2nd_{order response surface model. ”1” means the highest priority. . . 90}

5.8 Optimal recipes for the case of maximized voltage gain calculated by the 2nd_{order response surface model. ”1” means the highest priority. . . 91}

5.9 Optimal results for the case of satisfied all specifications calculated by the 2nd_{order response surface model. ”1” means the highest priority. . . 92}

5.10 Optimal results for the case of minimized noise figure calculated by the2nd order response surface model. ”1” means the highest priority. . . 92

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LIST OF TABLES xix

5.11 Optimal results for the case of maximized voltage gain calculated by the 2nd _{order response surface model. ”1” means the highest priority. . . 93}

5.12 Optimal results for the case of satisfied all specifications by running circuit simulator. ”1” means the highest priority. . . 93

5.13 Optimal results for the case of minimized noise figure by running circuit simulator. ”1” means the highest priority. . . 94

5.14 Optimal results for the case of maximized voltage gain by running circuit simulator. ”1” means the highest priority. . . 94

6.1 Sensitivity analysis for LNA circuit calculated from the response surface model which is obtained from circuit simulator by varying VB1. Calculated mean and standard deviation for seven circuit performances are shown. . . . 108

6.2 Sensitivity analysis for LNA circuit calculated from response surface mod-els and obtained from circuit simulator by varied 10 factors, displaying calculated mean and standard deviation for seven circuit performances. . . . 109

7.1 The levels of each factor for 6T and 4T SRAM cells . . . 118

7.2 The calculated results of SNM response surface model for the 6T and 4T SRAM cells using CCF design. . . 119

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xx LIST OF TABLES

7.3 Accuracy verification of the response values calculated from the response surface model and obtained from circuit simulator for 6T and 4T SRAM cells. . . 121 7.4 Comparison of the sensitivity of the SNM for 6T SRAM cell between the

sensitivity of the SNM for 4T SRAM cell. The mean of L₁, L₂, andL₃

is set to be its nominal values 65 nm, respectively; and VDD is set to be its nominal value 1.2 V. The standard deviation is 3.3 % for each nominal value. We generate 500 normally and independently distributed pseudo-random numbers for these four parameters. . . 125

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List of Figures

2.1 The proposed main computational procedure for IC design optimization in this work. . . 9 2.2 A central composite design of two factors. . . 14 2.3 Comparison of the three types of central composite designs. . . 14 2.4 A flowchart of the stepwise regression algorithm used in our work. . . 23 2.5 An example of residual normal probability plot. . . 29 2.6 An example of scatter plot of predicted values versus residuals. . . 29 2.7 Individual desirability functions for the simultaneous optimization. . . 30 3.1 The explored LNA circuit in our experiment. . . 37 3.2 An illustration of spurious-free dynamic range with the noise floor and IIP3. 41 3.3 A flow of circuit simulation. . . 43 3.4 A RF model applied in this work. . . 44 4.1 A half-normal plot for the effect of S11. . . 51

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xxii LIST OF FIGURES

4.2 A half-normal plot for the effect of S12. . . 52 4.3 A half-normal plot for the effect of S21. . . 53 4.4 A half-normal plot for the effect of S22. . . 54 4.5 A half-normal plot for the effect of K. . . 55 4.6 A half-normal plot for the effect of NF. . . 56 4.7 A half-normal plot for the effect of IIP3. . . 57 4.8 Residual normal probability plots for (a) S11 and (b) 1/S11. . . 68 4.9 Residual scatter plots for (a) S11 and (b) 1/S11. . . 69 4.10 A model adequacy checking for S12. . . 70 4.11 A model adequacy checking for S21. . . 71 4.12 A model adequacy checking for S22. . . 72 4.13 A model adequacy checking for K. . . 73 4.14 A model adequacy checking for NF. . . 74 4.15 Residual normal probability plots for (a) IIP3 and (b) log(IIP 3 + 11.1265). 75 4.16 Residual scatter plots for (a) IIP3 and (b) log(IIP 3 + 11.1265). . . 76 4.17 A scatter plot calculated from the response surface model versus values

obtained from circuit simulator for S11. . . 79 4.18 A scatter plot calculated from the response surface model versus values

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LIST OF FIGURES xxiii

4.19 A scatter plot calculated from the response surface model versus values obtained from circuit simulator for S21. . . 80

4.20 A scatter plot calculated from the response surface model versus values obtained from circuit simulator for S22. . . 81

4.21 A scatter plot calculated from the response surface model versus values obtained from circuit simulator for K. . . 81

4.22 A scatter plot calculated from the response surface model versus values obtained from circuit simulator for NF. . . 82

4.23 The scatter plots of values calculated from response surface models versus values obtained from circuit simulator for IIP3. . . 83

5.1 Comparison of original case and three optimized cases for the result of S11 response. A zoom-in plot for the operation frequency. . . 97

5.3 [Comparison of original case and three optimized cases for the result of S21 response. A zoom-in plot for the operation frequency. . . 99

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xxiv LIST OF FIGURES

5.5 Comparison of original case and three optimized cases for the result of K response. A zoom-in plot for the operation frequency. . . 101

5.6 Comparison of original case and three optimized cases for the result of NF response. A zoom-in plot for the operation frequency. . . 102

5.7 Comparison of original case and three optimized cases for the result of IIP3 response. A zoom-in plot for the operation frequency. . . 104

6.1 Statistical distribution of the model for S11, which is calculated by the sensitivity analysis and using the full2nd _{order response surface model by}

varying VB1. . . 109

6.2 Statistical distribution of the model for S12, which is calculated by the sensitivity analysis and using the full2nd _{order response surface model by}

6.3 Statistical distribution of the model for S21, which is calculated by the sensitivity analysis and using the full2nd order response surface model by varying VB1. . . 110

6.4 Statistical distribution of the model for S22, which is calculated by the sensitivity analysis and using the full2nd order response surface model by varying VB1. . . 111

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LIST OF FIGURES xxv

6.5 Statistical distribution of the model for K, which is calculated by the sen-sitivity analysis and using the full 2nd order response surface model by varying VB1. . . 111

6.6 Statistical distribution of the model for NF, which is calculated by the sen-sitivity analysis and using the full 2nd _{order response surface model by}

6.7 Statistical distribution of the model for IIP3, which is calculated by the sensitivity analysis and using the full2nd_{order response surface model by}

7.1 A circuit of 6T SRAM cell used in our circuit simulation. . . 116

7.2 A circuit of 4T SRAM cell used in our circuit simulation. . . 117

7.3 A 3D plot of SNM for 6T and 4T SRAM cells with respect to L₁and L₂. . 120

7.4 A model adequacy checking for 6T and 4T SRAM cells. . . 122

7.5 A scatter plot calculated from the response surface model versus values obtained from the circuit simulator. . . 123

7.6 A comparison of the sensitivity of SNM for 6T SRAM cell and the sensi-tivity of SNM for 4T SRAM cell. . . 126

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xxvi LIST OF FIGURES

E.1 Statistical distribution of the model for S11, which is calculated by the sensitivity analysis and using the full2nd order response surface model by varying 10 factors. . . 189

E.3 Statistical distribution of the model for S21, which is calculated by the sensitivity analysis and using the full2nd _{order response surface model by}

varying 10 factors. . . 190

E.5 Statistical distribution of the model for K, which is calculated by the sen-sitivity analysis and using the full 2nd _{order response surface model by}

varying 10 factors. . . 191

E.6 Statistical distribution of the model for NF, which is calculated by the sen-sitivity analysis and using the full 2nd order response surface model by varying 10 factors. . . 191

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LIST OF FIGURES xxvii

E.7 Statistical distribution of the model for IIP3, which is calculated by the sensitivity analysis and using the full2ndorder response surface model by varying 10 factors. . . 192

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Chapter 1 Introduction

Integrated circuits (ICs) market has become so intense that designers have adopted various optimization strategies in an effort to reduce the development time and to improve circuit performance. It is increasingly important to design robust circuits that would minimize fluctuations of the circuit performance. Designers usually do many try-and-error experi-ments to achieve specifications. One common practice is to guess the improved settings of the control factors using engineering judgment, and then conducts a paired comparison with the starting conditions. The guess-and-test cycle is repeated until an improvement has been obtained, the deadline has been reached, or the budget has been exhausted. This practice relies heavily on luck. It is inefficient and time-consuming [1].

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2 Chapter 1 : Introduction

Due to the excessive time and high costs associated with physical experiments, design-ers have applied Simulation Program with Integrated Circuit Emphasis (SPICE) to simu-late the circuit performance and predict the circuit characteristics. Different computational methods together with the circuit simulation tools to achieve optimization and sensitivity analysis have been of great interests.

In this thesis, a statistical approach is systematically developed for the circuit optimiza-tion and the sensitivity analysis in the low noise amplifier (LNA) and static random access memory circuit carried out as examples. Based on the screening design, the central com-posite design, a SPICE simulator, the response surface model, and the optimization using desirability function, the circuit performances have been optimized with respect to differ-ent specified constraints. For example, for the studied LNA circuit, they are (1) shifting the input return loss (S11) to the specific target; (2) shifting the output return loss (S22) to the specific target; (3) shifting the reverse isolation (S12) to the specific target; (4) maximizing the voltage gain (S21); (5) moving the stability factor (K) to the specific target; (6) moving the noise figure (NF) to the specific target; and (7) moving the third order intercept point (IIP3) to the specific target. Furthermore, the statistical approach also applies systemat-ically to 6T and 4T static random access memory (SRAM) cells and we investigate the sensitivity of the static noise margin (SNM) for 4T and 6T SRAM cells.

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1.1 : Motivation 3

1.1 Motivation

When circuit designers encounter the optimization problem, they often solve it according to their experiences. However, to extract a proper parameter setting of the VLSI circuit is a difficult problem and the empirical knowledge is needed [2][3][4][5][6]. If the circuit designers set the parameters corresponding to the experiences based on empirical formulas, an optimization procedure is needed to loop for times to get acceptable results.

So far, many researches have pointed out the methodologies for digital circuit optimiza-tion. Those methodologies are based on conventional optimization techniques which are in turn based on developed various local solution properties and they are ineffective or lack of accuracy [7][8][9][10]. These optimization problems often appear with high-dimensional and nonlinear state, and we provide a systematic method to explore this problem from computational statistical point of view.

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1.2 Literature Review

Generally speaking, there are four types of optimization approaches: the brute force method, numerical optimization method, evolutionary algorithm method, and Monte Carlo statisti-cal method. We discuss each method in brief.

(1) Brute force method

Brute force method is a traditional method for solving problem. This is a method that we try each possible solution one by one when we can not find the solution directly.

(2) Numerical optimization method

The Gauss-Newton method, for example, is a basic algorithm for solving nonlinear opti-mization problem, and Levenberg-Marquardt (LM) method is a quasi-Newton method to accelerate the Gauss-Newton method [16][17][18]. They start with an initial guess, and follow the direction of the normal of the gradient to find the optimal solution.

(3) Evolutionary algorithm method

Genetic algorithm (GA), for example, is a global search algorithm based on Darwinian survival of the fittest approach [19]. It has been proved having a capability of domain inde-pendent [20] and is an effective search method for large space problem [21]. The method could be adopted in many fields, such as combinatorial and numerical optimizations [22], supervised and unsupervised learning [23], and molecular computing [24]. In microelec-tronics, many works had been done on various VLSI circuit designs, such as cell placement

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1.2 : Literature Review 5

[25], channel routing [26], and model parameters extraction [27].

In addition, neural network (NN) is an artificial intelligent algorithm that mimics the behavior of human brain firstly established by McCulloch and Pitts in 1943. This model was first considered to be binary devices with fixed thresholds which is able to perform simple logic, such as unit and intersection. Currently NN has been wildly used in digital signal processing, such as eigen-state problem [29], image process, audio pattern recogni-tion [30], and feature classificarecogni-tion. Due to the strong capability, there are some research using NN for solving numerical problem like ordinary/partial differential equations [31] and other numerical methods [32]. Moreover, it is also applied in parameters extraction [33].

(4) Monte Carlo statistical method

Statistical simulation methods may be contrasted to conventional numerical discretization methods, which typically are applied to ordinary or partial differential equations that de-scribe some underlying physical or mathematical system. In many applications of Monte Carlo, the physical process is simulated directly, and there is no need to even write down the differential equations that describe the behavior of the system. The only requirement is that the physical (or mathematical) system be described by probability density functions [34][35].

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others, but is a basic and direct method. Traditional numerical method like LM method that is necessary for a good initial value and easily trapped into local optima. However, compared with the global optimization technique such as genetic algorithm method, the LM method finds a solution rapidly.

1.3 Objectives

In this work, we will provide a computational statistical methodology to study the spec-ification problem of circuits. We take a popular used circuit, such as low noise ampifier (LNA) circuit as be an example firstly. Here we want to optimize seven circuit perfor-mances to each specific value: (1) input return loss (S11); (2) output return loss (S22); (3) reverse isolation (S12); (4) voltage gain (S21); (5) stability factor (K); (6) noise figure (NF); and (7) the third order intercept point (IIP3). The object of this work is trying to con-struct the response surface model and to obtain the optimal recipes. In our process of the methodology, we verify the response surface models which are the relation of circuit pa-rameters and circuit performance, then the model will reflect realistic circuit performance. Furthermore, by using desirability function with seven performance constraints supplies us optimal solutions. Finally, we perform the sensitivity analysis with the constructed model, they help us understand whether the distribution of the seven circuit performances are in their specific values that we assigned. Our second application is 6T and 4T SRAM cells.

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1.4 : Outline of the Thesis 7

We construct the response surface models and investigate the sensitivity of the SNM for 6T and 4T SRAM cells.

1.4 Outline of the Thesis

This thesis is organized as follows. In Chap. 2, the statistic methodology and the procedure of methodology in this work will be introduced in detail. The application of the statisti-cal method to a low noise amplifier will be discussed in Chap. 3. The results of design of experiment which contain screening design, central composite design, construction of response surface model, model checking, and accuracy verification are shown in Chap. 4. The three optimized cases which are satisfied all specifications, minimized noise figure, and maximized voltage gain are provided in Chap. 5. Finally the outcomes of the LNA circuit sensitivity analysis have been shown in Chap. 6. The other application of the sta-tistical method to static random access memory will be discussed in Chap. 7. Finally we draw conclusions and suggest future work.

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Chapter 2 Statistical Methodology

In this chapter, we introduce the content of main methodology developed in this work in the following sections. A methodology flow is shown in Fig. 2.1, and then two designs will be discussed. First, screening design, in this step we merely use fewer experiments for screening the most important factors of the circuit parameters. After determining the important factors, we will execute the other design, central composite design. Next we construct the response surface models which are used to find the parameters to optimize circuit performance. In addition, we will describe some related applications of this work.

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9 Screening design Central composite design Model construction Model checking Circuit simulation

Output optimal recipes

Transformation or adjustment of the parameter range No Optimization and/or sensitivity analysis Accuracy verification Satisfaction of both? No Yes

Achieve the target?

Yes

Figure 2.1: The proposed main computational procedure for IC design optimization in this work. First we use fewer experiments to select the important factors by screening design. Then we execute central composite design and construct model. The model adequacy checking is necessary to check the model assumption, and the accuracy verification is to check the values that we are interested in the accuracy of the model within our high and low level settings. Finally, we use the model for optimization or sensitivity analysis. If we don’t achieve the target we will adjust the parameter range and repeat the flow chart which restarts at the step of central composite design

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10 Chapter 2 : Statistical Methodology

2.1 Screening Design

Screening design usually leads to an experiment which is designed to investigate these fac-tors with a view toward eliminating some unimportant ones. In other words, we determine significant factors by screening design. To determine factor’s significance, two-level frac-tional factorial design or Plackett-Burman design is ideally suited for screening design [36]. In short, screening designs are economically experimental plans that focus on determining the relative significance of many main effects with resolution III or IV (but the designs of this case require more runs than a resolution III design) [36].

Two-level fractional factorial design can reasonably assume that high-order interactions are negligible. We can run only a fraction of the complete factorial experiment to obtain in-formation on the main effects and low-order interactions. For example, in one-half fraction of the23design (23−1design ), A and BC are aliases, B and AC are aliases, C and AB are aliases, where A, B, and C are factors. When designs with resolution III, main effects are aliases with two-factor interactions and two-factor interactions may be aliased with each other. Sometimes designs with resolution IV are also used for screening designs. In this design main effects are aliased with, at worst, three-factor interactions. This is better from the confounding viewpoint, but the designs require more runs than a resolution III design.

Plackett-Burman design, attributed to Plackett and Burman (1946) [37], is two-level fractional designs for studying up to k= N − 1 variables in N runs, where N is a multiple

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2.2 : Central Composite Design 11

of 4. In a Plackett-Burman design, main effects are heavily confounded with two-factor in-teractions in general. For example, N = 12, every main effect is partially aliased with every two-factor interaction. Each main effect is partially aliased with 45 two-factor interactions. And the plus and minus signs are:

K = 11, N = 12 + + − + + + − − − + − . (2.1)

When we analyze data from screening designs, the use of an error mean square obtained by pooling high order interactions is inappropriate occasionally. To overcome this problem a half-normal probability plot of the estimates of the effects is suggested. The half-normal plot consists of the point:

(Φ−1_{(0.5 + 0.5[i − 0.5]/I), |ˆθ|}

(i)|), (2.2)

for i = 1, . . ., I. TheΦ is the cumulated density function of the standard normal distribution. If factors are unimportant, the effects with mean zero and variance σ2will tend to fall along a straight line on this plot, whereas important factors will not lie along the straight line [38].

2.2 Central Composite Design

The central-composite design (CCD) is perhaps the most common experimental design used to generate second-order response models. These designs combine a two-level full factorial or fractional factorial design of nf runs with 2k axial runs and nc center runs

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to estimate curvature, where k represents the number of control factors [38]. Figure 2.2 illustrates a CCD for two factors. The axial points represent new extreme values for each factor in the design. There is three varieties of CCD which are CCC, CCI, and CCF.

The central composite circumscribed (CCC) designs are the original form of the central composite design. The axial points at some distance α from the center is based on the properties desired for the design and the number of factors in the design. The axial points establish new extremes for the low and high settings for all factors. Figure 2.3 illustrates a CCC design. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. Augmenting an existing factorial or resolution V fractional factorial design with axial points can produce this design [36].

For those situations in which the limits specified for factor settings are truly limits, the central composite inscribed (CCI) design uses the factor settings as the axial points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by α to generate the CCI design) [36]. This design also requires 5 levels of each factor.

The other special design is called the face centered cube (CCF) design. In this design the axial points are at the center of each face of the factorial space, so α = ±1. If the diamond points move to the face in the cube, then the design is CCF. This variety requires

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2.3 : Models Construction 13

3 levels of each factor. Augmenting an existing factorial or resolution V design with ap-propriate axial points can also produce this design.

The diagrams in Fig. 2.3 illustrate the three types of central composite designs for two factors. Note that the CCC explores the largest process space and the CCI explores the smallest process space. Both the CCC and CCI are rotatable designs, but the CCF is not. In the CCC design, the design points describe a circle circumscribed about the factorial square. For three factors, the CCC design points describe a sphere around the factorial cube. To maintain rotatability, the value of α depends on the number of experimental runs in the factorial portion of the central composite design:

α= [nc]

1

4_, _(2.3)

where ncis the number of experimental runs in the factorial portion of the central composite

design. However, the factorial portion can also be a fractional factorial design of resolution V [36, 39].

2.3 Models Construction

It is necessary to develop an approximate model for the true response surface. If n obser-vations are collected in an experiment, the model for them takes the form [38]:

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14 Chapter 2 : Statistical Methodology Center points Cube points Axial points Center points Cube points Axial points Center points Cube points Axial points

Figure 2.2: A central composite design of two factors. The design includes one center point, four cube points, and four axial points.

CCC CCF CCI

Figure 2.3: Comparison of the three types of central composite designs.

where y= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ y₁ y₂ .. . yn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , X = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 x11 x12. . . x1k 1 x21 x22· · · x2k .. . ... ... ... 1 xn1 xn2· · · xnk ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , β= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ β₀ β₁ .. . βk ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ε= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ε₁ ε₂ .. . εn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .

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In general, y is an n× 1 vector of the observations, X is an n × p matrix of the levels of the independent variables, β is a p× 1 vector of the regression coefficients, and ε is an n × 1 vector of random errors.

We want to find the least squares estimators, ˆβ, that minimizes

L=

n

i=1

ε2_i = εTε= (y − Xβ)T(y − Xβ). (2.5)

As the result of our calculation, the least squares estimator of β is

ˆβ = (XT

X)−1XTy. (2.6)

The fitted regression model is

ˆy = X ˆβ. (2.7)

The difference between the responses yiand the fitted value ˆyiis a residual, say ei = y − ˆy,

The vector of residual is denoted by:

e= y − ˆy. (2.8)

To check the normality assumption is by preparing a normal probability plot of the residual values. If the assumption holds, this plot will resemble a straight line. If the assumption is violated, a non-linear data transformation (e.g., y = log(y)) may be applied and new mod-els are generated in an attempt to improve model adequacy [38]. A second plot showing the residual values versus the predicted response values is used to verify if the variance of

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the original observation is constant. A random scattering of the residual values indicates that no correlation exists between the observed variance and the mean level of the response [39].

To develop an estimator of this parameter consider the sum of squares of the residuals, say SSE = n i=1 (yi− ˆyi)2 = n i=1 e2_i = eTe. (2.9)

Equation (2.9) is called the error or residual of squares, and it has n− p degrees of freedom associated with it. It can be shown that

E(SSE) = σ2(n − p), (2.10)

so an unbiased estimator of σ2 is given by

ˆσ2 ₌ SSE

n− p. (2.11)

To determine if there is a linear relationship between the response variable y and a subset of the regressor variables x₁, x₂,· · · , xk is the test for significance of regression.

The appropriate hypotheses are [38]:

H₀ : β₁ = β₂ = · · · = βk= 0,

H₁ : βj = 0 for at least one j. (2.12)

If we reject H₀, it implies that at least one of the regressor variables x₁, x₂,· · · , xk

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squares due to residual, say

SST = SSR+ SSE. (2.13)

A relatively simple procedure is performed to check for model significance in relation to random error. This test involves calculating the test statistic:

F₀ = M SR M SE = SSR/k SSE/(n − k − 1) = 1 k n j=1(ˆyi− y)2 1 n−k−1 n j=1(yi − ˆyi)2 , (2.14)

where y is the average of measured response values. yi, ˆyi, and n are the ith measured

response, the ith predicted response, and the number of simulated runs, respectively [38]. If this statistic exceeds the corresponding value of the F distribution value (Fα,k,n−k−1), the

response model is considered significant in relation to random error.

A second statistic, the coefficient of multiple determination R2 is defined as:

R2 = SSR SST = 1 − SSE SST = 1 − n i=1(yi− ˆyi)2 n i=1(yi− y)2 . (2.15)

R2 measures the amount of reduction in variability of the response y achieved, using the input factors x₁, x₂, . . . , xk. From Eq. (2.13) we see that R2 varies from zero to one

[38][39]. However, a large value of R2 does not necessarily imply that the regression model is good one. Adding a variable to the model will always increase R2, regardless of whether the additional variable is statistically significant or not. About this problem, some regression model builders prefer to use an adjusted R2statistic defined as

R2_adj = 1 − SSE/(n − k − 1)

SST/(n − 1) = 1 −

n− 1

n− k − 1(1 − R

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In general, the adjusted R2 statistic will not always increase as variables are added to the model. In fact, if unnecessary terms are added, the value of R2_adj will often decrease.

2.4 Variable Selection

In response surface work it is customary to construct the full model corresponding to the situation at hand. That is, in steepest ascent we usually build the full first-order model, and in the analysis of a second-order model we usually construct the full quadratic. An experimenter may encounter situations where the full model may not be appropriate; that is, a model based on a subset of the regressors in the full model may be superior. Variable selection or model-building techniques may be used to identify the best subset of regressors to include in a regression model [39].

Variable selection is determined by statistical analysis of the generated response surface models. Input factors showing a significant effect on an individual response can be system-atically determined using statistical techniques. Variations of these significant input factors will produce the greatest fluctuations in device performance. This analysis is extremely useful in understanding what areas of manufacturing require greater control.

(1) Half-normal plot and t test

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2.4 : Variable Selection 19

on the half-normal plot of model coefficients. This method is originally proposed for ana-lyzing two-level factorial experiments applicable in cases where no degrees of freedom are available for estimating the variance of an error term. The effects are plotted on half-normal probability paper, those standing apart being identified as potentially real effects [40]. Prob-ability plotting may also be used for experiments having three level. One approach is to express the effects with linear and quadratic components, and construct the normal proba-bility plot of those components standardized to have the same variance [41][42].

The half-normal plots are informal graphical methods involving visual judgment. A formal test of effect significance is called t test for the least squares estimate ˆβ. It can be

shown that the least squares estimate ˆβ has a multivariate normal distribution with mean

vector β and variance-covariance matrix σ2(XTX)−1, i.e.,

ˆβ ∼ MN(β, σ2_(XT_X₎−1_), _(2.17)

where M N stands for multivariate normal. The (i, j)th entry of the variance-covariance matrix is Cov( ˆβi, ˆβj) and the jth diagonal element is Cov( ˆβj, ˆβj) = V ar( ˆβj). Therefore,

the distribution for the individual ˆβjis N(βj, σjj2 (XTX)−1), which suggests that for testing

the null hypothesis

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the following t statistic be used:

ˆβj

ˆσ2

jj(XTX)−1

∼ tN−p−1 (under H0). (2.19)

Under H₀, it has a t distribution with N − p − 1 degrees of freedom.

(2) Stepwise regression

Alternative of variable selection is called stepwise regression. It is one of various meth-ods for evaluating only a small number of subset regression models by either adding or deleting regressors one at a time. Stepwise regression is a popular combination of proce-dures forward selection and backward elimination [38].

The procedure of the forward selection begins with the assumption that there are no regressors in the model other than the intercept. An effort is made to find an optimal subset by inserting regressors into the model one at a time. The first regressor selected for entry into the equation is the one that has the largest simple correlation with the response variable

y. Suppose that this regressor is x₁. This is also the regressor that will produce the largest value of the F-statistic for testing significance of regression. This regressor is entered if the F-statistic exceeds a preselected F-value, say FIN (or F-to-enter). The second regressor

chosen for entry is the one that now has the largest correlation with y after adjusting for the effect of the first regressor entered (x₁) on y. We refer ro these correlations as partial correlations. They are the simple correlations between the residuals from the regression

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x₁, say ˆxj = ˆα0j + ˆα1jx1, j = 2, 3, . . . , K.

Suppose that at Step 2 the regressor with the highest partial correlation with y is x₂. This implies that the largest partial F-statistic is

F = SSR(x2|x1) M SE(x1, x2)

. (2.20)

If this F-value exceeds FIN, then x₂ is added to the model. In general, at each step the

regressor having the highest partial correlation with y (or equivalently the largest partial F-statistic given the other regressors already in the model) is added to the model if its partial F-statistic exceeds the preselected entry level FIN [38]. The procedure terminates

either when the partial F-statistic at a particular step does not exceed FIN or when the last

candidate regressor is added to the model.

Forward selection begins with no regressors in the model and attempts to insert vari-ables until a suitable model is obtained. Backward elimination attempts to find a good model by working in the opposite direction. That is, we begin with a model that includes all K candidate regressors. Then the partial F-statistic (or a t-statistic, which is equivalent) is computed for each regressor as if it is the last variable to enter the model. The smallest of these partial F-statistics is compared with a preselected value, FOU T (or F-to-move); and

if the smallest partial F-value is less than FOU T, that regressor is removed from the model.

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this new model calculated, and the procedure repeated. The backward elimination algo-rithm terminates when the smallest partial F-value is not less than the preselected cutoff value FOU T [38].

Backward elimination is often a very good variable selection procedure. It is particu-larly favored by analysts who like to see the effect of including all the candidate regressors, just so that nothing obvious will be missed. The two procedures described above suggest a number of possible combinations. One of the most popular is the stepwise regression algorithm and the flowchart is shown in Fig. 2.4 This is a modification of forward selection in which at each step all regressors entered into the model previously are reassessed via their partial F- or t-statistics. A regressor added at an earlier step may now be redundant because of the relationship between it and regressors now in the equation. If the partial F-statistic for a variable is less than FOU T, that variable is dropped from the model.

Stepwise regression requires two cutoff values, FIN and FOU T. Several analysts prefer

to choose FIN = FOU T, although this is not necessary. Sometimes we choose FIN >

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Calculate correlation matrix

Choose X which is the largest correlation between X and Y

to build regression model

Discard non-significant X Choose X which partial F ratio is the largest

No suitable regression model Get regression model No No No Yes Yes Explanation of X is significant?

Does there exist any other X?

Is every partial F ratio in the model significant?

Yes

Figure 2.4: A flowchart of the stepwise regression algorithm used in our work.

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2.5 Model Adequacy Checking

To checking the fitted model is an adequate approximation to the true system or not is al-ways necessary. Also, we must verify that none of the least squares regression assumptions are violated. In this section we present several techniques for checking model adequacy [38].

(1) The normality assumption : The residuals are defined by Eq. (2.8), and they play an

very important role in determining model adequacy. An useful method is to construct a normal probability plot of the residuals, as in Fig. 2.5. If the residual normal probability plot is approximately along a straight line, then the normality assumption of residuals is satisfied. When this plot indicates problems with the normality assumption, we often trans-form the response variable as a remedial measure [38][43]. Transtrans-formations are used for three purposes: stabilizing response variance, making the distribution of response variable closer to the normal distribution, and improving the fit of the model to the data.

We introduce transformation of the response variables called Box-Cox Method. The Box-Cox transformation is a particulary useful family of transformations. It is defined as:

yλ = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yλ−1 λ , λ= 0 ln y, λ= 0 , (2.21)

where y is the response variable and λ is the transformation parameter. When λ is selected by the Box-Cox method, the experimenter can analyze the data using yλ as the response,

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2.5 : Model Adequacy Checking 25

unless of λ = 0, in which he can use ln y. It is perfectly acceptable to use yλ as the actual response, although the model parameter estimates will have a scale difference and origin shift in comparison to the results obtained using yλ _{(or ln y) [43].}

An approximate100(1 − α) percent confidence interval for λ can be found by calculat-ing:

SS∗ = SSE(λ)(1 +

t2_α/_2,ν

ν ), (2.22)

where ν is the number of freedom. Plotting a graph of SSE(λ) versus λ, and then by

locating the points on the λ axis where SS∗cuts the curve SSE(λ), we can read confidence

limits on λ directly from the graph. If this confidence interval includes the value λ = 1, this implies that the datas do not support the need for transformation.

(2) Plot of residuals versus predicted value : If the model is correct and if the assumptions

are satisfied, the residuals should be unrelated to any other variable including the predicted response. A simple check is to plot the residuals versus the predicted value ˆy. This plot should not reveal any obvious pattern, as in Fig. 2.6. A defect that occasionally shows up on this plot is nonconstant variance. Nonconstant variance also arises in cases where the data follow a nonnormal, skewed distribution because in skewed distributions the variance tends to be a function of the mean.

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Considerable research has been devoted to the selection of an appropriate transforma-tion. If experimenters know the theoretical distribution of the observations, they may uti-lize this information in choosing a transformation. For example, if the observations follow the Poisson distribution, the square root transformation y∗_ij = √yij or y∗ij =

1 + yij is

appropriate. If the data follow the lognormal distribution, the logarithmic transformation

y_ij∗ = log yij is appropriate. When there is no obvious transformation, the experimenter

usually empirically seeks a transformation that equalizes the variance regardless of the value of the mean. Another approach is to select a transformation that minimizes the inter-action mean square, resulting in an experiment that is easier to interpret. Transformations made for inequality of variance also affect the form of the error distribution. In most cases, the transformation brings the error distribution closer to normal [43].

2.6 Desirability Function

The desirability function is a useful tool to solve multiple responses. It was proposed by Derringer and Suich (1980) [43]. The desirability function di varies over the range

0 ≤ di ≤ 1, where the di = 0 is representing a completely undesirable of yi and di =

1 is representing a completely desirable or target response value of yi. After multiple

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2.6 : Desirability Function 27

combined using geometric mean to maximize the overall desirability D :

D= (d₁× d₂× . . . × dm)1/m, (2.23)

where m is the number of responses [43]. By the equation, if any di is equal to zero, then

the overall desirability is zero.

According to the specification for the responses, response is to be maximized, mini-mized, or achieved a target value. For the ith response yiis a maximum value,

di = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ˆyi < Li (ˆyi−Li T_i−L_i) s_, _L i ≤ ˆyi ≤ Ti 1, ˆyi > Ti . (2.24)

For the response yiis a minimum value,

di = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1, ˆyi < Ti (U_i_−ˆy_i Ui−Ti) s_, _T i ≤ ˆyi ≤ Ui 0, ˆyi > Ui . (2.25)

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For the response is achieved a target value,

di = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ˆyi < Li (ˆyi−Li Ti−Li) s_, _L i ≤ ˆyi ≤ Ti (U_i_−ˆy_i Ui−Ti) t_, _T i ≤ ˆyi ≤ Ui 0, ˆyi > Ui , (2.26)

where the weight s and t determine how important it is close to the target value. When the weight s = 1, t = 1 the desirability function is liner. Choosing s > 1, t > 1 means more important to be close the target value with the function that is concave, and choosing

0 < s < 1, 0 < t < 1 means this less important with the function that is convex. Li, Ui,

and Tiare the lower, upper, and target value, respectively.

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2.6 : Desirability Function 29 Studentized Residuals N o rm a l % P ro b a b ility -2.77 -1.36 0.04 1.44 2.84 1 5 10 20 30 50 70 80 90 95 99

Figure 2.5: An example of residual normal probability plot. If the residual is close to the line, it will satisfy normality

assumption. The studentized residuals are the standardized residuals adjusted for the distance from the average X value.

Predicted S tuden ti zed R e si dual s -3.02 -1.52 -0.01 1.49 3.00 42.88 57.67 72.46 87.25 102.04

Figure 2.6: An example of scatter plot of predicted values versus residuals. This plot should not reveal to any obvious patterns. The studentized residuals are the standardized residuals adjusted for the distance from the average X value.

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30 Chapter 2 : Statistical Methodology -0.2 0 0.2 0.4 0.6 0.8 1 1.2 s = 0.1 s = 1 s = 10 T UU

(b)

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 s = 0.1 s = 1 s = 10 t = 1 t = 10 t = 0.1 L T U

(c)

0 1 1.2 s = 0.1 s = 1 s = 10 L T -0.2 0.2 0.4 0.6 0.8

(a)

Figure 2.7: Individual desirability functions for the simultaneous optimization (a) objective is to maximize response, (b) objective is to minimize response, and (c) objective is for response which is assumed to be as close as possible to the target. The weights s and t determine how important it is close to the target value, and L, U, T are the lower, upper, and target value, respectively.

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2.7 : Other Design Methods 31

2.7 Other Design Methods

There are different types of design of experiment that have been widely applied to other designs such as Taguchi method and mixture design. We briefly describe their validation.

2.7.1 Taguchi Method

Many of the DOE concepts were popularized by Taguchi’s contributions to the method-ology of off-line quality design. The basis for his approach is to minimizing the loss to society that occurs when a products performance varies from a customer-specified target [44][45]. Taguchi’s ideas for parameter and tolerance design have evolved into what indus-try labels Design of Experiments for robust product design. Taguchi introduced the DOE techniques to engineering for quality improvement. In the past, enhancements to the DOE technique have been used on a production line or laboratory to derive empirical models and optimize a given process. Some of the well-known Taguchi orthogonal arrays were given earlier when three-level, mixed-level and fractional factorial designs were discussed.

The aim of parameter design is to make a product or process less variable (more robust) in the face of variation over which we have little or no control. Taguchi advocated using inner and outer array designs to take into account noise factors (outer) and design factors (inner). We could have used fractional factorials for either the inner or outer array de-signs, or for both. The tolerance design of the design process concentrates on the selective

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reduction of tolerances to reduce quality loss at the expense of increasing manufacturing costs.

2.7.2 Mixture Design

In a mixture experiment, the independent factors are proportions of different components of a blend. For example, if you want to optimize the tensile strength of stainless steel, the factors of interest should be the proportions of iron, copper, nickel, and chromium in the alloy. The fact that the proportions of the different factors must sum to 100 % complicates the design as well as the analysis of mixture experiments.

In mixture problems, the purpose of an experiment is to model the blending surface with some form of mathematical equation so that: (1) predictions of the response for any mixture or combination of the ingredients can be made empirically;(2) and some measure of the influence on the response of each component singly and in combination with other components can be obtained [36].

2.7.3 Comparison with the Popular Designs

DOE using Taguchi approach has become a much more attractive tool to practicing engi-neers and scientists. The objective of Taguchi approach is to obtain reproducible results and

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2.8 : Summary 33

Table 2.1: Difference between Taguchi approach and classical DOE.

Taguchi approach Classical DOE

Standard approach Methods are not standardized Smaller number of experiments Larger number of experiments

Standard method of noise factor No standardized method of noise treatment Seeks to find stable condition Develops models

Used to solve engineering problems Used to solve scientific experiments

robust products. The objective of classical DOE is to gather scientific knowledge about fac-tor effects and their interactions. Difference between Taguchi approach and classical DOE is shown in Tab. 2.1 [46]. In this thesis, it is suitable for us to use the classical DOE to investigate the problem due to our data type.

2.8 Summary

In this chapter, we introduce the statistical methodology which is used in this work. Screen-ing design is the first step in this work to select the significant factors. After this step, we have three type central composite design and choose one type among them to construct the response surface model. The basic of the response surface model and adequacy check-ing are then introduced. Then we discuss desirability function which is used to solve the multiple responses and according to this index we can optimize successfully. Finally we compare the difference between Taguchi approach and classical DOE, and we note that the

計算統計方法在積體電路最佳化及敏感度分析之研究

國 立 交 通 大 學

統計學研究所

碩士論文

計算統計方法在積體電路設計最佳化及敏感度分析之研究

Computational Statistics Approach to Integrated Circuit

Design Optimization and Sensitivity Analysis

研 究 生：羅婉文

指導教授：洪慧念 博士

李義明 博士

計算統計方法在積體電路設計最佳化及敏感度分析之研究

Computational Statistics Approach to Integrated Circuit Design Optimization

and Sensitivity Analysi

s

研 究 生：羅婉文 Student：Wan-Wen Lo

指導教授：洪慧念 博士 Advisor：Dr. Hui-Nien Hung

李義明 博士 Advisor：Dr. Yiming Li

國 立 交 通 大 學

統計學研究所

碩 士 論 文

中華民國九十五年七月

c

Copyright by Wan-Wen Lo 2006

All Rights Reserved

計算統計方法在積體電路設計最佳化及敏感度分析之研究

學生：羅婉文

指導教授：洪慧念 博士

李義明 博士

國立交通大學 統計研究所 碩士班

摘

要

Computational Statistics Approach to Integrated Circuit Design

Optimization and Sensitivity Analysis

Student：Wan-Wen Lo

Advisor：Dr. Hui-Nien Hung

Dr. Yiming Li

Institute of Statistics

National Chiao Tung University

Abstrac

t

誌

謝

Contents

List of Tables

List of Figures

Chapter 1

Introduction

1.1

Motivation

1.2

Literature Review

1.3

Objectives

1.4

Outline of the Thesis

Chapter 2

Statistical Methodology

2.1

Screening Design

2.2

Central Composite Design

2.3

Models Construction

2.4

Variable Selection

2.5

Model Adequacy Checking

2.6

Desirability Function

(b)

(c)

(a)

2.7

Other Design Methods

2.7.1

Taguchi Method

2.7.2

Mixture Design

2.7.3

Comparison with the Popular Designs

國立交通大學

研究生：羅婉文

指導教授：洪慧念博士

李義明博士

研究生：羅婉文 Student：Wan-Wen Lo

指導教授：洪慧念博士 Advisor：Dr. Hui-Nien Hung

李義明博士 Advisor：Dr. Yiming Li

國立交通大學

碩士論文

指導教授：洪慧念博士

李義明博士

國立交通大學統計研究所碩士班