Hierarchical statistical characterization of mixed-signal circuits using behavioral modeling
10 Nov 1996-pp 374-380
TL;DR: A methodology for hierarchical statistical circuit characterization which does not rely upon circuit-level Monte Carlo simulation is presented and permits the statistical characterization of large analog and mixed-signal systems.
Abstract: A methodology for hierarchical statistical circuit characterization which does not rely upon circuit-level Monte Carlo simulation is presented. The methodology uses principal component analysis, response surface methodology, and statistics to directly calculate the statistical distributions of higher-level parameters from the distributions of lower-level parameters. We have used the methodology to characterize a folded cascode operational amplifier and a phase-locked loop. This methodology permits the statistical characterization of large analog and mixed-signal systems, many of which are extremely time-consuming or impossible to characterize using existing methods.
Citations
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01 Jan 2013
TL;DR: This work proposes to construct physically consistent closed-form density functions by two monotone interpolation schemes, and determines the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator.
Abstract: Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellen t performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input a-priori. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physi- cally consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.
Cites background or methods from "Hierarchical statistical characteri..."
...Our method can be employed to handle a wide variety of surrogate models, including device-level models for SPI CElevel simulators [1]–[3], circuit-level performance mode ls for behavior-level simulation [8], as well as gate-level stati tical models for the timing analysis of digital VLSI [16]–[18]....
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...Related work in this direction includes the statistical ana lysis of phase-lock loops [8] and the statistical timing analysis of digital VLSI [16]–[18]....
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...In a statistical behavior-level simulator [8],̂ xi is the performance metric of a small circuit block (e....
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...Typical surrogate models include linear (quadratic) response surface models [8], [28]–[32], truncated general iz d polynomial chaos representations [1], [2], smooth or nonsmooth functions, stochastic reduced-order models [11], [ 4], [33], and some numerical packages that can rapidly evaluate fi(~ ξi) (e....
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01 Jan 2001
TL;DR: A novel and effective method for predicting the distribution of MOSFET device characteristics, which also enables us to specify the most typical process conditions for any device characteristics with only a few calculations.
Abstract: We have developed a novel and effective method for predicting the distribution of MOSFET device characteristics, which also enables us to specify the most typical process conditions for any device characteristics. In our approach, the distribution of the device characteristic caused by the fluctuation of every single process are calculated and then merged. Comparison with the result of the measured data has established that our method is accurate and practical. 1. Background Predicting accurate distributions of device characteristics using TCAD at the device development phase is indispensable for the concurrent designing of cell libraries and circuits with appropriate performance margins. While designing, the saturated drain current of NMOS and PMOS (IdsN and IdsP) pair is used as the reference of the best/worst performance conditions. By measuring the IdsN-IdsP pairs of manufactured chips, it has been found that their distribution assumes an oval shape, as shown in Fig.1. Process conditions which lead to particular device characteristics, the F F and SS points in Fig.1 for example, should be specified for accurate prediction of device characteristics such as Id-Vg, Id-Vd and C-V. Monte Carlo simulations (Kunitomo et al. 1999) and response surface methods (Felt et al. 1996), which are well-known methodologies for predicting the distribution, have a problem in that they cannot specify the typical process conditions which lead to particular device characteristics. We have developed a new methodology for predicting the distribution of device characteristics, which also enables us to specify the most typical process conditions for any device characteristics with only a few calculations. In our approach, to obtain the total distribution and process conditions, arrays which represent the distribution of the device characteristic caused by the fluctuation of every single process are prepared and then merged. 2. Methodology for Predicting the Distribution of Device Characteristics Before applying our method, well-calibrated process and device simulators should be prepared. The method consists of the following four steps: (step 1) Processes which have significant effects on the device characteristic, saturated drain current IdsN and IdsP in this case, are selected by calculating sensitivities of every process fluctuation to thc cha,ra.cteristic. From this result, the distribution of every selected process is mapped onto the distribution of the device characteristic. (step 2) For every process selected, a two dimensional array with IdsN and IdsP for row a,nd column is prepared. Elements of the arrays are tilled with probability of occurrence of the corresponding IdsN-IdsP value pairs calculated from the distribution of the device characteristic. (step 3) Arrays for every significant process specified in step 2 are merged. Elements of the merged array are calculated using the following equation: J'p~,pz(4j) = X7n,TL(ppl (m,n) * pp2(i m , j n,)) where Pp(i, j) is tlie probability of occurrence of the (i, j ) element, with the origin specified at nominal values of IdsN and IdsP, caused by the fluctuation of process p. This operation represents tha.t assuming process p l and p2 are independent, every element of thc array caused by process p2 is distributed following the distribution of another process p l . In the computational irnplementa,tion of this operation, every array element has an attribute list, where the corresponding process name, value and probability of occurrence are described. These lists are then combined and passed t o the resultant array during the operation, which enables us to predict the most typical process conditions for the specified IdsNIdsP value pair. (step 4) To specify an appropriate 30 or any di~tribut~ion area frorn the resultant array, elements with a higher probability of occurrence are selected from the array and added until the sum of the probability reaches the specified rate. When the 3a area is specified, the values of the array elements are added until the silm reaches 99.7%. 3. Application Example We have applied this met,hod to predicting the distribution of the IdsN-IdsP characteristics of a 0.18pm generation lLlOSFET and t,hen evaluated our method by comparing the resultant distribution with the measured data, a.nd data calculated using Montc Carlo simulation. 3.1. Specification of Device Characteristics Distribution We have analyzed the sensitivity of a device characteristic of saturated drain current (Ids5 and IdsP) to every process fluctuation, and three major processes whose fluctuations have significant effect on the characteristic are selected. The nomi~lal values a,nd standard deviation ( 3 0 ) of the processes and their effect on the characteristic are shown in Table 1. Though every process fluctuation is assunled to follow Gaussian distribution, distribution of the characteristic caused by Lg fluctuation does not because of the non-linearity of the drain current vs. gate length relation. We specified two separa,te Gaussian distributions of the characteristic, one for the case gate length is manufact~ired longer than the nominal, and the other for shorter. Gate length mismatch represents the difference between the gate lengths of paired NMOS and PMOS; the distribution of the characteristic caused by the mismatch depends on the gate lengths of the pair because of short channel effects. We specified the distribution of the characteristic caused by gate length mismatch as a function of Lg. 3.2 Calculation Results and Their Accuracy in Re-calculation Calculation results are shown in Fig.:! with In, 2a and 30 distributions indicated. Most typical process conditions that lead to FF, SS, FS and SF points on the 3a line are extracted and shown in Table 2. With these process conditions, device characteristics are calculated again and compared with the original prediction (Fig.3 and Table 3). All differences are less than 2%, which indicates the degree of accuracy of our method for the extraction of process conditions which lead to particular device characteristics. 3.3 Comparison with Measured Data and Monte Carlo Simulation The 3a line derived using our method is shown with the calculated points using Monte Carlo simulation (1,000 points, Fig.4) and the measured points of manufactured chips (11,649 points, Fig.5). Compared with Monte Carlo simulation, 99.3% of calculated points are within our 30 line and its distribution indicates good agreement with our method. As for the measured data, 99.3% of the measured points are within our 3a line and its distribution tends to shift slightly in the direction of less Ids, compared with our method. These comparison results suggest that our method provides sufficient accuracy in the development phase. 4. Conclusion We have developed an efficient method to predict the distribution of device characteristics. Using this method, not only the distribution but also the typical process conditions for any particular device characteristics can be extracted from the result. Comparison with the result of Monte Carlo simulation and with the measured data has established that our method is accurate and practical.
Proceedings Article•
01 Jan 2000TL;DR: In this article, the authors present a statistical analysis method which bridges the statistical information between process-level and system-level information to evaluate the effect of process variation at the system level.
Abstract: In this paper we present a statistical analysis method which bridges the statistical information between process-level and system-level. This enables us to evaluate the effect of process variation at the system level. Also, we can derive constraints on the process variation from a performance requirement. We show an example of the hierarchical statistical analysis applied to a Phase Locked Loop (PLL) circuit.
01 Jan 2007
TL;DR: A general hierarchical method for efficient statistical analysis of performance parameter variations for complex circuits/systems is presented and a case study on a 4th order continuous-time Delta Sigma modulator is conducted.
Abstract: Statistical analysis has become increasingly impor- tant with increasing process parameter variations in manufactur- ing. Monte Carlo method has been most popular for statistical analysis, but it is not efficient for complex circuits/systems due to overwhelming computational time. In this paper, we present a general hierarchical method for efficient statistical analysis of performance parameter variations for complex circuits/systems and conduct a case study on a 4th order continuous-time Delta Sigma modulator. At circuit-level, we use response sur- face modeling method to extract quadratic models of circuit- level performance parameters in terms of process parameter variations. Then, at system-level, we use behavioral models to extract statistical distribution of the overall system performance parameter. The method can achieve a good tradeoff between computational efficiency and accuracy.
Cites background or methods from "Hierarchical statistical characteri..."
...Hierarchical statistical analysis has been used in a few examples [4] [ 6 ] [7]....
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...[ 6 ] uses a hierarchical method for statistical analysis of lock time variation of (PLL) Phase Locked Loop....
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...Also, it is not limited to linear models and it randomly samples the process parameters instead circuit-level performance parameters, therefore avoids the computation of mean, variance and co-variance of circuit-level performance parameters [4] [ 6 ]....
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...Compared to other hierarchical methods [4] [ 6 ], the proposed method is free of mean, variance and correlation computation and is not limited to linear models....
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References
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Book•
29 Aug 1995TL;DR: Using a practical approach, this book discusses two-level factorial and fractional factorial designs, several aspects of empirical modeling with regression techniques, focusing on response surface methodology, mixture experiments and robust design techniques.
Abstract: From the Publisher:
Using a practical approach, it discusses two-level factorial and fractional factorial designs, several aspects of empirical modeling with regression techniques, focusing on response surface methodology, mixture experiments and robust design techniques. Features numerous authentic application examples and problems. Illustrates how computers can be a useful aid in problem solving. Includes a disk containing computer programs for a response surface methodology simulation exercise and concerning mixtures.
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...The non-Monte Carlo techniques described in this paper utilize response surface methodology (RSM) [6]....
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13 Mar 1991
TL;DR: In this paper, the authors present a directory of Symbols and Definitions for PCA, as well as some classic examples of PCA applications, such as: linear models, regression PCA of predictor variables, and analysis of variance PCA for Response Variables.
Abstract: Preface.Introduction.1. Getting Started.2. PCA with More Than Two Variables.3. Scaling of Data.4. Inferential Procedures.5. Putting It All Together-Hearing Loss I.6. Operations with Group Data.7. Vector Interpretation I : Simplifications and Inferential Techniques.8. Vector Interpretation II: Rotation.9. A Case History-Hearing Loss II.10. Singular Value Decomposition: Multidimensional Scaling I.11. Distance Models: Multidimensional Scaling II.12. Linear Models I : Regression PCA of Predictor Variables.13. Linear Models II: Analysis of Variance PCA of Response Variables.14. Other Applications of PCA.15. Flatland: Special Procedures for Two Dimensions.16. Odds and Ends.17. What is Factor Analysis Anyhow?18. Other Competitors.Conclusion.Appendix A. Matrix Properties.Appendix B. Matrix Algebra Associated with Principal Component Analysis.Appendix C. Computational Methods.Appendix D. A Directory of Symbols and Definitions for PCA.Appendix E. Some Classic Examples.Appendix F. Data Sets Used in This Book.Appendix G. Tables.Bibliography.Author Index.Subject Index.
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01 Jan 1971
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...6 is used to compute cov yi; yj [13]....
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...Note that for any given coefficients in a quadratic equation, A is uniquely determined [13]....
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01 Jan 1964
TL;DR: The general nature of Monte Carlo methods can be found in this paper, where a short resume of statistical terms is given, including random, pseudorandom, and quasirandom numbers.
Abstract: 1 The general nature of Monte Carlo methods.- 2 Short resume of statistical terms.- 3 Random, pseudorandom, and quasirandom numbers.- 4 Direct simulation.- 5 General principles of the Monte Carlo method.- 6 Conditional Monte Carlo.- 7 Solution of linear operator equations.- 8 Radiation shielding and reactor criticality.- 9 Problems in statistical mechanics.- 10 Long polymer molecules.- 11 Percolation processes.- 12 Multivariable problems.- References.
3,226 citations