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Proceedings ArticleDOI

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.

AbstractA 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.

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Citations
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Proceedings ArticleDOI
07 Nov 2004
TL;DR: An asymptotic probability extraction method, APEX, for estimating the unknown random distribution when using nonlinear response surface modeling, which uses a binomial moment evaluation to efficiently compute the high order moments of the unknown distribution and applies moment matching to approximate the characteristic function of the random circuit performance by an efficient rational function.
Abstract: While process variations are becoming more significant with each new IC technology generation, they are often modeled via linear regression models so that the resulting performance variations can be captured via normal distributions. Nonlinear (e.g. quadratic) response surface models can be utilized to capture larger scale process variations; however, such models result in non-normal distributions for circuit performance which are difficult to capture since the distribution model is unknown. In this paper we propose an asymptotic probability extraction method, APEX, for estimating the unknown random distribution when using nonlinear response surface modeling. APEX first uses a binomial moment evaluation to efficiently compute the high order moments of the unknown distribution, and then applies moment matching to approximate the characteristic function of the random circuit performance by an efficient rational function. A simple statistical timing example and an analog circuit example demonstrate that APEX can provide better accuracy than Monte Carlo simulation with 10 samples and achieve orders of magnitude more efficiency. We also show the error incurred by the popular normal modeling assumption using standard IC technologies.

110 citations


Journal ArticleDOI
TL;DR: This paper develops an efficient analysis of variance-based stochastic circuit/microelectromechanical systems simulator to efficiently extract the surrogate models at the low level and employs tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points.
Abstract: Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient analysis of variance-based stochastic circuit/microelectromechanical systems simulator to efficiently extract the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is efficiently simulated by our simulator at the cost of only 10min in MATLAB on a regular personal computer.

99 citations


Cites methods from "Hierarchical statistical characteri..."

  • ...1 stochastic testing [15] Method 1 [32] Monte Carlo Monte Carlo Method 2 Alg....

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  • ...If Method 1 [32] is used, Monte Carlo has to be repeatedly used for each MEMS capacitor, leading to extremely long CPU time due to the slow convergence....

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  • ...In our preliminary conference paper [1], this method was employed to simulate a low-dimensional stochastic oscillator with 9 random parameters, achieving250× speedup over the hierarchical Monte-Carlo method proposed in [32]....

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  • ...In Method 1, both low-level and high-level simulations use Monte Carlo, as suggested by [32]....

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  • ...In such example, our method is over 92× faster than the hierarchical Monte Carlo method developed in [32], and is about 14× faster than the method that uses ANOVA-based solver at the low level and Monte Carlo at the high level....

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Journal ArticleDOI
TL;DR: A simple, yet reliable methodology to expedite yield estimation and optimization of microwave structures by exploiting the almost linear dependence of the feature points on the designable parameters of the structure.
Abstract: In this paper, we propose a simple, yet reliable methodology to expedite yield estimation and optimization of microwave structures. In our approach, the analysis of the entire response of the structure at hand (e.g., $S$ -parameters as a function of frequency) is replaced by response surface modeling of suitably selected feature points. On the one hand, this is sufficient to determine whether a design satisfies given performance specifications. On the other, by exploiting the almost linear dependence of the feature points on the designable parameters of the structure, reliable yield estimates can be realized at low computational cost. Our methodology is verified using two examples of waveguide filters and one microstrip hairpin filter and compared with conventional Monte Carlo analysis based on repetitive electromagnetic simulations, as well as with statistical analysis exploiting linear response expansions around the nominal design. Finally, we perform yield-driven design optimizations on these filters.

99 citations


Journal ArticleDOI
TL;DR: The APEX begins by efficiently computing the high-order moments of the unknown distribution and then applies moment matching to approximate the characteristic function of the random distribution by an efficient rational function, and is proven that such a moment-matching approach is asymptotically convergent when applied to quadratic response surface models.
Abstract: While process variations are becoming more significant with each new IC technology generation, they are often modeled via linear regression models so that the resulting performance variations can be captured via normal distributions. Nonlinear response surface models (e.g., quadratic polynomials) can be utilized to capture larger scale process variations; however, such models result in nonnormal distributions for circuit performance. These performance distributions are difficult to capture efficiently since the distribution model is unknown. In this paper, an asymptotic-probability-extraction (APEX) method for estimating the unknown random distribution when using a nonlinear response surface modeling is proposed. The APEX begins by efficiently computing the high-order moments of the unknown distribution and then applies moment matching to approximate the characteristic function of the random distribution by an efficient rational function. It is proven that such a moment-matching approach is asymptotically convergent when applied to quadratic response surface models. In addition, a number of novel algorithms and methods, including binomial moment evaluation, PDF/CDF shifting, nonlinear companding and reverse evaluation, are proposed to improve the computation efficiency and/or approximation accuracy. Several circuit examples from both digital and analog applications demonstrate that APEX can provide better accuracy than a Monte Carlo simulation with 104 samples and achieve up to 10times more efficiency. The error, incurred by the popular normal modeling assumption for several circuit examples designed in standard IC technologies, is also shown

79 citations


Journal ArticleDOI
TL;DR: The comparison with Monte Carlo simulations performed by neglecting the effect of mismatch confirmed that local device variations play a crucial role in the design and sizing of the clock distribution network.
Abstract: In this paper, we analyze the impact of process variations on the clock skew of VLSI circuits designed in deep submicrometer technologies. With smaller feature size, the utilization of a dense buffering scheme has been proposed in order to realize efficient and noise-immune clock distribution networks. However, the local variance of MOSFET electrical parameters, such as V/sub T/ and I/sub DSS/, increases with scaling of device dimensions, thus causing large intradie variability of the timing properties of clock buffers. As a consequence, we expect process variations to be a significant source of clock skew in deep submicrometer technologies. In order to accurately verify this hypothesis, we applied advanced statistical simulation techniques and accurate mismatch measurement data in order to thoroughly characterize the impact of intradie variations on industrial clock distribution networks. The comparison with Monte Carlo simulations performed by neglecting the effect of mismatch confirmed that local device variations play a crucial role in the design and sizing of the clock distribution network.

65 citations


References
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Book
29 Aug 1995
TL;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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"Hierarchical statistical characteri..." refers methods in this paper

  • ...The non-Monte Carlo techniques described in this paper utilize response surface methodology (RSM) [6]....

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Journal ArticleDOI

3,788 citations


"Hierarchical statistical characteri..." refers methods in this paper

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13 Mar 1991
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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"Hierarchical statistical characteri..." refers background or methods in this paper

  • ...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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Book
01 Jan 1964
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.

3,222 citations