## Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance (2004)

Venue: | IEEE ICCAD |

Citations: | 31 - 7 self |

### BibTeX

@INPROCEEDINGS{Li04asymptoticprobability,

author = {Xin Li and Jiayong Le and Padmini Gopalakrishnan and Lawrence T. Pileggi},

title = {Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance},

booktitle = {IEEE ICCAD},

year = {2004},

pages = {2--9}

}

### Years of Citing Articles

### OpenURL

### 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 novel 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 4 samples and achieve orders of magnitude more efficiency. We also show the error incurred by the popular Normal modeling assumption using standard IC technologies. 1.

### Citations

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Citation Context ...in Section 3.2 we will link APEX to traditional probability theory and explain why it can be used to efficiently approximate PDF/CDF functions. 3.1 Mathematical Formulation We define the time moments =-=[11]-=- for a given circuit performance f whose probability density function is pdf �f� as follows: ��1� k �� � �� k �f� mk � f � pdf �df (6) k! In (6), the definition of time moments is identical to the tra... |

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Citation Context ...duct yield or unnecessary over-design cannot be avoided if these process variations are not accurately modeled and analyzed within the IC design flow. Table 1. Technology parameters and 3� variations =-=[1]-=- Leff (nm) Tox (nm) Vth (mV) W (µm) H (µm) 250�80 5.0�0.40 500�50 0.80�0.20 1.2�0.3 180�60 4.5�0.36 450�45 0.65�0.17 1.0�0.3 130�45 4.0�0.39 400�40 0.50�0.14 0.9�0.27 100�40 3.5�0.42 350�40 0.40�0.12 ... |

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Citation Context ...xis, which is then accurately approximated by the impulse/step response of a causal LTI system. 4.3 Reverse PDF/CDF Evaluation In many practical applications, such as robust circuit optimization [4], =-=[14]-=-, the best-case performance (e.g. the 1% point on CDF) and the worst-case performance (e.g. the 99% point on CDF) are two important metrics to be evaluated. As discussed in Section 3.2, APEX matches t... |

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Citation Context ...ue to the nonlinear mapping between the process variations and the circuit performance f. The distribution of f is no longer Normal, as is the case for the linear regression model. The authors in [3]-=-=[5]-=- utilize Monte Carlo simulation to evaluate the probability distribution of f, which is computationally expensive. Note that the computation cost for this probability extraction is crucial, especially... |

12 |
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Citation Context ...es due to the nonlinear mapping between the process variations and the circuit performance f. The distribution of f is no longer Normal, as is the case for the linear regression model. The authors in =-=[3]-=--[5] utilize Monte Carlo simulation to evaluate the probability distribution of f, which is computationally expensive. Note that the computation cost for this probability extraction is crucial, especi... |

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Citation Context ...e large-scale variations in the deep sub-micron technologies, which can reach �35% as shown in Table 1, suggest applying high order regression models in order to guarantee high approximation accuracy =-=[4]-=--[7]. Using a high order response surface model, however, brings about new challenges due to the nonlinear mapping between the process variations and the circuit performance f. The distribution of f i... |

6 |
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Citation Context ....0�0.3 130�45 4.0�0.39 400�40 0.50�0.14 0.9�0.27 100�40 3.5�0.42 350�40 0.40�0.12 0.8�0.27 70�33 3.0�0.48 300�40 0.30�0.10 0.7�0.25 During the past decade, various statistical analysis techniques [1]-=-=[7]-=- have been proposed and utilized in many applications such as statistical timing analysis, mismatch analysis, yield optimization, etc. The objective of these techniques is to model the probability dis... |

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Citation Context ...dress these problems. The aforementioned moment-matching method was previously applied for IC interconnect order reduction [11], [12] and is related to the Padé approximation in linear control theory =-=[13]-=-. In the following section, we will explain why this momentmatching approach is an efficient way to approximate PDF/CDF functions. 3.2 Connection to Probability Theory In probability theory, given a r... |

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