## Defining Statistical Sensitivity for Timing Optimization of Logic Circuits with Large-Scale Process and Environmental Variations (2005)

### Cached

### Download Links

- [www.ece.cmu.edu]
- [users.ece.cmu.edu]
- [www.ece.cmu.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | IEEE International Conference on Computer-Aided Design |

Citations: | 20 - 3 self |

### BibTeX

@INPROCEEDINGS{Li05definingstatistical,

author = {Xin Li and Jiayong Le and Mustafa Celik and Lawrence T. Pileggi},

title = {Defining Statistical Sensitivity for Timing Optimization of Logic Circuits with Large-Scale Process and Environmental Variations},

booktitle = {IEEE International Conference on Computer-Aided Design},

year = {2005},

pages = {844--851}

}

### OpenURL

### Abstract

The large-scale process and environmental variations for today’s nanoscale ICs are requiring statistical approaches for timing analysis and optimization. Significant research has been recently focused on developing new statistical timing analysis algorithms, but often without consideration for how one should interpret the statistical timing results for optimization. In this paper [1] we demonstrate why the traditional concepts of slack and critical path become ineffective under large-scale variations, and we propose a novel sensitivity-based metric to assess the “criticality ” of each path and/or arc in the statistical timing graph. We define the statistical sensitivities for both paths and arcs, and theoretically prove that our path sensitivity is equivalent to the probability that a path is critical, and our arc sensitivity is equivalent to the probability that an arc sits on the critical path. An efficient algorithm with incremental analysis capability is described for fast sensitivity computation that has a linear runtime complexity in circuit size. The efficacy of the proposed sensitivity analysis is demonstrated on both standard benchmark circuits and large industry examples. 1.

### Citations

235 |
Multivariate Observations
- Seber
- 1984
(Show Context)
Citation Context ...i.e. zero mean and unit standard deviation), and M is the total number of these random variables. The independent random variables {ηi, i = 1,2,...,M} can be extracted by principle component analysis =-=[15]-=-, even if the original process parameters are correlated. Such a delay model in (25) is also used in many other statistical timing analysis algorithms, e.g. [10], [11]. Given the operation z = x+y or ... |

170 | Statistical timing analysis considering spatial correlations using a single pert-like traversal
- Chang, Sapatnekar
- 2003
(Show Context)
Citation Context ...racted by principle component analysis [15], even if the original process parameters are correlated. Such a delay model in (25) is also used in many other statistical timing analysis algorithms, e.g. =-=[10]-=-, [11]. Given the operation z = x+y or z = MAX(x,y) where x, y and z are approximated as (25), we define the sensitivity matrix Qz←x as: ⎡ ∂z0 ∂x0 ∂z0 ∂x1 L ∂z0 ∂xM ⎤ ⎢ ⎥ ⎢ ∂z1 ∂x0 ∂z1 ∂x1 L ∂z1 ∂xM Q... |

128 | First-order incremental blockbased statistical timing analysis
- Visweswariah, Ravindran, et al.
- 2004
(Show Context)
Citation Context ...qual to the probability that an arc sits on the critical path. There are two main advantages of our sensitivity concept for statistical timing analysis. Firstly, unlike the criticality computation in =-=[11]-=-, where independence is assumed between the criticality probabilities of two paths, our proposed sensitivitybased measure is not restricted to such an independence assumption. Secondly, from the compu... |

116 |
The greatest of a finite set of random variables
- Clark
- 1961
(Show Context)
Citation Context ...f the standard Normal distribution respectively, and: M ρ = ∑ i i 0 0 i= 1 2 ( x − y ) α = ( x − y ) ρ (29) Equations (28) and (29) can be derived by directly following the mathematic formulations in =-=[16]-=-. Due to the lack of space, the detailed proof of these equations is omitted here. It is worth noting that the sensitivity matrix Qz←y can be similarly computed using (27)-(29), since both the SUM and... |

103 | Statistical timing analysis for intra-die process variations with spatial correlations
- Agarwal, Blauuw, et al.
- 2003
(Show Context)
Citation Context ...cted based on their nominal delay values. In contrast, the block-based statistical timing analysis is more general, yet is limited by the variation modeling assumptions. In particular, the authors in =-=[9]-=--[12] demonstrate that since many circuit delays can be accurately approximated as Normal distributions, the spatial correlations and re-convergent fanouts can be handled efficiently for a block-based... |

75 | Block-based static timing analysis with uncertainty - Devgan, Kashyap |

74 | A general probabilistic framework for worst case timing analysis - Orshansky, Keutzer - 2002 |

47 | Statistical timing analysis using bounds and selective enumeration - Agarwal, Zolotov, et al. - 2003 |

46 | False-Path-Aware Statistical Timing Analysis And Efficient Path Selection For Delay Testing And Timing 159 - Liou, Krstic, et al. - 2002 |

45 | Statistical timing for parametric yield prediction of digital integrated circuits
- Jess, Kalafala, et al.
- 2003
(Show Context)
Citation Context ...ecently proposed for statistical timing analysis with consideration of large-scale variations [2]-[12]. Most of the proposed solutions fall into one of two broad categories: path-based approaches [2]-=-=[5]-=- and block-based approaches [6]-[12]. The path-based approaches can take into account the correlations from both path sharing and global parameters; however, the set of critical paths must be preselec... |

43 | Fast statistical timing analysis by probabilistic event propagation
- Liou, Cheng, et al.
- 2001
(Show Context)
Citation Context ...l timing analysis with consideration of large-scale variations [2]-[12]. Most of the proposed solutions fall into one of two broad categories: path-based approaches [2]-[5] and block-based approaches =-=[6]-=--[12]. The path-based approaches can take into account the correlations from both path sharing and global parameters; however, the set of critical paths must be preselected based on their nominal dela... |

38 | Stac: Statistical timing analysis with correlation
- Le, Li, et al.
- 2004
(Show Context)
Citation Context ...ons are considered at all levels of design hierarchy. Toward this goal, various algorithms have been recently proposed for statistical timing analysis with consideration of large-scale variations [2]-=-=[12]-=-. Most of the proposed solutions fall into one of two broad categories: path-based approaches [2]-[5] and block-based approaches [6]-[12]. The path-based approaches can take into account the correlati... |

19 | Uncertainty-aware circuit optimization
- Bai, Visweswariah, et al.
- 2002
(Show Context)
Citation Context ...babilities to be critical and all these paths must be selected for timing optimization. Even in nominal cases, many paths in a timing graph can be equally critical, which is so-called “slack wall” in =-=[13]-=-. This multiple-critical-path problem is more pronounced in statistical timing analysis, since more paths can have overlapped delay distributions due to large-scale process variations. In addition to ... |

1 |
Modeling and analysis of manufacturing variations
- Nasif
- 2001
(Show Context)
Citation Context ...iations are considered at all levels of design hierarchy. Toward this goal, various algorithms have been recently proposed for statistical timing analysis with consideration of large-scale variations =-=[2]-=--[12]. Most of the proposed solutions fall into one of two broad categories: path-based approaches [2]-[5] and block-based approaches [6]-[12]. The path-based approaches can take into account the corr... |