## Statistical timing yield optimization by gate sizing (2006)

Venue: | TCAD |

Citations: | 6 - 4 self |

### BibTeX

@ARTICLE{Sinha06statisticaltiming,

author = {Debjit Sinha and Narendra V. Shenoy and Hai Zhou and Senior Member and Senior Member},

title = {Statistical timing yield optimization by gate sizing},

journal = {TCAD},

year = {2006},

volume = {25},

pages = {1140--1146}

}

### OpenURL

### Abstract

Abstract—In this paper, we propose a statistical gate sizing approach to maximize the timing yield of a given circuit, under area constraints. Our approach involves statistical gate delay modeling, statistical static timing analysis, and gate sizing. Experiments performed in an industrial framework on combinational International Symposium on Circuits and Systems (ISCAS’85) and Microelectronics Center of North Carolina (MCNC) benchmarks show absolute timing yield gains of 30 % on the average, over deterministic timing optimization for at most 10 % area penalty. It is further shown that circuits optimized using our metric have larger timing yields than the same optimized using a worst case metric, for iso-area solutions. Finally, we present an insight into statistical properties of gate delays for a commercial 0.13- m technology library which intuitively provides one reason why statistical timing driven optimization does better than deterministic timing driven optimization. Index Terms—Gate sizing, optimization, statistical gate delay modeling, statistical timing analysis, timing yield, variability, VLSI. I.

### Citations

155 | First-order incremental block-based statistical timing analysis
- Visweswariah, Ravindran, et al.
- 2004
(Show Context)
Citation Context ... since it facilitates fast analytical evaluation. Timing analysis involves add and max operations. A max operation on Gaussian random variables is nontrivial. Chang et al. [3] and Visweswariah et al. =-=[5]-=- propose to approximate the maximum of multiple Gaussians with a Gaussian using Clark’s approach [7] to obtaining the max of two Gaussians. Pairwise max operations are, thus, employed in the computati... |

138 |
The Greatest of a Finite Set of Random Variables
- Clark
- 1961
(Show Context)
Citation Context ... max operation on Gaussian random variables is nontrivial. Chang et al. [3] and Visweswariah et al. [5] propose to approximate the maximum of multiple Gaussians with a Gaussian using Clark’s approach =-=[7]-=- to obtaining the max of two Gaussians. Pairwise max operations are, thus, employed in the computation of the maximum of multiple Gaussians, each of which involve approximations. However, none of the ... |

120 | Statistical timing analysis for intradie process variations with spatial correlations
- Agarwal, Blaauw, et al.
- 2003
(Show Context)
Citation Context ...ern deep submicrometer integrated circuits necessitates statistical approaches to timing analysis and optimization. Researchers have proposed multiple approaches to statistical static timing analysis =-=[2]-=-–[6] in the past few years. A majority of these approaches consider circuit component delays as Gaussian random variables since it facilitates fast analytical evaluation. Timing analysis involves add ... |

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

56 | Parameterized block-based statistical timing analysis with non-gaussian parameters, nonlinear delay functions
- Chang, Zolotov, et al.
- 2005
(Show Context)
Citation Context ... as a weighted linear sum of Gaussian random variables, the statistical timing yield improvement approach can be extended to handle nongaussian parameters and nonlinear delay functions as proposed in =-=[14]-=-. However, obtaining a simple metric for timing yield optimization would be a challenging problem. APPENDIX Using notations defined in (6), (7), and (9), we prove that the variance matching method in ... |

47 | Statistical timing analysis under spatial correlations
- Chang, Sapatnekar
- 2005
(Show Context)
Citation Context ...as Gaussian random variables since it facilitates fast analytical evaluation. Timing analysis involves add and max operations. A max operation on Gaussian random variables is nontrivial. Chang et al. =-=[3]-=- and Visweswariah et al. [5] propose to approximate the maximum of multiple Gaussians with a Gaussian using Clark’s approach [7] to obtaining the max of two Gaussians. Pairwise max operations are, thu... |

34 | S.Manne, “New Algorithms for Gate Sizing: A Comparative Study
- Coudert
- 1996
(Show Context)
Citation Context ... design, using as both the global cost function and the local cost function. The complexity of this algorithm using the best-fit polynomial is shown to be , where denotes the number of internal nodes =-=[13]-=-. The pseudocode of the SGGS algorithm is presented in Fig. 2. V. IMPLEMENTATION AND EXPERIMENTAL RESULTS The proposed statistical modeling, statistical timing analysis, and gate sizing routines are i... |

32 | Statistical gate sizing for timing yield optimization - Sinha, Shenoy, et al. - 2005 |

31 | Gate sizing using incremental parameterized statistical timing analysis
- Guthaus, Venkateswaran, et al.
- 2005
(Show Context)
Citation Context ...tric programming is proposed by Singh et al. [9]. They incorporate an uncertainty ellipsoid to model variations and attain to optimize circuit area under worst case timing constraints. Guthaus et al. =-=[10]-=- propose a gate sizing algorithm to optimize circuit area while satisfying a given timing yield target. They employ a sensitivity metric to select gates for resizing. Our experiments conclude that nod... |

21 |
Robust gate sizing by geometric programming
- Singh
(Show Context)
Citation Context ...on. Intra-die variability is considered, and gate delay variations are assumed to be 10% of their nominals. A robust gate sizing methodology based on geometric programming is proposed by Singh et al. =-=[9]-=-. They incorporate an uncertainty ellipsoid to model variations and attain to optimize circuit area under worst case timing constraints. Guthaus et al. [10] propose a gate sizing algorithm to optimize... |

17 | Sizing: a General Purpose Optimization Approach
- Coudert
- 1996
(Show Context)
Citation Context ...ective function attains to maximize the worst case slack. We design a statistical global gate sizing (SGGS) algorithm for timing yield optimization as an extension to the global gate sizing algorithm =-=[12]-=-. Our choice of the global sizing algorithm is motivated by results obtained by Coudert et al. [12], which show that this algorithm is superior to common greedy or genetic approaches to circuit optimi... |

15 |
Circuit optimization using statistical static timing analysis
- Agarwal, Chopra, et al.
- 2005
(Show Context)
Citation Context ...ical timing optimization have emerged recently. Agarwal et al. propose a sensitivity-based gate sizing algorithm, and faster approaches that perform sensitivity calculation based on slack computation =-=[8]-=-, to minimize Manuscript received November 17, 2005; revised March 28, 2006. This work was supported in part by the National Science Foundation under Grant CCR0238484. This paper is an extended versio... |

9 | STAC: Statistical timing with correlation - Le, Li, et al. |

7 | Advances in computation of the maximum of a set of random variables
- Sinha, Zhou, et al.
- 2006
(Show Context)
Citation Context ...ay accumulate errors and can significantly affect the accuracy of the final solution. We employ a greedy approach for smart pairwise max (min) operations based on the approximation error computations =-=[11]-=-. Slack estimation during timing analysis involves subtract operations which can be performed on the canonical forms of the timing distributions. A min operation on the slack distributions at the prim... |