Abstract
Transistor sizing is a classic computer-aided design problem that has received much attention in the literature. Given the increasing importance of process variation in deep sub-micron circuits and the wide-spread use of statistical methods, the sizing problem for such circuits warrants revisiting. This paper addresses transistor sizing as one that can be solved via interior point nonlinear optimization with an objective function that is directly dependent on statistical process variation. Our technique automatically adjusts transistor sizes to maximize parametric yield at a given timing performance, or to maximize performance at a required parametric yield. Our results show that for process variation sensitive circuits, consisting of thousands of independently tunable transistors, a statistically aware tuner can give more robust, higher yield solutions when compared to deterministic circuit tuning and is thus an attractive alternative to the Monte Carlo methods that are typically used to size transistors in such circuits. To the best of our knowledge, this is the first working system to optimize transistor sizes in custom circuits using a process variation aware tuner.
Similar content being viewed by others
References
Agarwal AB, Blaauw D, Zolotov V, Sundareswaran S, Zhao M, Gala K, Panda R (2002) Path-based statistical timing analysis considering inter- and intra-die correlations. In: Proc 2002 TAU (ACM/IEEE workshop on timing issues in the specification and synthesis of digital systems), Monterey, CA, December 2002, pp 16–21
Agarwal AB, Blaauw D, Zolotov V, Vrudhula S (2003a) Computation and refinement of statistical bounds on circuit delay. In: Proc. 2003 design automation conference, Anaheim, CA, June 2003, pp 348–353
Agarwal A, Blaauw D, Zolotov V (2003b) Statistical timing analysis for intra-die process variations with spatial correlations. In: IEEE international conference on computer-aided design, San Jose, CA, November 2003, pp 900–907
Agarwal A, Chopra K, Blaauw D, Zolotov V (2005) Circuit optimization using statistical timing analysis. In: Proc 2005 design automation conference, Anaheim, CA, June 2005, pp 321–324
Bard K, Dewey B, Hsu M, Mitchell T, Moody K, Rao V, Rose R, Soreff J, Washburn S (2007) Transistor-level tools for high-end processor custom circuit design at IBM. Proc IEEE 95(3):530–554
Berkelaar M, Jess J (1990) Gate sizing in mos digital circuits with linear programming. In: Proc Eur design automat conf, pp 217–221
Brayton RK, Hachtel GD, Sangiovanni-Vincentelli AL (1981) A survey of optimization techniques for integrated-circuit design. Proc IEEE 69(10):1334–1362
Chang H, Sapatnekar SS (2003) Statistical timing analysis considering spatial correlations using a single PERT-like traversal. In: IEEE international conference on computer-aided design, San Jose, CA, November 2003, pp 621–625
Chang H, Zolotov V, Visweswariah C, Narayan S (2005) Parameterized block-based statistical timing analysis with non-Gaussian and nonlinear parameters. In: Proc 2005 design automation conference, Anaheim, CA, June 2005, pp 71–76
Chen C-P, Chu CCN, Wong DF (1998) Fast and exact simultaneous gate and wire sizing by Lagrangian relaxation. In: IEEE international conference on computer-aided design, November 1998, pp 617–624
Chen C-P, Chu CN, Wong DF (1999) Fast and exact simultaneous gate and wire sizing by Lagrangian Relaxation. IEEE Trans Comput-Aided Des Integr Circuits Syst 18(7):1014–1025
Chen R, Zhang L, Zolotov V, Visweswariah C, Xiong J (2008) Static timing: back to our roots. In: Asia South Pacific design automation conference, Seoul, Korea, January 2008, pp 310–315
Chopra K, Shah S, Srivastava A, Blaauw D, Sylvester D (2005) Parametric yield maximization using gate sizing based on efficient statistical power and delay gradient computation. In: IEEE international conference on computer-aided design, San Jose, CA, November 2005, pp 1023–1028
Clark CE (1961) The greatest of a finite set of random variables. Oper Res March-April:145–162
Conn AR, Visweswariah C (2001) Overview of continuous optimization advances and applications to circuit tuning. In: Proc international symposium on physical design, Sonoma County, CA, April 2001, pp 74–81
Conn AR, Coulman PK, Haring RA, Morrill GL, Visweswariah C (1996) Optimization of custom MOS circuits by transistor sizing. In: IEEE international conference on computer-aided design, San Jose, CA, November 1996, pp 174–180
Conn AR, Haring RA, Visweswariah C, Wu CW (1997) Circuit optimization via adjoint Lagrangians. In: IEEE international conference on computer-aided design, San Jose, CA, November 1997, pp 281–288
Conn AR, Coulman PK, Haring RA, Morrill GL, Visweswariah C, Wu CW (1998) JiffyTune: circuit optimization using time-domain sensitivities. IEEE Trans Comput-Aided Des Integr Circuits Syst 17(12):1292–1309
Conn AR, Elfadel IM, Molzen WW Jr, O’Brien PR, Strenski PN, Visweswariah C, Whan CB (1999) Gradient-based optimization of custom circuits using a static-timing formulation. In: Proc 1999 design automation conference, New Orleans, LA, June 1999, pp 452–459
Davoodi A, Srivastava A (2005) Variability driven gate sizing for binning yield optimization. In: Proc design automat conf, pp 294–299
Director SW, Rohrer RA (1969) The generalized adjoint network and network sensitivities. IEEE Trans Circuit Theory CT-16(3):318–323
Feldmann P, Nguyen TV, Director SW, Rohrer RA (1991) Sensitivity computation in piecewise approximate circuit simulation. IEEE Trans Comput-Aided Des Integr Circuits Syst 10(2):171–183
Fishburn JP, Dunlop AE (1985) TILOS: A posynomial programming approach to transistor sizing. In: IEEE international conference on computer-aided design, November 1985, pp 326–328
Guthaus MR, Venkateswaran N, Sylvester D, Brown RB, Zolotov V (2005) Optimization objectives and variation models for statistical gate sizing. In: Great Lakes symposium on VLSI (GLSVLSI), Chicago, IL, April 2005, pp 313–316
Guthaus MR, Venkateswaran N, Visweswariah C, Zolotov V (2005) Gate sizing using incremental parameterized statistical timing analysis. In: IEEE international conference on computer-aided design, San Jose, CA, November 2005, pp 1029–1036
Hitchcock RB (1982) Timing verification and the timing analysis problem. In: Proc design automation conference, pp 594–604
Jacobs ETAF, Berkelaar MRCM (2000) Gate sizing using a statistical delay model. In: Design and test in Europe, pp 283–290
Jess JAG, Kalafala K, Naidu SR, Otten RHJM, Visweswariah C (2003) Statistical timing for parametric yield prediction of digital integrated circuits. In: Proc 2003 design automation conference, Anaheim, CA, June 2003, pp 932–937
Jess JAG, Kalafala K, Naidu SR, Otten RHJM, Visweswariah C (2006) Statistical timing for parametric yield prediction of digital integrated circuits. IEEE Trans Comput-Aided Des Integr Circuits Syst 25(11):2376–2392
Ketkar M, Kasamsetty K, Sapatnekar SS (2000) New class of convex functions for delay modeling and their application to the transistor sizing problem. IEEE Trans Comput-Aided Des 19(7):779–788
Li X, Le J, Celik M, Pileggi LT (2005) Defining statistical sensitivity for timing optimization of logic circuits with large-scale process and environmental variations. In: IEEE international conference on computer-aided design, San Jose, CA, November 2005, pp 844–851
Luo Z, Singh J, Nookala V, Sapatnekar S (2005) Robust gate sizing by geometric programming. In: Proc design automat conf, pp 315–320
Menezes N, Baldick R, Pileggi LT (1995) A sequential quadratic programming approach to concurrent gate and wire sizing. In: IEEE international conference on computer-aided design, November 1995, pp 144–151
Nassif SR (2001) Modeling and analysis of manufacturing variations. In: Custom integrated circuits conference, pp 223–228
Neiroukh S, Song X (2006) Improving the process-variation tolerance of digital circuits using gate sizing and statistical techniques. In: Proc of design automat and test Europ conf, pp 959–964
Nguyen TV, Feldmann P, Director SW, Rohrer RA (1989) SPECS simulation validation with efficient transient sensitivity computation. In: IEEE international conference on computer-aided design, November 1989, pp 252–255
Nguyen TV, Devgan A, Nastov OJ (1998) Adjoint transient sensitivity computation in piecewise linear simulation. In: Proc 1998 design automation conference, June 1998, pp 477–482
Orshansky M, Keutzer K (2002) A general probabilistic framework for worst case timing analysis. In: Proc 2002 design automation conference, New Orleans, LA, June 2002, pp 556–561
Pillage LT, Rohrer RA, Visweswariah C (1995) Electronic circuit and system simulation methods. McGraw-Hill, New York
Rao VB, Soreff JP, Brodnax TB, Mains RE (1999) EinsTLT: transistor-level timing with EinsTimer. In: Proc. TAU (ACM/IEEE workshop on timing issues in the specification and synthesis of digital systems), Austin, TX December 1999, pp 1–6
Sapatnekar SS, Rao VB, Vaidya PM, Kang SM (1993) An exact solution to the transistor sizing problem for CMOS circuits using convex optimization. IEEE Trans Comput-Aided Des Integr Circuits Syst CAD-12(11):1621–1634
Shenoy NV, Sinha D, Zhou H (2006) Statistical timing yield optimization. IEEE Trans Very Large Scale Integr (VLSI) Syst 14(10):1140–1146
Tennakoon H, Sechen C (2002) Gate sizing using Lagrangian relaxation combined with a fast gradient-based pre-processing step. In: Int conf on computer-aided design, pp 395–402
Tennakoon H, Sechen C (2005) Efficient and accurate gate sizing with piecewise convex delay models. In: Proc design automat conf, pp 807–812
Visweswariah C (1997) Optimization techniques for high-performance digital circuits. In: IEEE international conference on computer-aided design, San Jose, CA, November 1997, pp 198–205
Visweswariah C (2000) Formal static optimization of high-performance digital circuits. In: Proc TAU (ACM/IEEE workshop on timing issues in the specification and synthesis of digital systems), Austin, TX, December 2000, p 51
Visweswariah C (2003) Death, taxes and failing chips. In: Proc 2003 design automation conference, June 2003, Anaheim, CA, pp 343–347
Visweswariah C (2004) Statistical analysis and design: from picoseconds to probabilities. In: 17th symposium on integrated circuits and systems design (SBCCI), Porto de Galinhas, Brazil, September 2004. Invited tutorial
Visweswariah C, Conn AR (1999) Formulation of static circuit optimization with reduced size, degeneracy and redundancy by timing graph manipulation. In: IEEE international conference on computer-aided design, San Jose, CA, November 1999, pp 244–251
Visweswariah C, Rohrer RA (1991) Piecewise approximate circuit simulation. IEEE Trans Comput-Aided Des Integr Circuits Syst 10(7):861–870
Visweswariah C, Ravindran K, Kalafala K, Walker SG, Narayan S (2004) First-order incremental block-based statistical timing analysis. In: Proc 2004 design automation conference, San Diego, CA, June 2004, pp 331–336
Visweswariah C, Ravindran K, Kalafala K, Walker SG, Narayan S, Beece DK, Piaget J, Venkateswaran N, Hemmett JG (2006) First-order incremental block-based statistical timing analysis. IEEE Trans Comput-Aided Des Integr Circuits Syst 25(10):2170–2180
Wächter A (2002) An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA, January 2002
Wachter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57
Xiong J, Zolotov V, Visweswariah C, Venkateswaran N (2006a) Criticality computation in parameterized statistical timing. In: Proc 2006 design automation conference, San Francisco, CA, July 2006, pp 63–68
Xiong J, Zolotov V, He L (2006b) Robust extraction of spatial correlation. In: Proc international symposium on physical design, San Jose, CA, April 2006, pp 2–9
Xiong J, Zolotov V, Visweswariah C (2008) Incremental criticality and yield gradients. In: Design and test in Europe, Messe Munich, Germany, March 2008, pp 1130–1135
Zhang L, Hu Y, Chen CC (2005) Block based statistical timing analysis with extended canonical timing model. In: Asia South Pacific design automation conference, Shanghai, China, January 2005, pp 250–253
Zolotov V, Xiong J, Visweswariah C (2006) Computation of yield gradients from statistical timing analysis. In: Proc 2006 TAU (ACM/IEEE workshop on timing issues in the specification and synthesis of digital systems), San Jose, CA, February 2006, pp 125–130
Zuchowski PS, Habitz PA, Hayes JD, Oppold JH (2004) Process and environmental variation impacts on ASIC timing. In: IEEE international conference on computer-aided design, San Jose, CA, November 2004, pp 336–342
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beece, D.K., Visweswariah, C., Xiong, J. et al. Transistor sizing of custom high-performance digital circuits with parametric yield considerations. Optim Eng 15, 217–241 (2014). https://doi.org/10.1007/s11081-012-9208-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-012-9208-0