Eva Dyer
ABSTRACT
Accurate characterization of spatial variation is essential for statistical performance analysis and modeling, post-silicon tuning, and yield analysis. Existing approaches for spatial modeling either assume that: (i) non-stationarities arise due to a smoothly varying trend component or that (ii) the process is stationary within regions associated with a predefined grid. While such assumptions may hold when profiling certain classes of variations, a number of recent modeling studies suggest that non-stationarities arise from both shifts in the process mean as well as fluctuations in the variance of the process. In order to provide a compact model for non-stationary process variations, we introduce a new hybrid spatial modeling framework that models the spatially varying random field as a union of non-overlapping rectangular regions where the process is assumed to be locally-stationary within each region. To estimate the parameters in our hybrid spatial model, we develop a host of techniques to both estimate the change-points in the random field and to find an appropriate partitioning of the chip into disjoint regions where the field is locally-stationary. We verify our models and results on measurements collected from 65nm FPGAs.
- A. Agarwal, D. Blaauw, V. Zolotov, S. Sundareswaranand, M. Zhao, K. Gala, and R. Panda. Statistical delay computation considering spatial correlations. In ASPDAC, pages 271--276, 2003. Google Scholar
Digital Library
- K. Balakrishnan and D. Boning. Measurement and analysis of contact plug resistance variability. In Custom Integrated Circuits Conference (CICC), pages 415--422, 2009.Google ScholarCross Ref
- D. Boning, J. Chung, D. Ouma, and R. Divecha. Spatial variation in semiconductor processes. Process Control Diagnostics and Modeling in Semiconductor Manufacturing, 97(9):92--83, 1997.Google Scholar
- S. Borkar, T. Karnik, S. Narendra, J. Tschanz, A. Keshavarzi, and V. De. Parameter variations and impact on circuits and microarchitecture. In DAC, pages 338--342, 2003. Google ScholarDigital Library
- H. Chang and S. S. Sapatnekar. Statistical timing analysis considering spatial correlations using a single pert-like traversal. In ICCAD, page 621, 2003. Google ScholarDigital Library
- H. Chang and S. S. Sapatnekar. Prediction of leakage power under process uncertainties. ACM Trans. Des. Autom. Electron. Syst., 12(2):12, 2007. Google ScholarDigital Library
- L. Cheng, P. Gupta, C. Spanos, K. Qian, and L. He. Physically justifiable die-level modeling of spatial variation in view of systematic across wafer variability. In DAC, pages 104--109, 2009. Google ScholarDigital Library
- Q. Fu, W.-S. Luk, J. Tao, and C. Y. and X. Zeng. Characterizing intradie spatial correlation using spectral density method. In Symposium on Quality of Electronic Design, pages 718--723, 2008. Google ScholarDigital Library
- M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming. 2009.Google Scholar
- S. i. Ohkawa, H. Masuda, and Y. Inoue. A novel expression of spatial correlation by a random curved surface model and its application to LSI design. E91-A:1062-1070, 2008. Google ScholarDigital Library
- J. Jinjun, V. Zolotov, and H. Lei. Robust extraction of spatial correlation. TCAD, 26(4):619--631, 2007. Google ScholarDigital Library
- J. l. Starck, M. Elad, and D. Donoho. Image decomposition via the combination of sparse representations and a variational approach. IEEE Transactions on Image Processing, 14:1570--1582, 2004. Google ScholarDigital Library
- B. Liu. Spatial correlation extraction via random field simulation and production chip performance regression. In DATE, pages 527--532, 2008. Google ScholarDigital Library
- F. Liu. A general framework for spatial correlation modeling in VLSI design. In DAC, pages 817--822, 2007. Google ScholarDigital Library
- J.-H. Liu, M.-F. Tsai, L. Chen, and C. C.-P. Chen. Accurate and analytical statistical spatial correlation modeling for VLSI DFM applications. In DAC, pages 694--697, 2008. Google ScholarDigital Library
- M. Majzoobi, E. Dyer, A. Elnably, and F. Koushanfar. Rapid FPGA characterization using clock synthesis and signal sparsity. In ITC, 2010.Google Scholar
- M. Orshansky, S. R. Nassif, and D. Boning. Design for Manufacturability and Statistical Design: A Constructive Approach. Springer, 2007. Google ScholarDigital Library
- L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Phys. D, 60(1--4):259--268, 1992. Google ScholarDigital Library
- T. Sato, H. Ueyama, N. Nakayama, and K. Masu. Determination of optimal polynomial regression function to decompose on-die systematic and random variations. In ASPDAC, pages 518--523, 2008. Google ScholarDigital Library
- P. Sedcole and P. Y. K. Cheung. Within-die delay variability in 90nm FPGAs and beyond. In FPT, pages 97--104, 2006.Google ScholarCross Ref
- P. Sedcole, S. P. Wong, and P. Y. K. Cheung. Compensating for variability in FPGAs by re-mapping and re-placement. In FPL, pages 613--616, 2009.Google ScholarCross Ref
- M. Shiqian, Y. Wotao, Z. Yin, and A. Chakraborty. An efficient algorithm for compressed mr imaging using total variation and wavelets. In Computer Vision and Pattern Recognition Conference (CVPR), pages 1--8, 2008.Google ScholarCross Ref
- W. Zhang, W. Yu, Z. Wang, Z. Yu, R. Jiang, and J. Xiong. An efficient method for chip-level statistical capacitance extraction considering process variations with spatial correlation. In Proceedings of the conference on Design, automation and test in Europe (DATE), pages 580--585, 2008. Google ScholarDigital Library
Index Terms
- Hybrid modeling of non-stationary process variations
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