Statistical Model Checking of Mixed-Analog Circuits with an Application to a Third Order Δ − Σ Modulator

  • Edmund Clarke
  • Alexandre Donzé
  • Axel Legay
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5394)


In this paper, we consider verifying properties of mixed-signal circuits, i.e., circuits for which there is an interaction between analog (continuous) and digital (discrete) quantities. We follow the statistical Model Checking approach of [You05, You06] that consists of evaluating the property on a representative subset of behaviors, generated by simulation, and answering the question of whether the circuit satisfies the property with a probability greater than or equal to some value. The answer is correct up to a certain probability of error, which is pre-specified. The method automatically determines the minimal number of simulations needed to achieve the desired accuracy, thus providing a convenient way to control the trade-off between precision and computational cost. We propose a logic adapted to the specification of properties of mixed-signal circuits, in the temporal domain as well as in the frequency domain. Our logic is unique in that it allows us to compare the Fourier transform of two signals. We demonstrate the applicability of the method on a model of a third order Δ − Σ modulator for which previous formal verification attempts were too conservative and required excessive computation time.


Model Check Quantization Error Linear Temporal Logic Sequential Probability Ratio Test Linear Temporal Logic Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ASS96]
    Aziz, P.M., Sorensen, H.V., Van Der Spiegel, J.: An overview of sigma-delta converters. IEEE Signal Processing Magazine, 61–84 (January 1996)Google Scholar
  2. [BHHK03]
    Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time markov chains. IEEE Trans. Software Eng. 29(6), 524–541 (2003)CrossRefzbMATHGoogle Scholar
  3. [CB06]
    Ciesinski, F., Baier, C.: Liquor: A tool for qualitative and quantitative linear time analysis of reactive systems. In: QEST, pp. 131–132. IEEE, Los Alamitos (2006)Google Scholar
  4. [CG04]
    Ciesinski, F., Größer, M.: On probabilistic computation tree logic. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 147–188. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. [CY95]
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [DDM04]
    Dang, T., Donze, A., Maler, O.: Verification of analog and mixed-signal circuits using hybrid systems techniques. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 21–36. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. [dR]
    d’Amorim, M., Roşu, G.: Efficient monitoring of ω-languages. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 364–378. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. [EC08]
    Legay Edmund Clarke, A., Donzé, A.: Statistical model checking of mixedanalog circuits. In: Second Workshop on Formal Verification of Analog Circuits of CAV 2008 (July 2008)Google Scholar
  9. [FJ97]
    Frigo, M., Johnson, S.G.: The fastest Fourier transform in the west. Technical Report MIT-LCS-TR-728, Massachusetts Institute of Technology (September 1997)Google Scholar
  10. [GKR04]
    Gupta, S., Krogh, B.H., Rutenbar, R.A.: Towards formal verification of analog designs. In: ICCAD, pp. 210–217 (2004)Google Scholar
  11. [HLMP04]
    Hérault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. [KNP04]
    Kwiatkowska, M.Z., Norman, G., Parker, D.: Prism 2.0: A tool for probabilistic model checking. In: QEST, pp. 322–323. IEEE, Los Alamitos (2004)Google Scholar
  13. [LS06]
    Bauer, A., Leucker, M., Schallhart, C.: Monitoring of real-time properties. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 260–272. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. [MNP08]
    Maler, O., Nickovic, D., Pnueli, A.: Checking temporal properties of discrete, timed and continuous behaviors. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 475–505. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. [MPVRV01]
    Medeiro, F., Pérez-Verdú, B., Rodríguez-Vázquez, A.: Top-Down Design of High-Performance Sigma-Delta Modulators, ch. 2. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  16. [NM07]
    Nickovic, D., Maler, O.: Amt: A property-based monitoring tool for analog systems. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 304–319. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. [SH93]
    Zakhor, A., Hein, S.: On the stability of sigma delta modulators. IEEE Transactions on Signal Processing 41 (July 1993)Google Scholar
  18. [Smi97]
    Smith, S.W.: The scientist and engineer’s guide to digital signal processing. California Technical Publishing, San Diego (1997)Google Scholar
  19. [SVA04]
    Sen, K., Viswanathan, M., Agha, G.: Statistical model checking of black-box probabilistic systems. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 202–215. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. [SVA05]
    Sen, K., Viswanathan, M., Agha, G.: On statistical model checking of stochastic systems. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 266–280. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. [Wal45]
    Wald, A.: Sequential tests of statistical hypotheses. Annals of Mathematical Statistics 16(2), 117–186 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [YKNP06]
    Younes, H.L.S., Kwiatkowska, M.Z., Norman, G., Parker, D.: Numerical vs. statistical probabilistic model checking. STTT 8(3), 216–228 (2006)Google Scholar
  23. [You05]
    Younes, H.L.S.: Verification and Planning for Stochastic Processes with Asynchronous Events. Ph.D thesis, Carnegie Mellon (2005)Google Scholar
  24. [You06]
    Younes, H.L.S.: Error control for probabilistic model checking. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 142–156. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. [YS06]
    Younes, H.L.S., Simmons, R.G.: Statistical probabilistic model checking with a focus on time-bounded properties. Information and Computation 204(9), 1368–1409 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Edmund Clarke
    • 1
  • Alexandre Donzé
    • 1
  • Axel Legay
    • 1
  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations