Statistical Model Checking of Mixed-Analog Circuits with an Application to a Third Order Δ − Σ Modulator
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.
KeywordsModel Check Quantization Error Linear Temporal Logic Sequential Probability Ratio Test Linear Temporal Logic Formula
Unable to display preview. Download preview PDF.
- [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
- [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
- [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
- [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
- [GKR04]Gupta, S., Krogh, B.H., Rutenbar, R.A.: Towards formal verification of analog designs. In: ICCAD, pp. 210–217 (2004)Google Scholar
- [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
- [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
- [SH93]Zakhor, A., Hein, S.: On the stability of sigma delta modulators. IEEE Transactions on Signal Processing 41 (July 1993)Google Scholar
- [Smi97]Smith, S.W.: The scientist and engineer’s guide to digital signal processing. California Technical Publishing, San Diego (1997)Google Scholar
- [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
- [You05]Younes, H.L.S.: Verification and Planning for Stochastic Processes with Asynchronous Events. Ph.D thesis, Carnegie Mellon (2005)Google Scholar