Statistical Model Checking of Mixed-Analog Circuits with an Application to a Third Order ∆ - Σ Modulator ⋆ Edmund Clarke, Alexandre Donz´ e, Axel Legay School of Computer Science Carnegie Mellon University, Pittsburgh, PA 15213 {emc|adonze|alegay@cs.cmu.edu} Abstract. 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 sta- tistical Model Checking approach of [You05,You06] that consists of eval- uating the property on a representative subset of behaviors, generated by simulation, and answering the question of whether the circuit satis- fies the property with a probability greater than or equal to some value. The answer is correct up to a certain probability of error, which can be 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 computa- tional cost. We propose a logic adapted to the specification of properties of mixed-signal circuits, in the temporal domain as well as in the fre- quency domain. Our logic is unique in that it allows us to compare the Fourier transform of two signals. We also demonstrate the applicability of the method on a model of a third order ∆ - Σ modulator for which previous formal verification attempts were too conservative and required excessive computation time. 1 Introduction Given a property φ, the Probabilistic Model Checking Problem consists of check- ing whether a stochastic system satisfies φ with a probability greater than or equal to a certain threshold θ. This problem is generally solved with a numeri- cal approach that consists of computing the exact probability for the system to satisfy φ and by comparing the result to θ. The way the probability is computed ⋆ This research was sponsored by the GSRC (University of California) under contract no. SA423679952, National Science Foundation under contracts no. CCF0429120, no. CNS0411152, and no. CCF0541245, Semiconductor Research Corporation under contract no. 2005TJ1366, Air Force (University of Vanderbilt) under contract no. 18727S3, International Collaboration for Advanced Security Technology of the Na- tional Science Council, Taiwan, under contract no. 1010717, and a grant from the Belgian American Educational Foundation.