Semi-Symbolic Analysis of Mixed-Signal Systems including Discontinuities Carna Radojicic, Christoph Grimm, Javier Moreno and Xiao Pan Kaiserslautern University of Technology {radojicic, grimm, moreno, pan}@cs.uni-kl.de Abstract—The paper describes an approach for semi-symbolic analysis of mixed-signal systems that contain discontinuous func- tions, e.g. due to modeling comparators. For modeling and semi- symbolic simulation, we use extended Afﬁne Arithmetic. Afﬁne Arithmetic is currently limited to accurate analysis of linear func- tions and mild non-linear functions, but not yet discontinuities. In this paper we extend the approach to also handle discontinuities. For demonstration, we symbolically analyze a ΣΔ-modulator. I. I NTRODUCTION Embedded systems include an increasing number of mixed- signal circuits. Veriﬁcation is a challenge, because system properties result from interaction of analog circuits with pa- rameter variations and SW/DSP systems that compensate these variations. For veriﬁcation of corner cases multi-run simulation techniques like Monte-Carlo (e.g. [1]) and Worst Case analysis (e.g. [2], [3]) are used. However, they often require a very high number of simulation runs to validate all corners. Formal methods for analog or mixed-signal systems are a rather new ﬁeld of intensive research that could offer inter- esting alternatives to multi-run simulations. Formal approaches for analog/mixed-signal circuits and systems allow equivalence checking [4], model checking [5], or symbolic analysis [6] deal with pure continuous behavior of analog circuits. At a much higher level of abstraction, Henzinger [7] suggests modeling hybrid systems with linear hybrid automata on which algorithmic analysis can be performed. However, this method is limited to very abstract and simple systems. A less formal approach is semi-symbolic simulation [8] based on Afﬁne Arithmetic [9]) that we use in this work. For simulation we use (symbolic) afﬁne expressions instead of real values. Afﬁne terms describe the impact of different sources of uncertainty/deviation in a symbolic way. The method has two advantages: 1) Modeling parameter uncertainties and deviations as ranges allows us to simulate all possible corner cases in a single simulation run 2) Representation of uncertainties with deviation sym- bols allows clear traceability of a violation in system performances back to contributions of corners of (e.g. technological) parameters. In previous work, the semi-symbolic simulation has been used successfully with block diagram level models [8], and even circuits [10]. Afﬁne Arithmetic is also used in static analysis of ﬂoating point errors of DSP algorithms [11]. A problem not yet solved within such semi-symbolic simulation is the transition between discrete and continuous behavior. In [8], an ADC between discrete controller and continuous plant is modeled by adding quantization noise to a linear transfer function. Mod- eling/simulation of discontinuous functions with considering two (or more) separate cases was not possible. In this work we extend semi-symbolic simulation to handle discontinuities in such a way. In Section II we describe simulation based on Afﬁne Arithmetic Forms (AAF). In Section III we describe a new method for modeling/simulation of discontinuities. In Section IV we demonstrate applicability by analysis of a ΣΔ- modulator. II. SEMI -SYMBOLIC SIMULATION WITH AAF Afﬁne Arithmetic [9] allows accurate computation with ranges. In each afﬁne form, the inﬂuence of uncorrelated sources of uncertainty i to a value with the ‘ideal’, central value x 0 is represented by a sum of terms x i ε i . Noise (also:deviation) symbols ε i are unknown values from the interval [−1, 1] that are scaled by partial deviations x i : ˜ x = x 0 +  i∈N˜ x x i ε i ε i ∈ [−1, 1] . N ˜ x represents a set of natural numbers identifying deviation terms x i ε i in ˜ x. Linear operations are accurate, because noise terms keep correlation information. However, non-linear operations introduce more or less over-approximation that guarantees safe inclusions. For semi-symbolic analysis of circuits and systems, we model uncertainties and parameter variations of circuits and systems such as noise, aging, drift, offsets, etc. by noise symbols ε i scaled by x i . This gives us a symbolic description of all parameters that inﬂuence the traces for a given stimuli and parameter ranges. Fig. 1 for example shows semi-symbolic x 0 (t) ... ... t ˜ x (Spec. violation) x n (t)ε n /2 x 1 (t)ε 1 /2 x 1 (t)ε 1 /2 x n (t)ε n /2 diameters of ˜ x(t) plotted Symbolic representation of a sample by AAF Speciﬁcation of step response (tolerance scheme) Fig. 1: Semi-symbolic simulation results and it‘s visualization by plotting diameters of signals. 978-3-9815370-2-4/DATE14/©2014 EDAA