Adaptive Numerical Integration Methods for Deterministic Analysis of Non-Stationary Noise in Dynamic Integrated Circuits Amir Zjajo, Qin Tang, Michel Berkelaar, Nick van der Meijs Circuits and Systems Group Delft University of Technology Delft, The Netherlands Abstract—This paper reports a new step-size control strategy for adaptive numerical integration in time-domain noise analysis of non-linear dynamic integrated circuits with arbitrary excitations. A non-stationary stochastic noise process is described as an Itô system of stochastic differential equations and a numerical solution for such a set of equations is found. Statistical simulation of dynamic circuits fabricated in 45 nm CMOS process shows that the proposed numerical methods offer an accurate and efficient solution. Keywords-integration methods, noise analysis, computed aided design, sotchastic differential equations, dynamic integrated circuits I. INTRODUCTION Noise limitations are a fundamental issue for robust circuit design and its evaluation has been subject of numerous studies [1]. Correspondingly, a number of CAD tools have been suggested [2]-[3]. The most important types of electrical noise sources (thermal, shot, and flicker noise) in passive elements and integrated circuit devices have been investigated extensively, and appropriate models have been derived [1] as stationary and in [4] as non-stationary noise sources. The noise performance of a circuit can be analyzed in terms of small- signal equivalent circuits by considering each of the uncorrelated noise sources in turn and separately computing their contribution at the output. Unfortunately, this method is only applicable to circuits with fixed operating points and is not appropriate for noise simulation of circuits with changing bias conditions. A widespread approach for noise simulation in the time domain is Monte Carlo analysis. However, accurately determining the noise content requires a large number of simulations, so consequently, the Monte Carlo method becomes very cpu-time consuming if the chip becomes large. As a result, several methods, such as variance reduction techniques including importance sampling, stratified sampling, correlated sampling, and regression sampling are reported to improve the precision of the statistical estimate with a smaller set of random simulations. Similarly, statistical regression techniques such as response surface modeling have been applied to further reduce the number of random simulations. Nevertheless, the computational cost still remains high for large-scale circuits. In this paper, we treat the noise as a non-stationary stochastic process, and introduce an Itô system of stochastic differential equations (SDE) as a convenient way to represent such a process. We adapt model description as defined in [4], where thermal and shot noise are expressed as delta-correlated noise processes having independent values at every time point, modeled as modulated white noise processes. These noise processes correspond to current noise sources which are included in the models of the integrated-circuit devices. As numerical experiments suggest that both the convergence and stability analyses of adaptive schemes for SDEs extend to a number of sophisticated methods which control different error measures, we follow the adaptation strategy, which can be viewed heuristically as a fixed time-step algorithm applied to a time re-scaled differential equation. Additionally, adaptation also confers stability on algorithms constructed from explicit time-integrators, resulting in better qualitative behavior than for fixed time-step counter-parts [5]. Similarly, recognizing that the variance-covariance matrix when backward Euler is applied to such a matrix can be written in the continuous-time Lyapunov matrix form, we then provide a numerical solution to such a set of linear time-varying equations. II. STOCHASTIC MNA FOR TIME-DOMAIN NOISE ANALYSIS In general, for time-domain analysis, modified nodal analysis (MNA) leads to a nonlinear ordinary differential equation (ODE) or differential algebraic equation (DAE) system which, in most cases, is transformed into a nonlinear algebraic system by means of linear multistep integration methods [6]-[7] and, at each integration step, a Newton-like method is used to solve this nonlinear algebraic system. Therefore, from a numerical point of view, the equations modeling a dynamic circuit are transformed to equivalent linear equations at each iteration of the Newton method and at each time instant of the time-domain analysis. Thus, we can say that the time-domain analysis of a nonlinear dynamic circuit consists of the successive solutions of many linear circuits approximating the original (nonlinear and dynamic) circuit at specific operating points. Consider MNA and circuits embedding, besides voltage- controlled elements, independent voltage sources, the remaining types of controlled sources and noise sources. Combining Kirchhoff’s Current law with the element characteristics and using the charge-oriented formulation yields a stochastic differential equation of the form