Monte Carlo-Alternative Probabilistic Simulations for Analog Systems Rasit Onur Topaloglu University of California at San Diego Computer Science and Engineering Department La Jolla, CA, 92093 rtopalog@cse.ucsd.edu Abstract Probabilistic system simulations for analog circuits have traditionally been handled with Monte Carlo analysis. For a manufacturable design, fast and accurate simulations are necessary for time-to-market, design for manufacturability and yield concerns. In this paper, a fast and accurate proba- bilistic simulation alternative is proposed targeting the sim- ulation of analog systems. The proposed method shows high accuracy for performance estimation combined with a 100- fold reduction in run-time with respect to a 1000-sample Monte Carlo analysis. 1 Introduction Probabilistic simulation for analog systems is a necessity due to increased process variations and mismatch in new technologies. Although probabilistic simulation has long been an interest, today’s requirements necessitate faster and accurate simulation. Most of the time, Gaussian densities for the output pa- rameters are assumed [26]. Although Gaussian assumption might be sufficient for most input parameters, it is far from being accurate of an assumption for the output parameters for most analog blocks. Traditional techniques usually pro- vide the mean and the variance for an output parameter. Yet, capturing the exact shape of a continuous density at the out- put is important for various considerations such as yield es- timation. Speed and accuracy of today’s performance estimation techniques are lagging behind technology. Probabilistic simulation of large blocks brings significant time burdens on system designers. This has triggered us to come up with a technique to bring a solution to single input param- eter probabilistic simulations. Since individual blocks are highly important in analog systems, single parameter sim- ulations will give quite a deal of information. The whole system can then either be evaluated using single parameter as input and multiple parameters as output by selecting the dominant input parameter, or solving the system separately for more input parameters individually, followed by super- position, e.g. as usually done in sensitivity analysis [25]. The proposed estimation methods can be used for optimiza- tion, e.g. design parameter tuning, as well. Most multi- parameter optimization techniques also work on single pa- rameters at a time for improved convergence [20]. We first introduce the basis of the proposed technique, probability discretization and sample propagation imple- mented using a forward operation, where significant run- time improvement over Monte Carlo is gained through weight propagation and systematic sampling. Then, we propose the re-binning algorithm to convert the propagated samples to a continuous density. A traditional spline in- terpolation follows re-binning. Then we provide the per- formance estimation algorithm. We have used behavioral simulations on the jitter analysis of a phase-locked loop to validate the proposed computational technique. 2 Previous Work A number of approaches for probabilistic simulation of analog blocks has been proposed. [12] and [9] have used principal component analysis on the correlation matrix of process variations to reduce the number of variables that are responsible for mismatch. In [23], sensitivity analysis followed by response surface polynomial fitting and Monte Carlo sampling are implemented. [29] has used regression analysis including second order terms. [8] has used variance propagation. Particular interest in probabilistic simulation has been in the area of mismatch and process variation simulation. [21] has used Tailor series expansion. [6] has used sensitivity analysis and assumed that there are at least the same num- ber of output parameters as there are input parameters, all of which are Gaussian. [17] has applied principal component analysis to account for correlations between input param- eters. In [30], we have used hierarchical sensitivities for Proceedings of the 7th International Symposium on Quality Electronic Design (ISQED’06) 0-7695-2523-7/06 $20.00 © 2006 IEEE Authorized licensed use limited to: ADVANCED MICRO DEVICES. Downloaded on September 11, 2009 at 19:47 from IEEE Xplore. Restrictions apply.