A Geometric Programming-Based Worst Case Gate Sizing Method Incorporating Spatial Correlation

Abstract

We present an efficient optimization scheme for gate sizing in the presence of process variations. Our method is a worst case design scheme; however, it reduces the pessimism involved in traditional worst case methods by incorporating the effect of spatial correlations in the optimization procedure. The pessimism reduction is achieved by employing a bounded model for the parameter variations in the form of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uncertainty</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ellipsoid</i> , which captures the spatial correlation information between the physical parameters. The use of the uncertainty ellipsoid, along with the assumption that the random variables corresponding to the varying parameters follow a multivariate Gaussian distribution, enables us to size the circuits for a specified timing yield. Using a posynomial delay model, the delay constraints are modified to incorporate uncertainty in the transistor widths and effective channel lengths due to the process variations. The resulting optimization problem is relaxed to a geometric program and is efficiently solved using convex optimization tools. The effectiveness of our robust gate sizing scheme is demonstrated by applying the optimization on the ISCAS'85 benchmark circuits and testing the optimized circuits by performing Monte Carlo simulations to model the process variations. Experimental results show that the timing yield of the robustly optimized circuits improves manifold over the traditional deterministically sized circuits. For the same transistor area, the circuits sized by our robust optimization approach have, on average, 12% fewer timing violations as compared to the gate sizing solutions that are obtained via the traditional deterministically based guard-banding method.