A framework for statistical timing analysis using non-linear delay and slew models

Abstract

In this paper 1 we propose a framework for Statistical Static Timing Analysis (SSTA) considering intra-die process variations. Given a cell library, we propose an accurate method to characterize the gate and interconnect delay as well as slew as a function of underlying parameter variations. Using these accurate delay models, we propose a method to perform SSTA based on a quadratic delay and slew model. The method is based on efficient dimensionality reduction technique used for accurate computation of the max of two delay expansions. Our results indicate less than 4% error in the variance of the delay models compared to SPICE Monte Carlo and less than 1% error in the variance of the circuit delay compared to Monte Carlo simulations.

Adaptive Stochastic Collocation Method (ASCM) for Parameterized Statistical Timing Analysis with Quadratic Delay Model

“…In order to represent the spatial correlated intra-die process variations, a set of common random variables with independent identical normal distribution are employed using either PCA [3] or KL expansion [7]. As a consequence, the delays of gates and interconnects for SSTA, as well as the arrive times (ATs), can be modeled as parametric functions with respect to these common random variables.…”

“…The nonlinear delay models have been proposed recently to replace the linear canonical delay model (LCDM) for modeling the nonlinear affect of process variations on gate or interconnect delays and the arrive times (ATs) generated during the delay propagation in a circuit. These nonlinear delay models are the quadratic polynomials based on either the Tay-lor expansion [3][4][5] or the Homogeneous Chaos (Wiener-Askey Scheme) expansion (HCE) [7,8]. Compared with the Taylor expansion, Homogeneous Chaos expansion has exponential convergence rate, and is more promising for large process variations [9].…”

“…Compared with the Taylor expansion, Homogeneous Chaos expansion has exponential convergence rate, and is more promising for large process variations [9]. Some collocation methods, such as efficient collocation method in [8] and stochastic collocation method based on the Smolyak grid interpolation in [7], have been proposed for computing the unknown coefficients of the delay model with HCE.…”

“…Contrary to the MMM which only matches the first two order moments of the MAX, stochastic collocation method approximates the MAX problem in the full random space by minimizing the mean square error as shown later in this paper. SCM is firstly proposed in [7] for MAX approximation and also proposed in [11] for capacitance extraction under process variations. In [7], the SCM adopts quadratic delay model with HCE.…”

“…SCM is firstly proposed in [7] for MAX approximation and also proposed in [11] for capacitance extraction under process variations. In [7], the SCM adopts quadratic delay model with HCE. In order to approximate MAX with a HCE based quadratic polynomial, the stochastic Galerkin method is adopted in [7] for computing the unknown coefficients of the polynomial by solving a set of multi-dimensional integrations.…”