Path-Based Delay Variation Models for Parallel-Prefix Adders

Abstract

State-of-the-art static timing analysis algorithms can evaluate worst-case delay in statistical terms. In this paper, a modeling framework is introduced for the evaluation of the maximum-delay Cumulative Density Function (CDF) of an ensemble of parallel-prefix adder topologies. For moderate variations and close-to-nominal supply voltages, the maximum delay of parallel-prefix adders is practically determined by the maximum of a set of near-critical-delay paths around the nominal maximum-delay path. These paths end to the most significant and neighboring bit positions. Matrix-based path delay formulations are derived for the particular set of paths. The introduced matrix formulations are exploited to assess the maximum-delay CDF by means of a multivariate Gaussian CDF. To validate the accuracy of the introduced models, a quantitative comparison of the proposed probabilistic delay models against Spice-level Monte-Carlo simulations is offered for certain parallel-prefix adders. Threshold-voltage variations summarize several process-dependent variation-inducing mechanisms and are modeled as Gaussian variations, introduced to BSIM-4 transistor models for a 16-nm technology node. For the nominal voltage case and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10\%$</tex-math></inline-formula> threshold-voltage variations, the introduced models estimate the 0.95 timing yield point with a mean absolute error below <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1\%$</tex-math></inline-formula> compared to Spice-level simulations for the 16 bit-length case. Furthermore, an extension of the proposed approach to account for multiple end points is investigated that reduces the error for the estimation of maximum delay, demonstrated for a unit delay model and certain bit-lengths of Kogge-Stone adder.