Variogram based Robust Extraction of Process Variation Model

Aggressive device scaling has made it imperative to account for process variations in the design flow. A robust model of process variations is an essential requirement for any meaningful variation aware design analysis and optimization. Unfortunately the previous approaches on extracting spatial correlation function assume ergodicity and isotropy while estimating the inter-die(global) component of variation and spatial correlation function, respectively. We find that making such simplifying assumptions may result in significant estimation error. In this work to address these issues, we propose an alternative approach to extract spatial variation models based on the theory of spatial statistics. The proposed approach uses the concept of variogram function that represents how parameters can covary as a function of spatial distance. The variogram function provides us with a representation that is independent of the global component of variation. This allows us to directly estimate the within die component of variations and thus circumvents the need for making ergodicity assumption. We further show that using two dimensional variogram functions allows us to model geometrically anisotropic process variation data. Additionally, for extracting process variation models in the presence of significant measurement noise, we employ weighted least squares regression technique, which is known to be statistically more robust technique than the previously used ordinary least square technique. Our experimental results on extracting the process variation model model from a Monte-Carlo generated data-set corrupted with significant random noise validates the robustness of the proposed approach.