Robust extraction of spatial correlation

Increased variability of process parameters and recent progress in statistical static timing analysis make extraction of statistical characteristics of process variation and spatial correlation an important yet challenging problem in modern chip designs. Unfortunately, existing approaches either focus on extraction of only a deterministic component of spatial variation or do not consider actual difficulties in computing a valid spatial correlation function and matrix, simply ignoring the fact that not every function and matrix can be used to describe the spatial correlation. Based upon the mathematical theory of random fields and convex analysis, in this paper, we develop (1) a robust technique to extract a valid spatial correlation function by solving a constrained nonlinear optimization problem; and (2) a robust technique to extract a valid spatial correlation matrix by employing a modified alternative projection algorithm. Our novel techniques guarantee to extract a valid spatial correlation function and matrix that are closest to measurement data, even if those measurements are affected by unavoidable random noises. Experiment results based upon a Monte-Carlo model confirm the accuracy and robustness of our techniques, and show that we are able to recover the correlation function and matrix with very high accuracy even in the presence of significant random noises.