Robust extraction of spatial correlation

The increased variability of process parameters makes it important yet challenging to extract the statistical characteristics and spatial correlation of process variation. Recent progress in statistical static-timing analysis also makes the extraction important for modern chip designs. Existing approaches extract either only a deterministic component of spatial variation or these approaches do not consider the actual difficulties in computing a valid spatial-correlation function, ignoring the fact that not every function and matrix can be used to describe the spatial correlation. Applying mathematical theories from random fields and convex analysis' 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 from measurement data, even if those measurements are affected by unavoidable random noises. Experiment results, obtained from data generated by 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.