Robust extraction of spatial correlation

Abstract

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.

Exploiting correlation kernels for efficient handling of intra-die spatial correlation, with application to statistical timing

“…These variations often exhibit strong spatial correlation. Standard grid-based models assume model parameters (grid-size, regularity) in an ad hoc manner and can have high measurement cost [1]. The random field model [1][2] overcomes these issues.…”

“…[15] tries to answer the last question by proposing a Bayesian learningbased approach to estimating the correlation coefficients from measurements. Recently, [1] presented the criteria for a physically valid spatial correlation model and showed the practical problems with using a grid-based model. The paper proposed representing the random variations for each parameter (e.g.…”

“…The paper proposed representing the random variations for each parameter (e.g. V t , t ox ) as a two-dimensional random field, and provided a construction for a valid correlation model in the form of an isotropic correlation function (correlation kernel) K´x yµ 1 . K´x yµ returns the correlation between any two locations (x, y) on the chip.…”

“…[2] views the chip-wide variation as a stochastic process and proposes using the Karhunen-Loéve Expansion (KLE) to extract a small set of uncorrelated random variables, which can then be used as parameters for the gate timing models. However, the approach is restricted to a theoretically simple, but physically unrealistic, correlation kernel, and no generic method is proposed to handle any realistic kernel extracted from process data (e.g., as per [1]). …”

“…Standard grid-based models assume model parameters (grid-size, regularity) in an ad hoc manner and can have high measurement cost [1]. The random field model [1][2] overcomes these issues. However, no general algorithm has been proposed for the practical use of this model in statistical CAD tools.…”