Process variations have become a critical issue in performance verification of high-performance designs. We present a new, statistical timing analysis method that accounts for inter- and intra-die process variations and their spatial correlations. Since statistical timing analysis has an exponential run time complexity, we propose a method whereby a statistical bound on the probability distribution function of the exact circuit delay is computed with linear run time. First, we develop a model for representing inter- and intra-die variations and their spatial correlations. Using this model, we then show how gate delays and arrival times can be represented as a sum of components, such that the correlation information between arrival times and gate delays is preserved. We then show how arrival times are propagated and merged in the circuit to obtain an arrival time distribution that is an upper bound on the distribution of the exact circuit delay. We prove the correctness of the bound and also show how the bound can be improved by propagating multiple arrival times. The proposed algorithms were implemented and tested on a set of benchmark circuits under several process variation scenarios. The results were compared with Monte Carlo simulation and show an accuracy of 3.32% on average over all test cases.

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Statistical Timing Analysis for Intra-Die Process Variations with Spatial Correlations

Static timing analysis is a critical step in design of any digital integrated circuit. Technology and design trends have led to significant increase in environmental and process variations which need to be incorporated in static timing analysis. This paper presents a new, efficient and accurate block-based static timing analysis technique considering uncertainty. This new method is more efficient as its models arrival times as cumulative density functions (CDFs) and delays as probability functions (PDFs). Computationally simple expression are presented for basic static timing operations. The techniques are valid for any form of the probability distribution, though the use piecewise linear modeling of CDFs is highlighted in this paper. Reconvergent fanouts are handled using a new technique that avoids path tracing. Variable accuracy timing analysis can be performed by varying the modeling accuracy of the piecewise linear model. Regular and statistical timing on different parts of the circuit can be incorporated into a single timing analysis run. Accuracy and efficiency of the proposed method is demonstrated for various ISCAS benchmark circuits.

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Block-based Static Timing Analysis with Uncertainty

The growing impact of within-die process variation has created the need for statistical timing analysis, where gate delays are modeled as random variables. Statistical timing analysis has traditionally suffered from exponential run time complexity with circuit size, due to arrival time dependencies created by reconverging paths in the circuit. In this paper, we propose a new approach to statistical timing analysis which uses statistical bounds and selective enumeration to refine these bounds. First, we provide a formal definition of the statistical delay of a circuit and derive a statistical timing analysis method from this definition. Since this method for finding the exact statistical delay has exponential run time complexity with circuit size, we also propose a new method for computing statistical bounds which has linear run time complexity. We prove the correctness of the proposed bounds. Since we provide both a lower and upper bound on the true statistical delay, we can determine the quality of the bounds. If the computed bounds are not sufficiently close to each other, we propose a heuristic to iteratively improve the bounds using selective enumeration of the sample space with additional run time. The proposed methods were implemented and tested on benchmark circuits. The results demonstrate that the proposed bounds have only a small error, which can be further reduced using selective enumeration with modest additional run time.

Statistical timing analysis using bounds and selective enumeration

The growing impact of within-die process variation has created the need for statistical timing analysis, where gate delays are modeled as random variables. Statistical timing analysis has traditionally suffered from exponential run time complexity with circuit size, due to the dependencies created by reconverging paths in the circuit. In this paper, we propose a new approach to statistical timing analysis which uses statistical bounds and selective enumeration to refine these bounds. First, we provide a formal definition of the statistical delay of a circuit and derive a statistical timing analysis method from this definition. Since this method for finding the exact statistical delay has exponential run time complexity with circuit size, we also propose a new method for computing statistical bounds which has linear run time complexity. We prove the correctness of the proposed bounds. Since we provide both a lower and upper bound on the true statistical delay, we can determine the quality of the bounds. If the computed bounds are not sufficiently close to each other, we propose the use of a heuristic to iteratively improve the bounds using selective enumeration of the sample space with additional run time. The proposed methods were implemented and tested on benchmark circuits. The results demonstrate that the proposed bounds have only a small error, which could be further reduced using selective enumeration with modest additional run time.

A new approach for enhancing the process-variation tolerance of digital circuits is described. We extend recent advances in statistical timing analysis into an optimization framework. Our objective is to reduce the performance variance of a technology-mapped circuit where delays across elements are represented by random variables which capture the manufacturing variations. We introduce the notion of statistical critical paths, which account for both means and variances of performance variation. An optimization engine is used to size gates with the goal of reducing the timing variance along the statistical critical paths. We apply a pair of nested statistical analysis methods deploying a slower more accurate approach for tracking statistical critical paths and a fast engine for evaluation of gate size assignments. We derive a new approximation for the max operation on random variables which is deployed for the faster inner engine. Circuit optimization is carried out using a gain-based algorithm that terminates when constraints are satisfied or no further improvements can be made. We show optimization results that demonstrate an average of 72% reduction in performance variation at the expense of average 20% increase in design area.

/pdf/improving-the-process-variation-tolerance-of-digital-bsfmcm4dp6.pdf

Improving the Process-Variation Tolerance of Digital Circuits Using Gate Sizing and Statistical Techniques

High performance circuit design is becoming increasingly important in VLSI design. The most important problem faced in the design of these circuits is to meet a certain performance level. In the past few years CAD algorithms and tools have been well developed that improve the performance of logic circuits in the sense that the worst case delay is minimized. However, manufacturers recognize that worst case delay models are typically pessimistic and the manufactured ICs will have a range of performances reflecting the manufacturing variations. Thus, the real problem that needs to be solved in the performance optimization of these circuits is to maximize the percentage of fabricated circuits that will achieve a certain performance level, as opposed to minimizing the worst case delay which has been the focus thus far. We develop methods to improve the statistical timing behavior of a combinational logic circuit, given probability distributions for the gate and wire delays. This work uses a statistical timing analysis technique developed earlier to drive timing optimization in the right direction to achieve a prescribed goal with the least area overhead. >

H.-F. Jyu

Papers

Statistical timing optimization of combinational logic circuits