Post-Silicon Tunable (PST) buffers are widely adopted in high-performance integrated circuits to fix timing violations introduced by process variations. In typical optimization procedures, the statistical timing analysis of the circuits with PST clock buffers will be executed more than 2000 times for large scale circuits. Therefore, the efficiency of the statistical timing analysis is crucial to the PST clock buffer optimization algorithms. In this paper, we propose a stochastic collocation based efficient statistical timing analysis method for circuits with PST buffers. In the proposed method, we employ the Howard algorithm to calculate the clock periods of the circuits on less than 100 deterministic sparse-grid collocation points. Afterwards, we use these obtained clock periods to derive the yield of the circuits according to the stochastic collocation theory. Compared with the state-of-the-art statistical timing analysis method for the circuits with PST clock buffers, the proposed method achieves up to 22X speedup with comparable accuracy.

Statistical timing analysis for manufacturing variability requires modeling of spatially-correlated variation. Common grid-based modeling for spatially-correlated variability involves a trade-off between accuracy and computational cost, especially for PCA (principal component analysis). This paper proposes to spatially interpolate variation coefficients for improving accuracy instead of fining spatial grids. Experimental results show that the spatial interpolation realizes a continuous expression of spatial correlation, and reduces the maximum error of timing estimates that originates from sparse spatial grids For attaining the same accuracy, the proposed interpolation reduced CPU time for PCA by 97.7% in a test case.

Clock driver suffers from delay variation due to manufacturing and environmental variabilities as well as combinational cells. The delay variation causes clock skew and jitter, and varies both setup and hold timing margins. This paper presents a timing verification method that takes into consideration delay variation inside a clock network due to both manufacturing variability and dynamic power supply noise. We also discuss that setup and hold slack computation inherently involves a structural correlation problem due to common paths, and demonstrate that assigning individual random variables to upstream clock drivers provides a notable accuracy improvement in clock skew estimation with limited increase in computational cost. We applied the proposed method to industrial designs in 90 nm process. Experimental results show that dynamic delay variation reduces setup slack by over 500 ps and hold slack by 16.4 ps in test cases.

In order to improve the performance of the existing statistical timing analysis, slew distributions must be taken into account and a mechanism to propagate them together with delay distributions along signal paths is necessary. This paper introduces Gaussian mixture models to represent the slew and delay distributions, and proposes a novel algorithm for statistical timing analysis. The algorithm propagates a pair of delay and slew in a given circuit graph, and changes the delay distributions of circuit elements dynamically by propagated slews. The proposed model and algorithm are evaluated by comparing with Monte Carlo simulation. The experimental results show that the accuracy improvement in µ+3σ value of maximum delay is up to 4.5 points from the current statistical timing analysis using Gaussian distributions.

In this paper, we propose an Adaptive Stochastic Collocation Method for block-based Statistical Static Timing Analysis (SSTA). A novel adaptive method is proposed to perform SSTA with delays of gates and interconnects modeled by quadratic polynomials based on Homogeneous Chaos expansion. In order to approximate the key atomic operator MAX in the full random space during timing analysis, the proposed method adaptively chooses the optimal algorithm from a set of stochastic collocation methods by considering different input conditions. Compared with the existing stochastic collocation methods, including the one using dimension reduction technique and the one using Sparse Grid technique, the proposed method has 10x improvements in the accuracy while using the same order of computation time. The proposed algorithm also shows great improvement in accuracy compared with a moment matching method. Compared with the 10,000 Monte Carlo simulations on ISCAS85 benchmark circuits, the results of the proposed method show less than 1% error in the mean and variance, and nearly 100x speeds up.

As process technologies decrease below a hundred nanometers, the variability of circuit parameters increases, and statistical timing analysis, which analyzes the distribution of the critical delay of a circuit, is receiving a great deal of attention. In such statistical approaches, correlations between random variables are important to the accuracy of analysis. Since interconnect delays dominate in recent technology, their correlations are of primary concern in statistical timing analysis. In this paper, we propose an efficient algorithm for calculating correlation coefficients between Elmore interconnect delays with the use of Gaussian distributions. Our algorithm is efficient and yields reasonable results for correlations between interconnect delays of different nets. In order to evaluate the performance of the proposed algorithm, we show experimental results compared against Monte-Carlo simulations using SPICE.

Critical path selection is an indispensable step for AC delay test and timing validation. Traditionally, this step relies on the construction of a set of worse-case paths based upon discrete timing models. However, the assumption of discrete timing models can be invalidated by timing defects and process variation in the deep sub-micron domain, which are often continuous in nature. As a result, critical paths defined in a traditional timing analysis approach may not be truly critical in reality. In this paper, we propose using a statistical delay evaluation framework for estimating the quality of a path set. Based upon the new framework, we demonstrate how the traditional definition of a critical path set may deviate from the true critical path set in the deep sub-micron domain. To remedy the problem, we discuss improvements to the existing path selection strategies by including new objectives. We then compare statistical approaches with traditional approaches based upon experimental analysis of both defect-free and defect-injected cases.

This paper proposes a model to calculate statistical gate-delay variation caused by intra-chip and inter-chip variabilities. The variation of each gate delay directly influences the circuit-delay variation, so it is important to characterize each gate-delay variation accurately. Every transistor in a gate affects transient characteristics of the gate, so it is indispensable to consider an intra-gate variability for the modeling of gate-delay variation. This effect is not captured in a statistical delay analysis reported so far. Our model considers the intra-gate variability by sensitivity constants. We evaluate our modeling accuracy, and we show some simulated results of a circuit delay variation.

The main purpose of our method is to obtain realistic worst-case delay in statistical timing analyses. This paper proposes a method of statistical delay calculation based on measured intra-chip and inter-chip variabilities. We present a modeling and extracting method of transistor characteristics for the intra-chip variability and the inter-chip variability. In the modeling of the intra-chip variability, it is important to consider a gate-size dependence by which the amount of intra-chip variation is affected. This effect is not captured in a statistical delay analysis reported so far. Our method proposes a method for modeling of the device variability and statistical delay calculation with consideration of the size dependence, and uses a response surface method (RSM) to calculate a delay variation with low processing cost. We evaluate the accuracy of our method, and we show some experimental results the variation of a circuit delay characterized by the measured variances of transistor currents.

This paper discusses a statistical effect of performance optimization to uncertainty in circuit delay. Performance optimization has an effect of balancing the delay of each path in a circuit, i.e. the delay times of long paths are shortened and the delay times of short paths are lengthened. In these path-balanced circuits, the uncertainty in circuit delay, which is caused by delay calculation error, manufacturing variability, fluctuation of operating condition, etc., becomes worse by a statistical characteristic of circuit delay. Thus, a highly-optimized circuit may not satisfy delay constraints. In this paper, we demonstrate some examples that uncertainty in circuit delay is increased by path-balancing, and we then raise a problem that performance optimization increases statistically-distributed circuit delay.