Quadratic Timing Model for Statistical Circuit Analysis
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Solution Overview
Problem
Current statistical timing analysis methods, particularly block-based STA, fail to accurately account for global and path correlations in integrated circuits, leading to inaccurate timing predictions due to the destruction of correlation information during the MAX operation, which is critical for precise timing analysis, especially in modern high-speed and deep-sub-micron circuits.
Innovation Solution
The introduction of a quadratic timing model that represents signal delays with a more accurate nonlinear relationship between gate/wire delays and global variations, using a quadratic model that includes node sensitivity coefficients, global sensitivity coefficients, and second-order global sensitivity coefficients, along with a spatial correlation model that decays exponentially with distance, to preserve correlation information and improve timing prediction accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If block-based STA uses MAX operation to compute timing, then computational complexity is reduced to linear growth, but correlation information is destroyed leading to inaccurate timing predictions
Solution Approach 1:
The circuit timing analysis is segmented into multiple passes: a first forward pass computes earliest arrival times, a backward pass computes latest arrival times and identifies critical edges, and a second forward pass refines the analysis. This segmentation allows preservation of correlation information through path history tracking while maintaining computational efficiency through localized processing.
Solution Approach 2:
The patent introduces path history as an intermediary mechanism that carries correlation information through the circuit analysis. Each edge maintains a path history set that records predecessor edges, serving as a mediator that preserves statistical correlation data without requiring full circuit-state tracking, thus enabling accurate timing predictions with linear computational complexity.
2Measurement precision
If Monte Carlo simulation is used to compute statistical distribution of edge delays, then accurate timing distribution is obtained, but computational time becomes extremely long and impractical
Solution Approach 1:
The patent performs preliminary computation of mean and variance for each delay element before the main timing analysis. By pre-calculating these statistical parameters and using them in the block-based STA framework with path history tracking, the method achieves Monte Carlo-level accuracy without requiring repeated full-circuit simulations, thus dramatically reducing computational time while maintaining precision.
3Measurement precision
If path-based STA identifies statistically critical paths, then timing constraints are accurately analyzed, but computational complexity grows exponentially with circuit size
Solution Approach 1:
The patent applies local quality by focusing computational resources on critical edges and their immediate path histories rather than analyzing all possible paths globally. The backward pass identifies edges where timing constraints are most sensitive, and the forward pass then computes statistics only for these localized critical regions, maintaining exponential accuracy without exponential complexity growth.
Data Source
AI summary
Statistical timing analysis methods for circuits are described which compensate for circuit elements having correlated timing delays with a high degree of computational efficiency. An quadratic timing model is used to represent each delay element along a circuit path, wherein each element's delay has a first-order relationship to local variations and a second-order relationship to global variations. Propagation of the modeled delays through the circuit is efficiently done via straightforward ADD operations where an input propagates through another element in a circuit path, and via a MAX operation (or an approximation thereof) where two or more inputs merge at an intersection. The inputs to the MAX operator can be tested for gaussianity, and can be processed by the MAX operation (or its approximation) if they are substantially gaussian. Otherwise, they may be stored in a tuple for processing at later points along the circuit path.