EDA Cell Library Preconditioning for Nonlinear Timing Models
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Solution Overview
Problem
Conventional characterized cell libraries face challenges in accurately modeling performance parameters, particularly for smaller devices with nonlinear behavior, due to excessive computational and storage requirements, and limitations of high-order polynomials and piecewise polynomials.
Innovation Solution
Preconditioning of operational parameters using functions like exponential, logarithmic, and rational functions to transform nonlinear patterns, allowing for improved modeling accuracy without increasing the complexity or size of the mathematical model, by applying preconditioning functions to operational parameters before using polynomial models.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If high-order polynomials or piecewise polynomials are used to model nonlinear device behavior, then modeling accuracy is improved, but computational complexity and storage requirements increase excessively
Solution Approach 1:
The patent transforms the original operational parameters through mathematical transformations (logarithmic, exponential, or power function preconditioning) before applying polynomial modeling. This changes the parameter space to one where lower-order polynomials can achieve the same modeling accuracy, thereby reducing computational complexity while maintaining precision.
Solution Approach 2:
The patent introduces preconditioning functions as intermediary transformations between the original operational parameters and the polynomial model. These intermediary functions (logarithmic, exponential, or power transformations) serve as a bridge that linearizes or simplifies the nonlinear relationships, enabling accurate modeling with simpler polynomials.
2Measurement precision
If high-order polynomials or piecewise polynomials are used to model nonlinear device behavior, then modeling accuracy is improved, but storage requirements increase excessively
Solution Approach 1:
By transforming operational parameters through preconditioning functions before polynomial fitting, the patent reduces the number of polynomial terms needed to achieve accurate modeling. This parameter transformation approach compresses the information representation, thereby reducing storage requirements while maintaining modeling fidelity.
Solution Approach 2:
The preconditioning functions act as intermediaries that restructure the data representation. By applying logarithmic, exponential, or power transformations to operational parameters before polynomial modeling, the patent creates a more compact representation that requires less storage space while preserving all necessary information for accurate device behavior prediction.
3Device complexity
If conventional polynomial models are used for smaller devices with nonlinear behavior, then device complexity is reduced, but modeling accuracy deteriorates
Solution Approach 1:
The patent applies preliminary preconditioning transformations to operational parameters before polynomial modeling. This preliminary action (applying logarithmic, exponential, or power functions) prepares the data in advance to reveal underlying linear or simpler nonlinear relationships, enabling accurate modeling with low-complexity polynomials that would otherwise fail to capture device behavior.
Solution Approach 2:
The patent changes the representation of operational parameters through mathematical transformations before polynomial fitting. This parameter change strategy converts difficult-to-model nonlinear relationships into forms that can be accurately captured by simple polynomials, thereby achieving high modeling accuracy without increasing device complexity.
Data Source
AI summary
A characterized cell library for EDA tools includes one or more mathematical models for each cell, and one or more preconditioning functions (and/or inverse preconditioning functions) for each mathematical model. Each mathematical model represents a performance parameter (e.g., delay, power consumption, noise) or a preconditioned performance parameter of the cell. The preconditioning functions convert an operating parameter (e.g., input slew, output capacitance) associated with the performance parameter into a preconditioned input variable for the mathematical models. In doing so, the preconditioning functions allow for more accurate modeling of complex data relationships without increasing the complexity (e.g., order and number of coefficients) of the mathematical models. Also, because the cell library can be substantially similar to conventional polynomial-based cell libraries except for the inclusion of preconditioning functions, preconditioning does not significantly increase storage requirements and conventional EDA tools can be readily adapted to use the preconditioned cell library.