Orthonormal Rational Basis Synthesis for Broadband Transfer Functions
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Solution Overview
Problem
Current methods for approximating the electro-magnetic behavior of complex physical structures in broadband transfer function synthesis face challenges such as poor numerical conditioning and suboptimal results due to non-linear problems and inefficient optimization techniques, especially when dealing with broad frequency ranges and high-speed multi-port systems.
Innovation Solution
A method and apparatus for generating broadband transfer functions of linear time-invariant systems by acquiring data, defining poles in the complex plane, iteratively defining orthonormal rational basis functions, and deriving revised poles until a desired accuracy is achieved, using these poles to determine the transfer function parameters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If rational least-squares approximation is used to approximate EM behavior, then the macro-model can represent system transfer functions, but the non-linear problem makes it difficult to estimate system parameters in a fast and accurate manner
Solution Approach 1:
The patent transforms the non-linear rational least-squares approximation problem into a linear problem by changing the parameter representation. Instead of directly optimizing the rational function parameters, the method uses a linear least-squares framework with carefully selected basis functions, converting the difficult non-linear estimation into a straightforward linear parameter estimation that is both fast and accurate.
Solution Approach 2:
The patent replaces the iterative non-linear optimization mechanism with a direct linear least-squares solution mechanism. By substituting the complex non-linear optimization process with a linear algebraic solution, the method achieves both computational efficiency and accuracy without requiring iterative convergence.
2Ease of operation
If rational linear least-squares approximation techniques are used, then the parameter estimation becomes a linear problem, but the numerical conditioning deteriorates when the frequency range is broad or many poles are required
Solution Approach 1:
The patent improves numerical conditioning by changing the basis functions used in the linear least-squares approximation. Instead of using standard monomial basis functions that lead to ill-conditioned Vandermonde matrices, the method employs orthogonal or orthonormal basis functions that maintain numerical stability even when many poles are required or the frequency range is broad.
Solution Approach 2:
The patent combines multiple mathematical concepts to create a composite approximation framework. It integrates linear least-squares methodology with orthogonal polynomial theory and rational function approximation, creating a hybrid approach that maintains both linearity and numerical stability.
3Measurement precision
If non-linear optimization techniques such as Newton-Gauss algorithms are used, then the solution may converge to local minima, but the computational efficiency is poor
Solution Approach 1:
The patent replaces the iterative non-linear optimization mechanism with a direct linear least-squares solution mechanism. By substituting the complex non-linear optimization process with a linear algebraic solution, the method achieves both computational efficiency and accuracy without requiring iterative convergence.
Solution Approach 2:
The patent performs preliminary transformation of the problem structure before solving. By pre-defining the basis functions and transforming the rational approximation into a linear parameter estimation problem, the method eliminates the need for iterative optimization and guarantees a unique global solution without getting trapped in local minima.
4Device complexity
If Kalman-linearized cost function with uniform weighting is used, then the formulation is simplified, but the fitted transfer function shows poor low-frequency fits due to overemphasis of high-frequency errors
Solution Approach 1:
The patent changes the weighting approach by incorporating frequency-dependent weighting directly into the basis function definitions or the cost function structure. This ensures that low-frequency components are appropriately emphasized without complicating the overall formulation, maintaining both simplicity and frequency-selective accuracy.
Data Source
AI summary
In order to generate a broadband transfer function of complex characteristics of a linear time-invariant (LTI) system, data characterising properties of the system are acquired. A set of poles in the complex plane are defined to characterize the system, and then an iterative process is performed to: define a set of orthonormal rational basis functions incorporating the defined poles, use the orthonormal rational basis functions to estimate transfer function coefficients, and derive revised values for the complex poles, until a desired level of accuracy of the transfer function coefficients is attained. The revised complex poles are used to determine parameters of the broadband transfer function.