Nonlinear System Characterization via Vandermonde Matrix Inversion
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Solution Overview
Problem
Current techniques for analyzing the linear response of nonlinear systems are limited, often relying on low-level test signals and linear approximations that fail to accurately represent the system across a broader range of operating conditions.
Innovation Solution
A method involving a computing device that applies a series of test signals at full power to a nonlinear system, generating responses, constructing sets to characterize the system, and using a Vandermonde matrix and inverse Vandermonde matrix to separate nonlinear order responses, allowing for the determination of true linear responses and order-separated outputs.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If extremely low level test signals are used to measure linear response, then the measurement avoids nonlinear distortions, but the measurement does not represent the system's linear response under realistic operating conditions
Solution Approach 1:
The patent segments the system response into different nonlinear orders (first-order linear response, second-order nonlinear response, third-order nonlinear response, etc.) using higher-order spectral analysis. By applying multi-tone test signals and analyzing the frequency spectrum, the method separates and identifies each order's contribution, allowing extraction of the linear response component even when the system operates in its full nonlinear regime with realistic full-power signals.
2Adaptability or versatility
If full power input signals are used to characterize nonlinear systems, then the measurement includes all factors contributing to linear response under realistic conditions, but the nonlinear distortions contaminate the linear response measurement
Solution Approach 1:
The patent extracts the linear response component from the total system response by using higher-order spectral analysis. The method applies multi-tone test signals at full power and processes the output spectrum to isolate and extract only the first-order (linear) response component, removing the contamination from higher-order nonlinear distortions. This allows accurate linear response measurement under realistic full-power operating conditions.
3Ease of manufacture
If linear approximations are used to analyze nonlinear systems, then the analysis is mathematically tractable, but the analysis fails to accurately represent the system across a broader range of operating conditions
Solution Approach 1:
The patent changes the analysis parameter from small-signal linear approximations to higher-order spectral analysis that operates directly on full-power signals. By using multi-tone test signals and analyzing the frequency spectrum to identify different nonlinear orders, the method maintains mathematical tractability while accurately representing system behavior across the full range of operating conditions, not just near the linear operating point.
Data Source
AI summary
A software application characterizes a nonlinear system by applying a series of test signals at full power and performing an algorithm on the resulting outputs to determine the true linear response and the order-separated outputs. The application generates a baseline test signal, multiplied by a gain factor. The application inputs the test signal to the system to produce a response. The application then generates another test signal by multiplying the baseline signal by a different gain factor. The application iterates generating test signals by multiplying by differing gain factors. The application then constructs a Vandermonde matrix of the gain factors, computes the inverse Vandermonde matrix, and convolves the inverse Vandermonde matrix with a matrix of the system responses to each of the test signals. The elements of the resulting convolution represent the order-separated outputs including the linear response output of the nonlinear system at full power.