Swept Sine Analysis Phase Correction via Chebyshev Unmixing
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Solution Overview
Problem
Existing swept-sine analysis techniques for nonlinear systems suffer from phase spreading issues, leading to inaccurate measurements due to the logarithmic swept-sine signal, which masks the true-phase response and introduces cross harmonic energy, complicating the analysis and modeling of nonlinear systems.
Innovation Solution
The proposed method involves applying a test signal with a varying frequency sweep to a nonlinear system, constructing a model, inverting and convolving matrices to generate a phase-true response, and using a Chebyshev unmixing matrix to eliminate extraneous cross components, resulting in harmonically pure and phase-true diagonal Volterra filters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a logarithmic swept-sine signal is used for nonlinear system analysis, then the analysis can be performed efficiently with a single test signal, but phase spreading occurs that masks the true-phase response and introduces cross harmonic energy
Solution Approach 1:
The patent segments the contaminated harmonic responses by applying a Chebyshev unmixing matrix to separate cross harmonic energy components. This matrix segmentation allows extraction of pure diagonal Volterra kernel coefficients from the mixed responses, resolving the phase spreading issue while maintaining analysis efficiency.
Solution Approach 2:
The patent introduces an intermediary processing stage involving matrix inversion and convolution operations. The Chebyshev unmixing matrix acts as an intermediary that transforms the contaminated responses into pure harmonic components, enabling accurate phase response measurement without requiring multiple test signals.
2Reliability
If the full Volterra model is used to characterize nonlinear systems, then complete system characterization is achieved, but memory and computation requirements become prohibitive
Solution Approach 1:
The patent extracts only the diagonal elements from the full Volterra model, creating a diagonal Volterra model that captures the essential nonlinear characteristics while eliminating the computationally prohibitive off-diagonal elements. This extraction maintains sufficient accuracy for practical applications while dramatically reducing complexity.
Solution Approach 2:
The patent applies partial action by considering only the necessary diagonal components of the Volterra kernels rather than the complete model. This partial characterization provides sufficient accuracy for most practical nonlinear system analysis while avoiding the excessive computational burden of the full model.
3Device complexity
If diagonal Volterra modeling is used to reduce data content, then computational complexity is reduced, but accuracy and precision of results are reduced
Solution Approach 1:
The patent applies feedback through the matrix inversion process, where the inverted response matrix is convolved with the Cheb unmixing matrix to refine the harmonic response estimates. This feedback mechanism compensates for the precision loss inherent in diagonal modeling by iteratively improving the coefficient estimates.
Solution Approach 2:
The patent performs preliminary action by pre-computing the Chebyshev unmixing matrix and applying it to the diagonal Volterra model coefficients. This preliminary transformation prepares the data structure to maximize the precision benefits of diagonal modeling while minimizing information loss.
Data Source
AI summary
A software application uncovers phase spreading that corrupts the linear response filters of a nonlinear system and executes a corrective algorithm that generates true-phase filters of the response over all nonlinear orders used in the analysis. The application first outputs a signal with logarithmically changing frequency, as a stimulus to generate a set of Farina harmonic response filters. The application then constructs an identity model and excites the identity model with an identical input under identical conditions, to generate a set of spreading function filters. The application computes the inverse of the spreading function filters and convolves the inverse spreading function filters with the Farina harmonic response filters to generate a set of spreading corrected filters. Finally, the application constructs the unmixing Chebyshev matrix and convolves the spreading correction filters with the unmixing Chebyshev matrix to remove the spreading and generate harmonically pure and phase-corrected filters.