Lp-Norm Harmonic Regression Spectral Analyzer
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Solution Overview
Problem
Conventional spectral analysis techniques, such as periodograms and autoregressive models, lack robustness for heavy-tailed data, leading to poor performance due to outlier contamination and statistical variability, especially in detecting and estimating sinusoidal signals in heavy-tailed noise.
Innovation Solution
The method employs Lp-norm regression, specifically L1-norm (least absolute deviations) to derive a novel periodogram, known as the Laplace periodogram, which is less sensitive to outliers and statistical variability, providing improved detection power and accuracy for both light-tailed and heavy-tailed data.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional periodograms and autoregressive models are used for spectral analysis, then the analysis can be performed using standard methods, but the robustness deteriorates for heavy-tailed data due to outlier contamination
Solution Approach 1:
The patent changes the norm parameter from L2 (conventional least squares) to Lp where 0 < p < 2, specifically using L1-norm for least absolute deviations regression. This parameter change in the regression approach makes the spectral estimator robust to outliers and heavy-tailed noise, as the Lp-norm with p<2 reduces the influence of large residuals compared to the standard L2-norm
Solution Approach 2:
The patent substitutes the conventional Fourier transform-based periodogram calculation with an alternative regression-based approach. Instead of directly computing the periodogram via FFT, the method uses least absolute deviations regression to fit sinusoidal models to the data, thereby replacing the mechanical computation path with a more robust statistical estimation procedure
2Productivity
If conventional periodograms are computed using FFT, then the computation is efficient, but the statistical variability increases for heavy-tailed data
Solution Approach 1:
The patent modifies the computational approach by changing from L2-norm minimization to Lp-norm minimization (p<2) in the regression framework. This parameter change leads to more stable spectral estimates with reduced statistical variability for heavy-tailed data, while maintaining computational feasibility through efficient regression algorithms
3Ease of operation
If L2-norm least squares regression is used to derive periodograms, then the method is simple and computationally straightforward, but the detection power decreases for sinusoidal signals in heavy-tailed noise
Solution Approach 1:
The patent changes the norm parameter from p=2 (least squares) to p<2 (least absolute deviations or similar Lp-norm regression). This parameter modification improves detection power for sinusoidal signals in heavy-tailed noise by reducing the influence of outliers on the regression fit, while the method remains relatively simple and can be implemented using standard optimization techniques
Data Source
AI summary
A method of analyzing a spectrum of one-dimensional or multi-dimensional signal X(t) requires a number of steps including deriving coefficients [AN(ω), BN(ω)] of an Lp-norm harmonic regression of tie signal with 0<p≦̸∞ and pγ2, squaring the coefficients, summing the squared coefficients, and scaling the summed, squared coefficients with a constant c to realize a periodogram of X(t) as LN(ω)=c{[AN(ω)]+[BN(ω)]2}. The method may include receiving the signal X(t), storing the received signal X(t), and outputting the periodogram LN(ω). The method may still further include scanning to maximize the periodogram LN(ω) by identifying its largest peak(s) and comparing the amplitude of the identified largest peak(s) with a threshold to determine if the largest peak(s) is(are) attributable to a presence of a periodic signal. The coefficients are preferable derived from a time series signal X(t), t=1, 2, . . . , N, but may include receiving a continuous-time signal and converting it to the time series signal.