Operator-Theoretic Energy Functionals for Impulse-Excited Nonstationary Signal Analysis

Abstract

This study presents an operator theoretic framework for defect detection in impulse excited nonstationary systems. Measured responses are modeled as finite energy impulse responses perturbed by stochastic disturbances and represented in the Hilbert space L2(R). Time frequency representations are formulated as bounded linear analysis operators associated with continuous frames, enabling a consistent description of how structural perturbations redistribute transient signal energy. Within this formulation, a nonlinear Energy Concentration Index ECI is introduced to quantify localized transform domain energy over selected regions of the time frequency plane. The boundedness and continuity of the functional ensure that small physical variations in system parameters produce measurable changes in localized energy distribution. This property enables the construction of a statistical separability functional that links multi resolution energy geometry to classification performance. Based on these results, a compact Impulse Based Multi Resolution Energy Detector IMRED is derived. The analysis shows that variations in damping and resonant frequency produce systematic changes in time frequency coefficients and localized energy concentration. Experimental validation using impulse excited ceramic measurements demonstrates that the proposed descriptor captures defect induced structural differences with strong discriminative capability. The resulting IMRED statistic achieves an AUC of 0.908 and provides clearer class separation than global Fourier band energy measures and non optimized wavelet band aggregation. These results establish a direct relationship between impulse response modeling, localized energy geometry, and statistical decision mechanisms, providing a mathematically grounded basis for energy driven defect detection in structural monitoring applications.