(untitled) (Eureka Journal of Education & Learning Technologies, 2026)
Early identification of technical states in complex mechanical systems plays a crucial role in maintaining industrial safety and operational performance. Signals from strain gauges and vibration sensors typically exhibit multiscale, non-stationary, and stochastic traits, posing challenges for reliable detection via traditional spectral techniques. This research develops an advanced mathematical framework for signal diagnostics employing discrete wavelet transformation. The signal is modeled as a generative process within Hilbert space via stochastic differential equations, with impulsive faults represented through Dirac delta functions and jump-diffusion processes. Rigorous theorems validate the wavelet operator’s frame properties, energy invariance, and superior impulse localization. Wavelet-domain Neumann-Pearson and Bayesian detectors are derived demonstrating exponential decay in false alarm rates per large deviation theory. Numerical simulations affirm the method’s exceptional sensitivity at low signal-to-noise ratios. It establishes a versatile mathematical foundation for predictive maintenance and intelligent industrial monitoring systems.