Ali Exponential–Difference wavelet for robust signal denoising and localized anomaly detection under contaminated stochastic environments

Wavelet-based signal analysis has become a fundamental framework for multiscale processing because of its ability to jointly characterize temporal and spectral information. However, many classical wavelet families exhibit limited robustness when signals are contaminated by impulsive noise, heavy-tailed disturbances, or highly localized transient anomalies. This study introduces a novel continuous mother wavelet, the Ali Exponential–Difference Wavelet, which is analytically constructed from the difference between two exponentially decaying kernels with distinct localization parameters. Unlike conventional wavelets generated through recursive filter-bank structures, the proposed wavelet possesses a closed-form analytical representation that facilitates rigorous mathematical analysis and efficient implementation. Its theoretical validity is established by verifying the principal admissibility conditions, including zero mean, finite energy, square integrability, and a finite admissibility coefficient. The corresponding Fourier-domain representation demonstrates that the proposed wavelet functions are localized band-pass analyzing functions with strong transient sensitivity and smooth spectral attenuation. Building upon these theoretical properties, a robust multiscale framework for signal denoising and anomaly detection is developed using adaptive thresholding together with median absolute deviation-based wavelet-energy standardization. Extensive Monte Carlo simulations under Gaussian, Laplace, Student’s [Formula: see text], and impulsive contamination demonstrate competitive reconstruction accuracy, enhanced preservation of localized transient structures, and improved anomaly localization compared with the Haar, Daubechies-4, Symlet-4, Coiflet-1, and Mexican Hat wavelets. The practical applicability of the proposed framework is further demonstrated using electrocardiogram signals from the Massachusetts Institute of Technology–Beth Israel Hospital Arrhythmia Database, where it effectively preserves clinically important waveform characteristics while accurately identifying abnormal cardiac intervals.