The Method for Best Basis Selection in the case of Non-Dyadic Wavelets

Wavelet analysis has established itself as a robust and highly effective framework for the processing and characterization of non-stationary signals. While classical dyadic wavelet transforms are widely utilized due to their computational efficiency, non-dyadic (rational) wavelet transforms often provide a superior representation of signal singularities and complex oscillatory patterns. The proliferation of diverse wavelet functions necessitates a systematic approach to basis selection, which remains a critical task for maximizing feature extraction capabilities. This paper investigates fundamental approaches for evaluating the efficiency of wavelet bases, focusing on criteria derived from the energy distribution of decomposition coefficients, the similarity between the wavelet coefficients and the original signal, and mutual information metrics. The applicability and mathematical robustness of these evaluation methods are specifically examined in the context of non-dyadic wavelet transforms. To validate the investigated methodologies, an additive two-harmonic test signal is employed, subjected to four distinct types of interference (additive white Gaussian, impulse, pink, and multiplicative noise) under varying signal-to-noise ratios. Finally, a comprehensive Composite Quality Index (CQI) is proposed. By aggregating the considered energetic and information-entropic characteristics, this index provides a reliable criterion for selecting the optimal non-dyadic wavelet basis for specific signal processing tasks.