Hierarchical bandwidth-adaptive variational mode decomposition: algorithms and applications

Nonlinear and non-stationary signal analysis is often affected by mode mixing and limited parameter adaptivity. To address these issues, a hierarchical bandwidth-adaptive variational mode decomposition method, termed HBVMD, is proposed. HBVMD uses a top-down binary recursive strategy to divide the signal into low- and high-frequency components. At each node, a binary variational model is constructed. The recursion termination condition and decomposition depth are determined by a time-varying mutual information criterion and a modal energy threshold. Thus, the number of intrinsic modes does not need to be predefined. A dynamic dual-bandwidth penalty scheme is then introduced. This scheme applies stronger narrowband regularization to low-frequency components and allows transient-aware bandwidth relaxation for high-frequency components. The augmented Lagrangian step size is adaptively updated according to the local signal-to-noise ratio. Experiments on bearing fault datasets show that HBVMD achieves high diagnostic accuracy and reliability. The method performs particularly well in composite fault detection and feature extraction. These results indicate that HBVMD provides a robust and interpretable framework for industrial fault diagnosis.