Three-dimensional sparse random mode decomposition: From theory to application

Non-stationary signal decomposition faces significant challenges when handling modes with crossover instantaneous frequencies. While sparse random mode decomposition (SRMD) offers a novel approach through stochastic time-frequency representations, its two-dimensional framework struggles to disentangle overlapping frequency components. Conversely, the chirplet transform (CT) introduces a three-dimensional time-frequencychirp rate (TFCR) space to separate such components but suffers from reconstruction inaccuracies due to blurring effects. To address these limitations, this paper proposes a three-dimensional sparse random mode decomposition (3D-SRMD) method that combines SRMD with CT technique. In 3D-SRMD, the random features are lifted from a two-dimensional plane to a three-dimensional (3D) space by introducing one extra chirp rate axis. This enhancement provides an intuitive means of disentangling the frequency components overlapped in the low dimension. A novel random feature generation strategy is further designed to improve approximation accuracy and enhance mode separation capability by combining the 3D ridge detection method. Theoretical analysis reveals the separability of crossover components and derives an approximation bound for the proposed 3D sparse random feature model. Numerical experiments demonstrate the method's superiority over state-ofthe-art techniques in decomposing nonlinear and crossover frequency-modulated modes. This work bridges the gap between theoretical interpretability and practical effectiveness in handling complex multi-component signals.