Jump Plus AM-FM Mode Decomposition

Abstract: A novel method for decomposing a nonstationary signal into amplitude- and frequency-modulated (AM-FM) oscillations and discontinuous (jump) components is proposed. Current nonstationary signal decomposition methods are designed to either obtain constituent AM-FM oscillatory modes or the discontinuous and residual components from the data, separately. Yet, many real-world signals of interest simultaneously exhibit both behaviors i.e., jumps and oscillations. Currently, no available method can extract jumps and AM-FM oscillatory components directly from the data. In our novel approach, we design and solve a variational optimization problem to accomplish this task. The optimization formulation includes a regularization term to minimize the bandwidth of all signal modes for effective oscillation modeling, and a prior for extracting the jump component. Our method addresses the limitations of conventional AM-FM signal decomposition methods in extracting jumps, as well as the limitations of existing jump extraction methods in decomposing multiscale oscillations. By employing an optimization framework that accounts for both multiscale oscillatory components and discontinuities, our methods show superior performance compared to existing decomposition techniques. We demonstrate the effectiveness of our approaches on synthetic, real-world, single-channel, and multivariate data, highlighting their utility in three specific applications: Earth's electric field signals, electrocardiograms (ECG), and electroencephalograms (EEG).