  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).
