JOT: A Variational Signal Decomposition into Jump, Oscillation and Trend

We propose a two stages signal decomposition method which efficiently separates a given signal into Jump, Oscillation and Trend. While there have been numerous advances in signal processing in past few decades, they mainly aim to analyze the signal in terms of oscillating (underlying frequencies) or non-oscillating (underlying trend) features. Both traditional Time-Frequency analysis methods, like Short Time Fourier Transform, wavelet, and advanced ones, like Synchrosqueezing wavelet, Hilbert Huang Transform or IMFogram, can fail when abrupt changes and jump discontinuities appear in the signal. We present a variational framework separating piece-wise constant jump features as well as smooth trends and oscillating features of a given signal. In the first stage, a three component signal decomposition is applied, using sparsity promoting regularization, and Sobolev spaces of negative differentiability to model oscillations. In the second stage, components are refined using residuals of other components. The proposed method finds big and small jumps, is stable against high level of noise, is independent from the choice of basis functions, and does not have different level of decompositions which can be affected by large discontinuities. This variational framework is free from training in network-based approaches, and can be used for generating training data. The optimization problem is efficiently solved by an alternating minimization strategy. Applied as pre-processing for time-frequency analysis and Synchrosqueezing, it allows for improvements in results showing much clearer separation without artifacts. The proposed method is tested against synthetic data, where the ground truth is known, and real world data.