Short-time variational mode decomposition

Variational mode decomposition (VMD) and its extensions like Multivariate VMD (MVMD) decompose signals into ensembles of band-limited modes with narrow central frequencies using Fourier transformations. However, since these transformations span the entire time-domain signal, they are suboptimal for analyzing non-stationary time series. We introduce Short-Time Variational Mode Decomposition (STVMD), an innovative extension of VMD that incorporates Short-Time Fourier transform (STFT) to minimize the impact of local disturbances. STVMD segments signals into short time windows and converts these segments into the frequency domain. It then formulates a variational optimization problem to extract band-limited modes representing the windowed data. The optimization aims to minimize the sum of mode bandwidths across the windowed data, extending the cost functions used in VMD and MVMD. Solutions are derived using the alternating direction method of multipliers, ensuring extraction of modes with narrow bandwidths. STVMD is divided into dynamic and non-dynamic types, depending on whether central frequencies vary with time. Our experiments show non-dynamic STVMD matches VMD with properly sized time windows, while dynamic STVMD better accommodates non-stationary signals through reduced mode function errors and tracking of dynamic frequencies. This effectiveness is validated using steady-state visual-evoked potentials in electroencephalogram signals.