Short-time Variational Mode Decomposition

TL;DR

STVMD extends variational mode decomposition by integrating the Short-Time Fourier Transform to perform localized time-frequency analysis on windowed segments, yielding two variants: non-dynamic with fixed central frequencies and dynamic with time-varying central frequencies $\Omega$. The method minimizes the sum of bandwidths across windows and channels using ADMM, enabling accurate decomposition of non-stationary signals and effective tracking of evolving frequencies, as demonstrated on simulated data and real EEG (SSVEP) signals. Compared to VMD/MVMD, STVMD with appropriate windowing achieves comparable performance in stationary regimes and superior tracking in non-stationary scenarios, with dynamic STVMD showing lower reconstruction errors and better mode separation. While offering improved time-frequency localization and adaptability, STVMD introduces higher computational complexity and a trade-off between window length and frequency resolution, suggesting future work on adaptive parameter selection and machine learning-assisted tuning.

Abstract

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

Figures (12)

• Figure 1: The time-frequency spectrum of the input signal of non-dynamic STVMD when the length of Hamming window is 16, 32, 64, 128, respectively.
• Figure 2: Decomposed mode functions with VMD and non-dynamic STVMD. The input is a mixture of 20$Hz$ and 28$Hz$ sinusoid functions. The number of mode functions is 3, and the mode functions whose central frequencies are not zeros are illustrated in the figure. The $N$ refers to the length of the hamming window in STVMD.
• Figure 3: The input two time series with sampling rate 128$Hz$, which is used to validate the mode alignment of non-dynamic STVMD.
• Figure 4: The time-frequency spectrum of the two time series in Figure \ref{['fig:1.2a']}, Channel 1: left column, Channel 2: right column. The spectrum is calculated with STFT and $N$ is the length of time windows in STFT.
• Figure 5: The mode functions decomposed with MVMD and non-dynamic STVMD. $N$ is the length of time window used in the STVMD. Mode functions at frequencies 20$Hz$, 28$Hz$ and 36$Hz$ are given in this figure. The left three columns are decomposed from Channel 1 and the right three columns are decomposed from Channel 2.