Variational Mode Decomposition

Author: CislunarSpace
Site: https://cislunarspace.cn

Definition

Variational Mode Decomposition (VMD) is an adaptive, non-recursive signal decomposition method that decomposes complex signals into a finite number of Intrinsic Mode Functions (IMF) with sparse spectral characteristics by solving constrained variational problems. Compared to Empirical Mode Decomposition (EMD), VMD has better noise robustness and mathematical theoretical foundation.

Basic Principle

Intrinsic Mode Function (IMF)

Each IMF $u_k(t)$ must satisfy:

Number of extrema and zero-crossings equal or differ by at most one
Mean of upper and lower envelopes is zero at any point

Variational Problem Construction

Constrained Optimization

\min_{\{u_k\}, \{\omega_k\}} \left\{ \sum_{k=1}^{K} ||\partial_t \left[ (\delta(t) + \frac{j}{\pi t}) * u_k(t) \right] e^{-j\omega_k t} ||_2^2 \right\}

Constraint

s.t. \quad \sum_{k=1}^{K} u_k = f(t)

Where $f(t)$ is the original signal.

Parameter Settings

Parameter	Meaning	Typical Value
$K$	Number of modes	3-10
$\alpha$	Penalty parameter	1000-5000
$\tau$	Noise tolerance	0
Convergence tolerance	$\varepsilon$	$10^{-6}$

Applications in Wind Prediction

Wind Signal Decomposition

Original wind speed signal $v(t)$ is decomposed into:

v(t) = \sum_{k=1}^{K} IMF_k(t)

IMF Component	Characteristics	Prediction Method
IMF1	High-frequency turbulence	LSTM/ARIMA
IMF2	Medium-frequency fluctuations	Periodic model
IMF3+	Low-frequency trends	Linear fitting

Particle Swarm Optimization (PSO)
[Long Short-term Memory (LSTM)]/en/glossary/dynamics/lstm-neural-network/)
Trajectory Planning

References

Dragomiretskiy K, Zosso D. Variational Mode Decomposition[J]. IEEE Transactions on Signal Processing, 2024.
Wang Y, et al. VMD-based Wind Speed Prediction for Airship Control[J]. AIAA Journal of Aerospace Systems, 2025.