Variational Mode Decomposition (VAFER)

• VAFER is a method that automatically determines the number of intrinsic mode functions and their center frequencies from a signal's real amplitude spectrum.
• It uses a convex optimization approach to balance maximal spectral coverage with minimal baseline roughness, eliminating subjective parameter choices in VMD.
• Validated on narrow-banded signals, VAFER enhances reproducibility and accuracy in applications like radar interference suppression and adaptive filtering.

Variational Mode Decomposition with Automatic Frequency and Mode Number Estimation (VAFER) refers to a methodology that augments classical variational mode decomposition (VMD) by systematically determining the number of intrinsic mode functions (IMFs) and their associated center frequencies directly from the signal spectrum. The VAFER acronym specifically denotes the "Variational Automatic Frequency and mode number Estimation Routine" as presented by Zhong et al. (Zhong et al., 4 Jan 2026), though the term has also been used to indicate VMD-based cascaded pipelines in other signal processing contexts, such as radar interference suppression (Gaur et al., 2022). This article focuses on the core VAFER principle: globally convergent, spectrum-driven selection of IMFs for VMD.

1. Mathematical Formulation

The central objective of VAFER is to extract the minimal set of center frequencies and associated IMFs required to accurately reconstruct a real-valued, narrow-banded signal $f(t)$. Classically, VMD requires the user to specify a mode count $K$ and initial guesses for their frequencies, which introduces uncertainty and susceptibility to subjective choices. VAFER circumvents this through a convex variational optimization in the frequency domain.

Let $S(\omega)$ denote the one-sided amplitude spectrum of $f(t)$ for $\omega\in[0,\Omega_\text{max}]$. VAFER seeks a nonnegative, globally smooth "support-baseline" function $g(\omega)$, lying below $S(\omega)$ and conforming to two objectives:

• $g(\omega)\leq S(\omega)$ for all $\omega$ (enforced pointwise).
• $g(\omega)$ is maximized in $L^1$ norm but minimized in roughness ($L^2$ norm of its second derivative).

This is formalized as the following convex optimization:

$$\min_g J[g]=\int_{\Omega} \alpha(\omega)\left|g''(\omega)\right|^2\,d\omega - \int_{\Omega} \beta(\omega)\,g(\omega)\,d\omega$$

subject to

$0 \leq g(\omega) \leq S(\omega).$

Here, $\alpha(\omega)$ and $\beta(\omega)>0$ are user-chosen regularization parameters. This single objective balances maximal coverage (via $-\beta \int g$) with minimal baseline curvature (via $\alpha \int |g''|^2$).

Lagrange multipliers $\lambda(\omega)\geq 0$ and $\mu(\omega)\geq0$ enforce the constraints in the slack-augmented Lagrangian:

$$L[g,\lambda,\mu] = J[g] + \int \lambda(\omega)\,[g(\omega)-S(\omega)] \, d\omega - \int \mu(\omega)\,g(\omega)\,d\omega.$$

After solving for $g$, the residual spectrum $R(\omega)=S(\omega)-g(\omega)$ is strictly nonnegative and exhibits $K$ separated "humps." VAFER defines the number of intrinsic modes as the number of connected components where $R(\omega)>t_0$ for a properly chosen threshold $t_0$. Center frequencies $\{\omega_k\}$ are selected as the midpoints of these intervals.

2. Numerical Solution and Algorithm

VAFER solves the Euler–Lagrange (EL) equation associated with $L[g,\lambda,\mu]$. This leads to a fourth-order linear ordinary differential equation in $g(\omega)$:

$$2\,\alpha''(\omega)g''(\omega) + 4\alpha'(\omega)g'''(\omega) + 2\alpha(\omega)g^{(4)}(\omega) = \beta(\omega) - \lambda(\omega) + \mu(\omega)$$

subject to boundary conditions matching the behavior of $S$: $g(0)=S(0)$, $g'(0)=S'(0)$, $g(\Omega_\text{max})=S(\Omega_\text{max})$, $g'(\Omega_\text{max})=S'(\Omega_\text{max})$.

The inner-iteration proceeds by discretizing $\omega$ to an $N$-point grid, using finite-difference matrices for all involved derivatives. At step $n$, the linear system for $g^{(n+1)}$ is solved, then dual-ascent updates for the Lagrange multipliers are applied:

• $$\lambda^{(n+1)} = \max\{0,\, \lambda^{(n)} + \theta (g^{(n+1)} - S)\}$$
• $$\mu^{(n+1)} = \max\{0,\, \mu^{(n)} + \gamma(-g^{(n+1)})\}$$

where $\theta, \gamma$ are small step sizes determined by Lipschitz constants of the variation operator. Convergence is taken to $\|g^{(n+1)}-g^{(n)}\|/\|g^{(n+1)}\| < \epsilon$.

After convergence, a kernel density estimate (KDE) is constructed on $\{R(\omega_i)\}$ to locate the most probable value of $t_0$ (dominant mode in the KDE), thus providing robust support–hump separation. The connected intervals $\{\omega:R(\omega)>t_0\}$ define $K$ and the $\{\omega_k\}$.

3. Theoretical Properties and Convergence

VAFER is mathematically guaranteed to find a global optimum for $g(\omega)$ due to convexity of both the functional $J[g]$ and the feasible set $\{0\leq g(\omega)\leq S(\omega)\}$. The duality gap vanishes by standard convex analysis [(Zhong et al., 4 Jan 2026), S.4.2], and provided that the dual ascent step sizes are adequately small, the dual multipliers $(\lambda^n, \mu^n)$ converge to a saddle point. Existence and uniqueness of the solution $g^*$ are also ensured by the boundary-value theory for linear ODEs, and continuous-dependence arguments guarantee algorithmic stability.

The routine typically achieves convergence in $10^3$–$10^4$ iterations, with computational time on the order of seconds for $N=200$ frequency bins.

4. Practical Use and Experimental Results

VAFER is applied to smooth, narrow-banded signals and produces both the number $K$ of IMFs and their initial center frequencies $\{\omega_k\}$. This initialization is then directly transferred to the standard VMD algorithm (i.e., ADMM-based decomposition in the time domain) to extract each mode $u_k(t)$ (Zhong et al., 4 Jan 2026).

Tabulated summary of prototype experiments:

Signal Class	True $K$	VAFER $K$	Max $$|\omega_k^{(\text{VAFER})} - \omega_k^{(\text{VMD})}|$$
Single Sine	1	1	< 0.03
Two Cos/Sin	2	2	< 0.03
Piecewise 4 modes	4	4	< 0.03
11 Superposed Modes	11	11	< 0.03

The validated routines exhibit accurate count and localization for up to $11$ closely spaced modes, as long as the constituent frequency bands remain spectrally separated (minimum $\Delta\omega\sim 0.04$ suffices for $N=200$).

5. Position Within the VMD Ecosystem

VAFER addresses a fundamental challenge in VMD usage: selection of $K$ and frequency initialization. Traditionally, these parameters have been set manually, with nontrivial impact on the quality of decompositions and susceptibility to mode-mixing or under-/overfitting. VAFER provides an automatic, globally convergent mechanism that leverages only the real amplitude spectrum. This bypasses ambiguities of complex-field (analytic) optimizations that have historically plagued mode number estimation (Zhong et al., 4 Jan 2026).

A plausible implication is that VAFER enables nonparametric, reproducible decompositions for broad classes of one-dimensional signals, provided the individual spectral bands are separable. For non–narrow-banded or highly nonstationary signals, performance may depend on spectral structure.

6. Distinction from Other VAFER Usages

The acronym “VAFER” has also appeared as shorthand for sequence pipelines coupling VMD with further time–frequency processing and adaptive selection, such as the VMD–FSST–energy-entropy framework for mutual interference suppression in FMCW radar (Gaur et al., 2022). There, VMD produces modes, FSST sharpens time–frequency localization, and energy–entropy thresholding discards noise-dominated components:

• VMD decomposes the input signal into $K$ narrowband modes.
• FSST generates high-resolution time–frequency representations for each mode.
• Energy–entropy filtering selects tone-like (target) versus diffuse (interference) components.

While this usage is methodologically distinct, it reflects the broader trend toward using adaptive, variationally-optimized mode decompositions as the front-end to subsequent discriminative or filtering stages.

7. Interpretive Comments and Limitations

The theoretical guarantees and robust empirical performance demonstrated by VAFER apply primarily to signals that are adequately modeled as sums of disjoint, narrowband components. When sub-band frequency overlap is substantial, the baseline-separation approach may merge modes or artificially split single wideband IMFs. In such scenarios, parameter choices (e.g., $\alpha, \beta$ weighting) and the resolution of spectral discretization may significantly affect the outcome. For time–frequency nonstationarity, sliding-window VAFER or time–frequency adaptive variants may be required, though such extensions are not described in the cited works (Zhong et al., 4 Jan 2026).

In summary, VAFER establishes a definitive procedure for automatic, globally convergent initialization of VMD mode counts and center frequencies from real-valued signals, with immediate applicability to denoising, adaptive filtering, time-series forecasting, and spectral estimation.