Frequency Fourier and Excitation Module (FFEM)

• FFEM is a computational methodology that decomposes arbitrary excitation signals into Fourier components to facilitate algebraic synthesis of system responses.
• It is applied across domains such as non-ideal capacitive modeling, transformer vision for flare removal, and quantum dynamics via Floquet-Fourier expansion.
• The framework bypasses differential equations using Fourier synthesis, enabling precise extraction of excitation metrics and energy partitioning.

The Frequency Fourier and Excitation Module (FFEM) is a computational and algorithmic approach for extracting, transforming, and analyzing the frequency-domain response of systems subject to arbitrary excitations. FFEM is implemented in distinct domains, notably for modeling nonlinear energy storage devices (Allagui et al., 2023), as a spectral-context global module in transformer-based vision models (Zhu et al., 17 Dec 2025), and as a multiharmonic driver for quantum systems in the openMMF library (Sinuco-León, 2020). Its common foundation is the decomposition of excitation signals into Fourier components, system modeling via frequency-domain transfer functions, and synthesis of time-domain response and excitation metrics.

1. Mathematical Foundation: Fourier-Based Decomposition and System Response

At the core of FFEM is the representation of arbitrary input signals—periodic, quasi-periodic, or sufficiently sampled random—as discrete Fourier series or transforms. Let $v_i(t)$ denote the input, then

$$v_i(t) = \sum_{n=-\infty}^{\infty} V_n\,e^{j\,n\,\omega_0\,t}$$

where $V_n$ are the complex Fourier coefficients and $\omega_0$ is the fundamental frequency. The system’s frequency-domain representation, be it an impedance $Z(j\omega)$ for electrochemical devices (Allagui et al., 2023), a quantum Hamiltonian block structure (Sinuco-León, 2020), or a vision model’s spectral features (Zhu et al., 17 Dec 2025), interacts linearly with each harmonic. By superposition, the total response is the sum of per-harmonic responses governed by algebraic transfer functions. This framework allows for complete time-domain reconstruction, bypassing the need for explicit numerical integration of differential equations—a principle leveraged in all FFEM deployments.

2. FFEM in Electrochemical Device Modeling: Rs–CPE Circuits and Time-Domain Synthesis

In Allagui et al. (Allagui et al., 2023), FFEM provides an algebraic procedure to estimate the time-domain current, voltage, instantaneous power, and energy of non-ideal capacitive systems. For supercapacitor models, the system is represented as $Z_{\mathrm{tot}}(s) = R_s + Z_{\mathrm{CPE}}(s)$, with the constant phase element $Z_{\mathrm{CPE}}(s) = 1/(C_\alpha s^\alpha)$. The key steps are:

1. Fit $R_s$, $C_\alpha$, $\alpha$ from impedance spectroscopy.
2. Decompose excitation $v_i(t)$ into Fourier coefficients $V_n$.
3. Compute transfer functions $H_v(s), H_i(s)$ for each harmonic.
4. Synthesize time-domain voltage and current via truncated Fourier synthesis:

$$v_c(t) = \sum_{n=-N}^{N} H_v(jn\omega_0) V_n e^{jn\omega_0 t}$$

$$i_c(t) = \sum_{n=-N}^{N} H_i(jn\omega_0) V_n e^{jn\omega_0 t}$$

1. Calculate instantaneous power and total/partitioned energy via direct integration and further Fourier analysis.

This formalism rigorously splits dissipated and stored energy via cosine and sine terms, ensuring high fidelity reconstruction for arbitrary excitation waveforms when the frequency-domain parameters are valid over the excitation band.

3. FFEM in Transformer Vision Models: Spectral Local-Context Attention for Flare Removal

In SLCFormer (Zhu et al., 17 Dec 2025), FFEM is introduced as a global context pathway inside each Global-Local Transformer Block (GLTB), replacing traditional attention mechanisms. Its structure is as follows:

• Input tensor $X \in \mathbb{R}^{B \times C \times H \times W}$.
• Point-wise $1 \times 1$ convolution projects channels; split into local and global halves.
• Local branch: token-wise MLP and dilated convolution.
• Global branch: 2D FFT per channel, followed by $1 \times 1$ convolution and nonlinear activations in the frequency domain, batch normalization, and inverse FFT reconstruction.
• Channel fusion via concatenation and further convolutions, followed by a Squeeze-and-Excitation block for adaptive channel scaling.

The mathematical backbone relies on the 2D discrete Fourier transform per image channel:

$$\mathcal{F}(X)[u,v] = \sum_{x=0}^{H-1}\sum_{y=0}^{W-1} X[x,y]\,e^{-2\pi i(\frac{ux}{H}+\frac{vy}{W})}$$

$$\mathcal{F}^{-1}(F)[x,y] = \frac{1}{HW}\sum_{u=0}^{H-1}\sum_{v=0}^{W-1} F[u,v]\,e^{2\pi i(\frac{ux}{H}+\frac{vy}{W})}$$

Ablation studies confirm a measurable performance penalty in PSNR, SSIM, LPIPS, and G/S-PSNR when FFEM is removed, demonstrating its utility in modeling spatially nonuniform flare artifacts through global-frequency domain feature extraction.

4. FFEM in Quantum Dynamics: Floquet–Fourier Expansion and Excitation Computation

In openMMF (Sinuco-León, 2020), FFEM appears as a computational engine for polychromatically driven quantum systems. The problem is cast as follows:

• The time-dependent Hamiltonian is decomposed into static and multiharmonic terms.
• The time-evolution operator is expanded in a multidimensional Fourier (Floquet) series:

$$U_F(t) = \sum_{\vec{n}} \sum_{i,\bar{i}} U_{i,\bar i}^{\vec{n}} e^{-i(\vec{n} \cdot \vec{\omega}) t} |i\rangle \langle\bar i|$$

• The Floquet Hamiltonian is constructed as a block-banded matrix in extended Hilbert space, with each block corresponding to a fixed combination of indices of the fundamental driving frequencies.
• Diagonalization yields generalized dressed states and quasienergies.
• Physical observables—micromotion operators, time- and phase-averaged transition probabilities—are synthesized through algebraic sums over Fourier indices.

This procedure is accessible via Fortran90 object APIs, with corresponding C++ and Python wrappers, allowing for dense and sparse matrix representations and efficient scripting for varied driving spectra.

5. Algorithmic Structures and Software Interfaces

In all cases, FFEM procedures consist of algebraic steps that leverage Fourier analysis and linear system theory. In the SLCFormer model (Zhu et al., 17 Dec 2025), pseudocode explicitly demonstrates the channel-wise splitting, frequency-domain convolution, and excitation via SE blocks; in openMMF (Sinuco-León, 2020), routines such as MULTIMODEFLOQUETMATRIX and MULTIMODETIMEEVOLUTIONOPERATOR form the computational backbone in both dense and sparse formats. In electrochemical modeling (Allagui et al., 2023), responses are built via superposition and frequency transfer functions, requiring only measured impedance parameters and a Fourier-decomposed excitation as input.

Domain	Key FFEM Role	Reference
Supercapacitor modeling	Impedance-based time-domain synthesis	(Allagui et al., 2023)
Vision transformer	FFT-based global context and excitation in GLTB	(Zhu et al., 17 Dec 2025)
Quantum dynamics	Floquet Fourier expansion of time-evolution	(Sinuco-León, 2020)

6. Performance and Practical Considerations

The FFEM approach generalizes well across domains due to its reliance on linearity, superposition, and algebraic synthesis. In practical settings, key assumptions include:

• Appropriateness of the system’s frequency-domain model over the excitation spectrum.
• Truncation of Fourier series; convergence is ensured by fractional-order system responses or spectral roll-off in practice.
• Handling of non-periodic signals via DTFT/FFT for sufficiently sampled random or complex waveforms.
• Interface accessibility: FFEM implementations in openMMF are object-oriented (Fortran90), with scriptable C++ and Python wrappers; transformer vision models employ deep learning infrastructure compatible with contemporary frameworks.

Ablation studies in SLCFormer confirm FFEM’s contribution to quantitative and perceptual gains in vision tasks. In quantum and electrochemical domains, FFEM enables rapid, algebraic computation of excitation metrics, facilitating analysis, prediction, and design for complex, multi-modal systems.

7. Generalization and Significance

FFEM encapsulates a methodology of frequency-domain analysis, algebraic transformation, and excitation metric computation applicable wherever system response to complex drivers is non-trivial and analytically tractable via Fourier synthesis. Its flexibility across physical modeling, machine vision, and quantum simulation highlights its foundational role in computational science and engineering, with interfaces and structures adaptable to diverse signal, stimulation, and response paradigms.