SLCFormer: FFT-Based Signal Fusion

• SLCFormer is a deep model that integrates the Frequency Fourier and Excitation Module (FFEM) to process signals using a Fourier basis and excitation pathways.
• It leverages an FFT-based global context module combined with local enhancement for improved image quality and mitigation of spatial artifacts.
• Its framework extends to quantum dynamics, enabling algebraic reconstruction of time-domain responses and efficient simulation of driven systems.

The Frequency Fourier and Excitation Module (FFEM) is a class of modules and algorithmic procedures that leverage the periodic or multi-harmonic structure of signals, excitations, or features by expressing them in a Fourier basis and combining this with an explicit excitation (input) pathway. FFEM serves as a bridge between time/frequency domain signal representations and the computational frameworks—either physical models or deep neural networks—that process these signals for a range of applications, including system identification, quantum dynamics, and frequency-domain feature aggregation in vision models. The term is used non-identically across domains but is characterized by a core methodology: representing signals, dynamical system responses, or high-dimensional features in the Fourier basis, allowing algebraic modeling or efficient context aggregation with input-dependent response modulation.

1. FFEM in Physical System Identification and Impulse Response Analysis

In the context of electrochemical systems, particularly supercapacitors, FFEM refers to a methodology for reconstructing the time-domain response of a non-ideal device (e.g., a supercapacitor modeled as an $R_s$–CPE circuit) to arbitrary excitation signals. Here, the excitation $v_i(t)$ is decomposed into a Fourier series,

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

where $V_n$ are complex Fourier coefficients and $\omega_0$ is the fundamental frequency. The system's transfer functions $H_v(s)$ and $H_i(s)$ (for voltage division and current response, respectively) are parameterized in the frequency domain using measured impedance spectroscopy data: $$H_v(s) = \frac{1}{1 + R_s C_\alpha s^\alpha}, \quad H_i(s) = \frac{C_\alpha s^\alpha}{1 + R_s C_\alpha s^\alpha}$$ For each harmonic $n$, the response to $V_n$ is computed using $H_v(jn\omega_0), H_i(jn\omega_0)$, then responses are summed to reconstruct time-domain quantities such as current, voltage, instantaneous power, and integrated energy. This approach is algebraic and avoids direct numerical solution of differential equations, making it robust for arbitrary periodic or sufficiently sampled non-periodic inputs. Key outputs such as dissipated (resistive) and stored (reactive) energies can be extracted from Fourier analysis of the instantaneous power signal. This method generalizes to any linear, time-invariant system with a known frequency-domain transfer function and can be extended by substituting more complex impedance models as needed (Allagui et al., 2023).

2. FFEM in Deep Visual Transformers: Spectral-Context Aggregation

Within transformer architectures for image processing, FFEM was introduced as a module that replaces costly multi-head self-attention with an explicit fast Fourier transform (FFT)-based global context path, lightweight spatial enhancement, and channel excitation. Specifically, in SLCFormer, FFEM operates on input features $X \in \mathbb{R}^{B \times C \times H \times W}$ and consists of the following pipeline:

• Point-wise projection via $1 \times 1$ convolution.
• Channel split into “local” and “global” branches.
• Local Enhancement Module (LEM): Recovers spatial details using parallel MLP and dilated convolution paths.
• Frequency Context Module (FCM): Applies 2D FFT channel-wise, processes in frequency space with $1 \times 1$ convolutions and activations, then returns to the spatial domain via inverse FFT, with skip connections preserving information.
• Output fusion and a Squeeze-and-Excitation (SE) block for channel reweighting.

Mathematically, the FCM core is: $$F_\text{freq} = \mathcal{F}(F_\text{global}),\; F' = \text{Conv}_{1 \times 1}(F_\text{freq}) + F_\text{freq},\; F'' = \mathrm{GELU}(\text{Conv}_{1 \times 1}(F')),$$

$$G_\text{out} = F_\text{global} + \mathcal{F}^{-1}(\mathrm{ReLU}(\mathrm{BN}(F'')))$$

with $\mathcal{F}$ denoting the 2D FFT. Ablation demonstrates that removing FFEM from the SLCFormer model reduces PSNR, SSIM, and visual perceptual quality metrics, validating its role as a critical context aggregation block in handling spatially variant flare artifacts (Zhu et al., 17 Dec 2025).

3. FFEM Methodology in Multi-Harmonic Quantum Dynamics

The openMMF library illustrates FFEM methodology in quantum system simulation, whereby time-evolution of a system under multi-frequency drive is recast into a multidimensional Fourier (Floquet) expansion. For a Hamiltonian with static and $N$-mode periodic driving,

$$H(t) = \sum_{i,j} E_{ij} |i\rangle \langle j| + \sum_{\ell=1}^{N} \sum_{n \in \mathbb{Z}} V_{ij}^{\ell, n} e^{i n \omega_\ell t} |i\rangle\langle j| + \text{h.c.}$$

The solution ansatz is expansion of the unitary evolution operator in a multidimensional Fourier basis indexed by tuples of harmonic numbers, reducing the original time-dependent Schrödinger equation to a block-banded, time-independent matrix eigenproblem (the Floquet Hamiltonian). This approach allows the computation of dressed energy levels, micromotion operators, and averaged state populations with arbitrary spectral content. The software exposes Fortran, C++, and Python APIs for constructing these frequency-domain system representations, diagonalizing them, and evaluating excitation/transition observables (Sinuco-León, 2020).

4. Computational Algorithmic Structure and Implementation

Across domains, FFEM involves the following computational structure:

FFEM Domain	Decomposition	Processing in Fourier Domain	Output Reconstruction
Physical Modeling	Fourier series	Harmonic-wise transfer fn.	Time-domain voltage, current, power
Visual Transformers	2D FFT	Convolutions/nonlinearities	Feature, spatial-frequency fusion
Quantum Dynamics	Multivariate Fourier	Floquet matrix diagonalization	Micromotion/observable time series

In each instantiation:

• Input signals or features are projected onto a harmonic or frequency basis.
• Each component is subjected to domain-specific transformation (physical transfer function, neural convolution/activation, Hamiltonian diagonalization).
• Outputs are reconstructed by summing over harmonics or using spectral fusion, yielding interpretable physical or learned representations.

Python and Fortran pseudo-code for these pipelines can be found in the referenced source materials, with attention to batch processing, numerical convergence of the Fourier expansion, and residual/skip pathway design.

5. Assumptions, Validity, and Practical Considerations

FFEM methodologies generally assume:

• Linearity and time-invariance: Necessary for convolution/superposition in Fourier space (physical modeling and quantum simulation cases).
• Fourier adequacy: Signal or feature content must be sufficiently captured by a truncated number of Fourier components, with truncation parameters chosen to satisfy convergence based on bandwidth or model cutoffs.
• Model validity: Transfer functions or Hamiltonian parameterizations are assumed valid over the frequency band of interest; mis-modeling leads to physical inaccuracy or feature bias.

For neural FFEM usages, particular concerns include artifacts due to boundary conditions of the FFT, the balance between local enhancement and global aggregation, and the allocation of model capacity between residual, local, and global pathways.

6. Impact, Applications, and Quantitative Evaluation

FFEM approaches provide substantial benefits in their respective domains:

• Physical systems: The algebraic (no ODE integration) computation of non-ideal system responses to arbitrary waveforms enables fast, generalizable characterization and energy/power assessment of devices such as supercapacitors (Allagui et al., 2023).
• Machine vision: FFT-based global context and explicit excitation in SLCFormer yield measurable improvements in PSNR, SSIM, LPIPS, and domain-specific PSNR metrics for nighttime flare removal, with ablation showing non-trivial performance degradation when FFEM is omitted (Zhu et al., 17 Dec 2025).
• Quantum simulation: Multifrequency Floquet analysis enables tractable, general-purpose evaluation of strongly driven, polychromatically excited quantum systems with efficient computation of observables and state populations (Sinuco-León, 2020).

A plausible implication is that FFEM-like modules—by treating signals or features in both frequency and excitation channels—enable efficient, interpretable, and scalable architecture design in both physical simulation and deep learning environments, particularly for problems governed by rich spectral structure or global context dependencies.