Multi-Domain Frequency Analysis

• Multi-domain frequency analysis is a method for decomposing signals into distinct frequency subspaces using transforms like DFT, wavelets, and STFT.
• It leverages adaptive band partitioning and hybrid domain architectures to extract robust, latent features from multi-modal data including images, time series, and biomedical signals.
• Empirical studies demonstrate that integrating multi-domain features improves performance and generalization in applications such as vision, fault diagnosis, and wireless communications.

Multi-domain frequency analysis is a general paradigm for extracting, representing, and learning from latent structure in data by explicitly leveraging properties of signals in multiple frequency subspaces or transform domains. This approach is particularly powerful in scenarios involving complex or nonstationary signals, domain generalization across multiple data distributions, multirate systems, biomedical time series, functional data, and modern deep learning for vision, audio, and sensor tasks. Multi-domain frequency analysis systematically decomposes data (e.g., images, time series, vibrations, biomedical signals) into constituent frequency bands, subspaces, or spectral modes and uses these as either direct features for modeling or as a principled foundation for domain-robust augmentation and representation learning. State-of-the-art frameworks encompass classical Fourier or wavelet-based analyses, adaptive data-driven frequency partitioning, hybrid domain network architectures, spectral mode extraction via segmentation, and generalized waveform processing across time–frequency–delay–Doppler–chirp domains.

1. Mathematical Principles of Multi-Domain Frequency Decomposition

At the core of multi-domain frequency analysis is the systematic partitioning of the frequency domain using transform techniques. For image data, the 2D @@@@1@@@@ (DFT) is a canonical operator, transforming an image $x \in \mathbb{R}^{C \times H \times W}$ into the spectral domain:

$$X(u,v) = \sum_{h=0}^{H-1} \sum_{w=0}^{W-1} x(h,w) \, e^{-2\pi i(u h / H + v w / W)}$$

For time series, analogous 1D transforms apply, e.g., $X(f) = \sum_t x(t) e^{-j 2\pi f t}$ (Tu et al., 1 Feb 2025).

Band-specific extraction is achieved by constructing filters in the frequency or time-frequency domain. For images, radial Gaussian band-pass and band-stop filters define spectral slices; for time series and vibration signals, amplitude and phase spectra are separated:

$$F^{\mathrm{pass}}_i(u,v) = \exp\left( -\frac{(\sqrt{u^2 + v^2} - c_i)^2}{2 \sigma_i^2} \right)$$

$$F^{\mathrm{stop}}_i(u,v) = 1 - F^{\mathrm{pass}}_i(u,v)$$

Each sub-band or mode can be reconstructed by applying the inverse transform to the filtered spectrum, yielding $x^{\mathrm{pass}}_i$ and $x^{\mathrm{stop}}_i$ in the image domain, or their equivalents in the time-series domain (Yang et al., 2023).

Extensions to multicomponent or nonstationary signals leverage short-time Fourier transforms (STFTs) and wavelets to analyze time-varying spectra (Zhou et al., 16 Jul 2025, Wang et al., 9 Jan 2026). In control and communication theory, transform-domain representations such as time–frequency (Gabor), delay–Doppler, and chirp/affine domains are unified under generalized pre/post-processing of discrete-time Fourier matrices (Arous et al., 14 Oct 2025).

2. Multi-Domain Learning Architectures and Hybrid Representations

State-of-the-art architectures exploit complementary representations learned from multiple transformed domains. In deep vision models for domain generalization, each frequency sub-band is input independently to parallel branches (e.g., ResNet-18 backbones) with separate learnable parameters. Feature pairs are aggregated, often with consistency constraints (e.g., via cosine similarity losses ensuring semantic equivalence across bands):

$$L_{\mathrm{cons}}(z^{\mathrm{pass}}, z^{\mathrm{stop}}) = 1 - \frac{(z^{\mathrm{pass}})^\top z^{\mathrm{stop}}}{\|z^{\mathrm{pass}}\| \, \|z^{\mathrm{stop}}\|}$$

Hybrid speech enhancement models process audio via both time-domain and frequency-domain branches (e.g., TasNet-style encoder–decoder, U-Net on spectrogram) and average final outputs for robust denoising (Kim et al., 2018).

Multi-axis frequency representation learning fuses spatial and frequency features via axiswise 2D DFTs combined with spatial detail (e.g., depthwise-separable convolutions), further refined by dual attention (spatial and channel) for biomedical image segmentation tasks (Lu et al., 19 Sep 2025). In multi-source generalization, learnable domain tokens are integrated with frequency-decomposed features and convexly fused at test time using domain-adaptive similarity (Wang et al., 9 Jan 2026).

Table: Key Example Architectures

Task/Domain	Decomposition Technique	Fusion Mechanism
Vision DG (Yang et al., 2023)	Gaussian band/pass-stop in 2D DFT	Paired ResNet branches, feature concat
Fault Diagnosis (Tu et al., 1 Feb 2025)	Fourier amplitude/phase decomposition	FSIM (freq-spatial blocks), manifold triplet loss
Image Segmentation (Lu et al., 19 Sep 2025)	Multi-axis 2D DFT, MEWB	Dual attention (DA⁺) fusion
Retinal Vessel Segmentation (Wang et al., 9 Jan 2026)	Learnable wavelets, SDM	Frequency-adaptive domain fusion (FADF)

3. Frequency Mode Extraction, Adaptive Band Partitioning, and Signal Segmentation

Adaptive multi-domain frequency analysis often requires data-driven identification of meaningful frequency bands/modes and the localization of structural breaks or changes. In nonstationary time series, piecewise-constant models for the time-varying spectral density matrix are adopted, and L₂-norm discrepancy measures detect breakpoints:

$$D(\omega) = \frac{1}{\delta} \int_0^1 \int_0^\delta \| g(u, \omega - \lambda) - g(u, \omega + \lambda) \|_F^2 d\lambda \, du$$

Breakpoints in spectral behavior are ranked and tested by nonparametric bootstrap techniques, locating changes that preserve nonstationary structure (Sundararajan et al., 2023). For functional time series (e.g., multichannel EEG), scan statistics and inchworm search algorithms extend this logic to operator-valued spectra, yielding personalized, data-adaptive bands (Bagchi et al., 2021).

STFT-based time-frequency mode decomposition reframes nonstationary signal analysis as image segmentation, where modes manifest as contiguous high-energy regions in the spectrogram. Automated pipelines employ smoothing, adaptive thresholding, connected-component labeling, and region size filtering to extract and reconstruct these modes without iterative parameter sweeps (Zhou et al., 16 Jul 2025).

4. Multi-Domain Frequency Analysis in Control and Communications

In sampled-data and multirate control systems, frequency domain analysis is complicated by aliasing and the lack of a frequency separation principle. Frequency-lifting transforms the multirate system’s frequency response into a multivariable, time-invariant representation, enabling the performance frequency gain (PFG) to be computed from a single identification experiment:

$$\mathcal{P}\left( e^{j \omega_0 T_h} \right) = \sqrt{ \sum_{f=0}^{p-1} \left| \widetilde{M}_{f+1,1} (e^{j \omega_0 T_h}) \right|^2 }$$

Dynamic domain selection between time–frequency, delay–Doppler, and chirp/affine representations enables adaptive waveform processing in advanced wireless networks. Pre- and post-processing matrices implement transformations appropriate to channel conditions and system requirements, generalizing OFDM, OTFS, and AFDM (Arous et al., 14 Oct 2025). Cross-domain waveform strategies unlock degrees of freedom for wireless physical layer adaptation, beamspace multiplexing, and IoT applications.

5. Hybrid Domain Likelihoods and Unified Time–Frequency Statistical Inference

Statistical models for hybrid time–frequency analysis incorporate per-event or per-sample weights stemming from full-likelihood models of the data (e.g., photon weights in gamma-ray astronomy). The unified formalism aggregates weighted events into arbitrary time spans for unbinned time-domain (e.g., Bayesian blocks, adaptive light curves) and frequency-domain (weighted periodograms, FFTs) analyses (Kerr, 2019). Profile-likelihood estimators and iterative reweighting are used to search for and characterize both temporal variability and periodicity across a vast dynamic range of scales, with explicit handling of exposure and instrumental artifacts.

6. Applications, Empirical Findings, and Implications

Empirical studies across domains demonstrate that multi-domain frequency analysis uncovers latent, domain-invariant features, enables adaptive detection of critical frequency bands or modes, and improves generalization to unseen domains:

• In single-domain visual DG, slicing training samples into frequency subbands exposes hitherto unused invariant cues, yielding 8–18% mean accuracy gain over strong baselines (Yang et al., 2023).
• In multi-source mechanical fault diagnosis, frequency-augmented data augmentation and manifold triplet loss produce accuracy gains from ~58% (ResNet-18) to >84% (FARNet) (Tu et al., 1 Feb 2025).
• Biomedical and functional data analyses yield data-adaptive bands aligned with established physiological rhythms (e.g., alpha in EEG), but with person-specific band boundaries that are statistically validated (Sundararajan et al., 2023, Bagchi et al., 2021).
• STFT-based mode segmentation recovers true mode count under noise and achieves low reconstruction error without prior parameter knowledge or significant computation (Zhou et al., 16 Jul 2025).
• In wireless communications, adaptive transform-domain switching and frequency-lifting provide capacity/BER gains under doubly-dispersive channels and enable scaling in IoT/reconfigurable networks (Arous et al., 14 Oct 2025).

Common implications across studies include the necessity of multi-resolution, multi-axis, or hybrid feature extraction for robust generalization; the utility of adaptive, data-driven subband design; and the computational advantages of non-iterative signal segmentation. Limitations noted include empirical filter selection, increased parameter and computation cost in dual-branch models, and challenges in separating overlapping but non-crossing spectral modes (Yang et al., 2023, Zhou et al., 16 Jul 2025).

7. Future Directions and Open Challenges

Future research directions identified in the literature include:

• End-to-end or learnable frequency filter/band selection for adaptive subspace decomposition (Yang et al., 2023).
• Extension to non-radial or anisotropic sub-band designs and hybrid (wavelet + Fourier) decompositions to capture richer structure (Lu et al., 19 Sep 2025).
• Domain-adaptive fusion via frequency-similarity metrics for test-time unlabeled domain adaptation (Wang et al., 9 Jan 2026).
• Computational scaling for 3D/volumetric/multimodal biomedical data, with harmonized spatial-frequency representations (Lu et al., 19 Sep 2025).
• Generalization of frequency-lifting and mode decomposition frameworks to multi-rate, nonlinear, or nonstationary systems in control, sensor networks, and communications (Zhou et al., 16 Jul 2025, Arous et al., 14 Oct 2025).
• Unified statistical frameworks for multi-domain likelihood analysis combining both time and frequency uncertainty (Kerr, 2019).

This body of research establishes multi-domain frequency analysis as a key methodological axis for future advances in signal processing, machine learning, control, and scientific data analysis.