Frequency Features: Concepts & Applications

• Frequency features are quantitative descriptors of periodic content extracted from signals and graphs, capturing key characteristics like centroids, bandwidths, and modulation rates.
• They are derived using techniques such as Fourier transforms, filter banks, wavelets, and graph spectral analysis, which reveal hidden spectral patterns across diverse data types.
• Their integration enhances classification accuracy and facilitates advanced applications in areas like speech recognition, medical imaging, and astrophysical analysis while addressing challenges like noise sensitivity.

Frequency features are quantitative representations of oscillatory content or periodic structure extracted from signals, fields, or graphs. They underpin a wide spectrum of scientific and engineering domains, including signal processing, audio analysis, communications, power system stability, vision, bioinformatics, and astrophysics. Formally, frequency features may refer to (a) local or global measures of frequency content in a temporal or spatial signal, (b) analytic properties such as centroids, bandwidths, or modulation rates, (c) Fourier or spectral filter responses, (d) frequency-derived statistical or geometric descriptors, and (e) domain-specific constructs such as graph spectral features, cycle frequencies, or frequency-domain network activations. Their extraction and interpretation are typically conditioned by the data structure (time series, image, network), the physical meaning of frequency, and the specific application constraints.

1. Mathematical Definitions and Classes of Frequency Features

Frequency features are diverse, but share a reliance on a transformation or representation mapping raw data into a frequency or spectral domain. Typical classes include:

• Global Frequency Measures: Mean, median, or percentile frequencies over intervals, e.g., instantaneous frequency $f(n)$ or median $\mathrm{freq}_\text{med}$ for audio (Stowell et al., 2013).
• Frequency Centroids: The “center of gravity” of a spectrum, such as the Mel-band frequency centroid, defined by

$$\mathrm{FC}_k = \frac{\sum_{f} f\,|S(f)|^{2}\,W_k(f)}{\sum_{f} |S(f)|^{2}\,W_k(f)}$$

where $S(f)$ is the spectrum and $W_k(f)$ is the bandpass filter (Berjon et al., 2022).

• Band Energy and Bandwidth: Total energy in frequency bands (e.g., MFCC bins) and bandwidth $=$ difference between upper and lower frequency percentiles (Stowell et al., 2013).
• Spectral Peak Sequences: Temporal tracking of top spectral peaks per frame; these yield trajectory-based features such as periodicity, zero-crossing rate, centroid gradient (Bhattacharjee et al., 2018).
• Frequency Modulation (FM) Features: Instantaneous frequency rates and higher-order descriptors extracted from analytic transforms or Teager–Kaiser energy operators, capturing time-localized modulation (Rodomagoulakis et al., 2018, Stowell et al., 2013).
• Graph Frequency Features: Eigenmode decompositions of signals defined on graphs; projections onto Laplacian eigenvectors yield low/mid/high-graph-frequency components (Adel et al., 2023).
• Cycle/Frequency Domain Features: Signatures of cyclostationarity (e.g., periodicity in modulated signals), measured as peaks in the cyclic spectrum at particular “cycle frequencies” (0903.1183).
• Explicit/Implicit High-Frequency Image Features: Derivatives or band-pass responses capturing edges, textures, or up-sampling artifacts, as well as transformer block outputs specialized for frequency discrimination (Qiao et al., 2023, Zafari et al., 2023, Qi et al., 25 Jun 2025, Zhao et al., 19 Jun 2025).
• Astrophysical Frequency Features: Bandwidth, central frequency, and Gaussian parameters of burst spectra in FRBs (Hu et al., 23 Mar 2025); modulation spectra in pulsar profiles (Chang et al., 6 May 2025).

2. Feature Extraction Methodologies

A broad set of signal processing and machine learning methodologies are used to extract frequency features, often tailored to the data domain:

• Fourier and Time–Frequency Analysis: Standard FFTs for spectra, STFT for frame-wise frequency tracks, or DCT block transforms for localized frequency content (Bhattacharjee et al., 2018, Qiao et al., 2023).
• Filter Banks and Wavelets: Multiresolution decompositions (e.g., Daubechies wavelets for pitch/formant extraction (Goki et al., 2022)), Gabor filterbanks for FM tracking in speech (Rodomagoulakis et al., 2018).
• Analytic Optimization: Explicit tracking via notch filters adaptively locking to signal harmonics (e.g., ANF for siren tracking (Damiano et al., 2024)).
• Graph Spectral Transforms: Laplacian eigendecomposition and application of bandwise filters for graph spectral analysis (Adel et al., 2023).
• CNN and Transformer Feature Maps: Frequency-aware architectures using (a) frequency-sorted convolutional channels (Zafari et al., 2023), (b) domain-specific frequency/channel attention or branch separation (e.g., SFNet, MADNet (Qi et al., 25 Jun 2025, Zhao et al., 19 Jun 2025)), or (c) spatial mapping of physically grounded features onto 2D tensors (e.g., electrical distance–preserving input for power grid frequency prediction (Lin et al., 2019)).
• Peak–Valley Parametrization and Statistical Tests: Empirical detection and quantification of transient high-frequency features in noisy signals by parameterizing spectral peaks and measuring statistical significance (Mezache et al., 2019).

These extraction pipelines are often deeply integrated with downstream classification (e.g., SVMs on SPS features (Bhattacharjee et al., 2018), CNNs on combined MFCC/FC (Berjon et al., 2022), bottleneck DNN fusion with MFCC/FM (Rodomagoulakis et al., 2018)).

3. Application Domains and Use Cases

Frequency features have demonstrated effectiveness in domains including:

Domain	Example Features	Role/Outcome
Audio recognition/speech/music	SPS, FC, FM, MFCC	Classification, robust transcription
Speech processing (pitch/formant)	Wavelet bands, centroid, FM	Speaker ID, word/phoneme recognition
Distant/Noisy Speech Recognition	MMD/ESA FM, CIF, DNN bottleneck	Improved WER, noise robustness
Power System Stability	Inertia center, RoCoF, nadir, response coefficient	Fault response prediction
Graph bioinformatics (e.g., cancer)	GFT, bandpass/bandstop features	Stage/type discrimination, biomarker discovery
Cognitive radio/spectrum sensing	Cycle frequencies, cyclic spectrum peak	Primary user detection
Remote sensing/forensics	Hyperspectral DFT/DCT features, channel attention	GAN/diffusion artifact identification
Image coding/denoising	Head-level frequency separation, binary masking	Rate/distortion improvement
Astrophysics	Gaussian spectrum fit, bandwidth–frequency scaling	FRB energetics, emission modeling
Pulsar astronomy	Modulation frequency spectrum, component indices	Emission geometry, plasma model constraint

In all, frequency features enable informative, physically meaningful descriptors aligned with both human interpretability and statistical discriminability across tasks (Stowell et al., 2013, Berjon et al., 2022, Adel et al., 2023, Chang et al., 6 May 2025).

4. Quantitative Impact and Statistical Performance

Empirical studies consistently show that frequency features, when properly designed and integrated, yield superior or complementary performance to purely time/space/energy-based features:

• Classification Gains: SPS-SCG features deliver F-scores ≥0.98 for speech/music, outperforming MFCCs alone (Bhattacharjee et al., 2018); frequency-centroid augmentation gives +2–6% recognition accuracy in noisy, accented speech, especially at low SNRs (Berjon et al., 2022).
• DNN Fusion: In distant-speech recognition, deep fusion of FM and MFCC features yields up to 10% relative WER reduction in high-reverberation scenarios (Rodomagoulakis et al., 2018).
• Power System Prediction: CNN frequency-feature prediction achieves 56% lower MAE than MLPs, and reliably generalizes with small/balanced data (Lin et al., 2019).
• Graph Spectral Filtering: Cancer gene network frequency features increase F-statistics up to 30× over raw expression, sharply improving class separability (Adel et al., 2023).
• Remote Sensing Forensics: Frequency-aware SFNet gives 4–15% accuracy boost over best prior methods for GAN/diffusion forgery detection (Qi et al., 25 Jun 2025).
• Image Denoising/Compression: Frequency separation yields 0.15–0.2 dB PSNR increase (MADNet (Zhao et al., 19 Jun 2025)), and frequency-disentangled coding provides 5–7% bitrate reduction at equal distortion (FDHA (Zafari et al., 2023)).
• Astrophysics: Correcting FRB frequency features for selection effects recovers intrinsic energy–bandwidth laws, resolving prior conflicts over observed spectra (Hu et al., 23 Mar 2025).

5. Challenges, Controversies, and Open Problems

Despite the clear utility of frequency features, several challenges remain:

• Context Conditioning: Many frequency features are meaningful only relative to the data’s structure (e.g., time, spatial, graph, cyclic, or multivariate domains).
• Feature Leakage/Sensitivity: Some methods (e.g., matching pursuit for FM estimation in birdsong) are brittle to noise and compression, whereas spectrogram or DDM-derived statistics are robust (Stowell et al., 2013).
• Interpretability vs. Performance: High-dimensional, fused deep features often provide best performance but are harder to ascribe to physical content (e.g., transformer head decompositions or attention-based branches (Zafari et al., 2023, Zhao et al., 19 Jun 2025)).
• Domain-Specific Ambiguity: In pulsar astrophysics, shifts in broadband modulation with frequency can defy simple geometric or polarization models (Chang et al., 6 May 2025). In bioinformatics, the link between frequency bands and actionable biomarkers, though empirically strong, demands further biological elucidation (Adel et al., 2023).
• Observational Incompleteness: In astrophysical settings, strong selection effects and detection thresholds can distort the observed distribution of frequency features, requiring careful inversion (Hu et al., 23 Mar 2025).

6. Recent Advances and Future Directions

The past five years have seen the emergence of hybrid models explicitly fusing spatial, frequency, and domain-adapted features:

• Adaptive and Learnable Frequency Filtering: Binary masks (Zhao et al., 19 Jun 2025), attention-based frequency mapping (Zafari et al., 2023, Qi et al., 25 Jun 2025), and self-supervised spectral separation are integrated directly into end-to-end architectures.
• Graph and Non-Euclidean Domains: Graph Fourier analysis is now a front-line tool for analyzing functional genomic data, social networks, and traffic, with filter designs adapting to graph topology (Adel et al., 2023).
• Small-Data and Domain Adaptation: Frequency-tracking features (e.g., ANF-based (Damiano et al., 2024)) provide dramatic gains in data efficiency and cross-domain robustness, reducing model size and sample requirements.
• Cross-Domain Fusion: Joint spatial–frequency architectures are now de facto in remote sensing, medical imaging, and security—for forgery detection, denoising, and anomaly identification (Qi et al., 25 Jun 2025, Zhao et al., 19 Jun 2025).
• Statistical Validation and Feature Selection: New hypothesis-testing frameworks rigorously quantify the presence of nontrivial frequency features in strongly nonstationary, noisy environments (Mezache et al., 2019).

7. Summary Table: Key Frequency Feature Types and Representative Contexts

Feature Type	Mathematical Construct	Key Domains	Example arXiv IDs
Spectral centroid, bandwidth	Weighted mean, quantile differences	Audio, speech, word recognition	(Berjon et al., 2022, Bhattacharjee et al., 2018)
Frequency modulation/descriptors	FM rate, chirp rate, DCT	Speech, birdsong, signal classification	(Rodomagoulakis et al., 2018, Stowell et al., 2013)
Explicit/implicit HF image features	Gradients, DCT, binary frequency mask	Depth, denoising, forensics, coding	(Qiao et al., 2023, Zafari et al., 2023, Qi et al., 25 Jun 2025, Zhao et al., 19 Jun 2025)
Graph spectral features	Laplacian projection/filtering	Gene networks, systems biology	(Adel et al., 2023)
Cycle-domain/cyclostationary	Cyclic spectrum, max cyclic feature	Cognitive radio, modulation recognition	(0903.1183)
Wavelet band features	DWT coefficients, scale selection	Speech pitch/formant extraction	(Goki et al., 2022)
Astrophysical frequency signatures	Gaussian-fit, bandwidth scaling	Fast radio bursts, pulsars	(Hu et al., 23 Mar 2025, Chang et al., 6 May 2025)

Frequency features continue to be a central instrument in both the understanding and exploitation of oscillatory and periodic properties of complex signals and systems, with ongoing methodological expansion into new data modalities and robust, interpretable architectures for learning from spectral information.