Waveform-Family Classification

• Waveform-family classification is the process of categorizing waveform signals into families using probabilistic, statistical, and data-driven methods to account for variability and noise.
• It employs diverse techniques such as time-frequency transforms, CNN-based feature extraction, and SVM ensembles to distinguish and cluster signals across different applications.
• Recent advancements integrate deep neural networks, open-set detection, and adaptive waveform design to achieve high accuracy in radar, biomedical, audio, and communication systems.

Waveform-family classification refers to the process of categorizing observed waveform signals into families or classes based on a combination of probabilistic, statistical, or data-driven models, taking into account both the shared structure and the variability inherent within each class. The term encompasses methodologies that range from generative models in biomedical and physical measurements, to discriminative approaches in radar, sonar, audio, and communication signal analysis. At its core, waveform-family classification seeks robust techniques to distinguish, cluster, and analyze signal forms despite intrinsic variations, noise, and context-dependent deformations.

1. Probabilistic and Statistical Frameworks for Waveform Modeling

Modern probabilistic models play a pivotal role in waveform-family classification, permitting the explicit incorporation of variability in segment duration, signal shape, and noise. Segmental Hidden Markov Models (segmental HMMs) generalize classic HMMs by allowing for segments with arbitrary duration distributions and by describing each segment's mean as a parametric regression (for example, linear functions of time), rather than as a constant means (Kim et al., 2012). For an observed waveform $y_{(r)}$ of length $d$ attributed to state $k$,

$y_{(r)} = X_{(r)} \beta_k + e_{(r)}$

where $X_{(r)}$ is a design matrix, $\beta_k$ the regression coefficients, and $e_{(r)} \sim \mathcal{N}(0, \sigma^2 I_d)$. Model likelihood is computed via marginalization over latent state sequences.

The introduction of random effects further generalizes the framework by capturing shape variability within a family—allowing each waveform instance to deviate locally from a global mean via random, normally distributed coefficients $u^{(i)}$. This hierarchy,

$$y^{(i)} = X^{(i)} (\beta_k + u^{(i)}) + e^{(i)}, ~~~ u^{(i)} \sim \mathcal{N}(0, \Psi_k)$$

decouples intra-class deformations from pure measurement noise, enabling more accurate learning and categorization (Kim et al., 2012).

2. Feature Extraction and Representation of Waveforms

Discriminating among waveform families relies crucially on constructing features or signatures that are invariant or robust to nuisance factors (such as amplitude scaling, phase shifts, local deformations). Approaches include:

• Higher-Order Statistical Cumulants: For communications and cognitive radio applications, a high-dimensional signature vector comprising normalized absolute higher-order cumulants up to the tenth order distinguishes modulations and classes robustly, even under frequency-selective and Doppler-fading conditions. This 20-dimensional “waveform signature” (WS) allows for direct L1 distance-based classification and is especially effective in blind environments (IV et al., 1 Apr 2024).
• Time-Frequency Transforms: For non-orthogonal multicarrier or radar signals, applying continuous wavelet transforms (CWTs) enables explicit manual extraction of time-frequency features. Summary statistics (e.g., variance, interquartile range) over the CWT output reduce dimensionality and emphasize class-distinguishing structures (Xu et al., 2020).
• CNN-Extracted Representations: In audio and RF waveform analysis, both raw waveforms and spectrally-derived images can be supplied directly into convolutional neural networks (CNNs). Small granularity filters in sample-level deep CNNs extract hierarchies of features capable of effective cross-domain discrimination and are visualizable via filter activation maximization (Lee et al., 2017, Elyousseph et al., 2021).
• Bitwise and Image-like Encodings: Direct conversion of PCM audio into multi-bit-plane pulse sequences or two-dimensional "bit pattern images" supports convolutional architectures that capture subtle dynamics with higher SNR robustness than both waveform and frequency-domain methods (Okawa et al., 2019).

3. Machine Learning Algorithms for Classification

Various learning-driven frameworks have been tailored to the specifics of waveform-family classification:

• Feed-forward Deep Neural Networks (DNNs): These can operate on power spectra (extracted via FFT), raw waveforms, or engineered features and are optimized with dropout and regularization to handle impairments like noise, fading, frequency/phase offsets, and IQ imbalance. DNNs with advanced activation functions (e.g., GELU), wide architectures, and large input feature size have demonstrated closed-set accuracy exceeding 83% at −10 dB SNR, with near-perfect results above 0 dB (Fredieu et al., 2021, Fredieu et al., 2021).
• Open-set and Anomaly Detection: Augmenting DNN classifiers with isolation forest (IF) models or autoencoders enables identification of “unknown” waveform families, crucial for real-world spectrum monitoring. IFs operate on compact (e.g., 32-dimensional) dense layer embeddings, trained per known class, and reject novel/unseen signals with ≈98% accuracy (Fredieu et al., 2021). CNN-based autoencoder reconstruction error (RMSE) is used for anomaly thresholding, supporting multi-region schemes to distinguish communication, radar, or spurious signals (Fredieu et al., 2021).
• Support Vector Machine (SVM) Ensembles: For multiclass problems where features are derived via explicit transforms (e.g., wavelet statistics), SVMs arranged in error-correcting output codes (ECOC) frameworks deliver high multiclass accuracy—especially in situations where feature extraction renders classes linearly separable (Xu et al., 2020).

4. Optimization, Adaptation, and Multi-family Waveform Design

Waveform-family classification is not purely a recognition task; optimal system function often involves waveform synthesis or selection to enhance downstream discrimination:

• Fourier-parameterized Multi-objective Design: In radar/sonar, families are generated using modulation functions expanded as Fourier series, with coefficients (or derived modulation indices) tuned via multi-objective optimization for low auto-correlation (ACF) and cross-correlation (CCF) sidelobes, constrained RMS bandwidth, and desired ambiguity function shape (e.g., thumbtack) (Hague, 2020).
• Adaptive Waveform Selection: Cognitive radar systems employ online learning and satisficing bandit algorithms (satisficing Thompson sampling) to optimize transmission waveform parameters such as bandwidth and slow-time unimodular code in situ, trading off classification performance and interference mitigation (Thornton et al., 2022).
• Continuous Family Generation: In continuous active sonar (CAS) or MIMO radar, subsets of nearly orthogonal waveforms within a family (e.g., modulated via parameter perturbation, reflection, and symmetry operations in the GSFM or MTSFM models) are employed to fill operational needs across long coherent processing intervals (Hague, 2018).

5. Evaluation, Accuracy, and Robustness

Classification systems are routinely validated on both simulated and real datasets:

• Biomedical and Fluid Data: Incorporating random effects in segmental HMMs for ECG and bubble-probe data led to ≈80% reduction in segmentation error and near-100% classification accuracy for certain waveform families, compared to fixed-parameter models (Kim et al., 2012).
• Radar and Communication Pulses: Discriminative DNNs and open-set feed-forward architectures achieved ≈100% classification above 0 dB SNR and robust detection of unknowns with IFs, maintaining ≈98% accuracy for known vs. unknown detection in SNRs above 0 dB, as well as competitive performance under multiple joint impairments (Fredieu et al., 2021, Fredieu et al., 2021, Wharton et al., 2021).
• Audio and RF: In audio event and music/speech discrimination, bit representation waveforms yielded an 88.4% accuracy on acoustic events—outperforming MFCC and power spectrum approaches—and retained high accuracy with improved robustness under noise and domain mismatch (Okawa et al., 2019). For RF signal classification, hybrid image CNNs combining time and frequency domain representations reached 100% accuracy compared to ≤92.5% for any single domain (Elyousseph et al., 2021).
• Gravitational Wave Templates: SVD-based interpolated and enriched-basis families have shown mismatches (faithfulness error) below 0.005, substantially better than non-calibrated phenomenological models (Cannon et al., 2012, Setyawati et al., 2018).

6. Challenges and Future Directions

Waveform-family classification remains challenged by several practical and theoretical issues:

• Variability and Deformations: Intra-class variability (shape, temporal, spectral) necessitates models that can factor or marginalize unwanted variations.
• Open-set and Anomaly Handling: Realistic systems require continuous rejection and updating in response to novel or adversarial waveform types encountered in the field.
• Computational Efficiency: Massive template banks or complex feature extractors place a premium on scalable algorithms; dimensionality reduction (e.g., PCA on cumulant signatures) and SVD-based basis compression are increasingly important (IV et al., 1 Apr 2024, Cannon et al., 2012).
• Physical Model Fidelity: For gravitational wave, radar, and sonar domains, neglecting higher-order effects (e.g., precession, higher harmonics) can result in substantial systematic bias, underscoring the importance of inclusive and physics-consistent families (Roy et al., 29 Apr 2025).
• Robustness to Impairments: The impact of noise, multipath, Doppler spread, hardware nonidealities, and domain shift necessitate robust architectures, regularization, and data augmentation.

This continues to drive research in both domain-informed waveform modeling, unsupervised and adaptive learning methods, and multi-objective waveform synthesis—aimed at maximizing classification performance, operational efficiency, and adaptability across the spectrum of physical and artificial signal sources.