Signal Processing

Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.

In numerical signal processing for electroacoustic composition, the progressive loss of specific development and research environments caused by the increasing use of digital market tools has favoured the dominance of the closed-architecture audio processor model. This model, while powerful, envisions the possibility of describing output data about its perceived characteristics, but at the cost of ignoring its internal process and interacting systems, which become complex, powerful environments but closed in an inscrutable black box, a loss we must consider. Any digital signal processing technique tells a story. Just as the words of a language incorporate social, historical and technical polysemic layers, a signal processor has its own story of implementation, a gradual technological achievement with its inevitable aesthetic consequences. Through the looking-glass of literature, one can access those environments with renewed awareness by reestablishing a scientific method and an attitude to research. In this specific case, starting from the case study of Manfred Schroeder's historical reverbs, we illustrate the process of building analytical evaluation tools, as well as practical implementation, at the basis of a conscious study path.

We introduce the neural echo as a tool for understanding the behavior of neural networks. It generalizes the model-based concepts of impulse responses, diffusion echoes, and filter echoes to learning-based methods. It provides local, space-adaptive impulse responses and filter kernels for a neural network, its so-called echoes. These echoes depend on the input image and can be visualized to understand the learned dynamics of the network via an affine mapping. Neural echoes build a bridge from classical signal processing to modern explainable AI. They are very general and can be applied to both image-to-image and classification networks, with convolutional or fully connected structure, of feedforward or recurrent type, including modern transformer networks. Network differentiability is not required. In the differentiable case, neural echoes comprise concepts based on the network Jacobian, such as saliency maps and the analysis of adversarial perturbations, as special instances. As a simple blueprint to explain our framework, we derive neural echoes for the denoising convolutional neural network (DnCNN). Our experiments suggest that this network weights pixels based on their spatial and gray value distances. This not only clarifies its behavior, but also shows that it can reproduce key concepts of classical model-based denoisers such as bilateral filtering.

Pretrained large language models (LLMs) have demonstrated in-context learning (ICL) capabilities for numerical inference over sequences serialized as text. Prior work has identified and characterized this form of numerical inference primarily through output-level evaluations such as prediction error. However, how numerical information is organized within LLM representations remains much less understood. To study this internal organization, we adopt a graph signal processing perspective in which attention induces a weighted graph over tokens, while token hidden states define signals on its nodes. Quantitative graph-spectral diagnostics and qualitative token-graph visualizations reveal that representations become more clearly differentiated by input dynamical complexity as context length increases. Simpler inputs produce attention-induced token graphs with stronger global connectivity and smoother, spectrally concentrated hidden-state signals, whereas more complex inputs produce more localized graphs and hidden-state signals with broader spectral support and greater high-frequency energy. Together, these findings point to systematic, context-dependent internal signatures associated with numerical ICL that are conserved across model families.

Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.

Evaluating camera image signal processing (ISP) pipelines requires measuring low-level artifacts introduced by operations such as denoising, demosaicing, tone mapping, and compression. Blind image quality assessment (IQA) techniques can grade visual quality without a reference, but they typically focus on semantic and high-level visual cues or human perceptual scores rather than the low-level image-processing artifacts introduced by camera pipelines. In contrast, full-reference metrics such as PSNR and SSIM measure pixel-level differences and structural similarity, while LPIPS measures perceptual similarity in deep feature space. However, these metrics require perfectly aligned image pairs, which are difficult to collect in practical settings. We propose a reference-free learning framework that estimates full-reference image quality metrics from a processed sRGB image and its ISO metadata. Our method predicts a proxy sRGB reference, which is then compared with the processed image to compute PSNR, SSIM, and LPIPS in their standard full-reference form. Our experiments show that the proxy-reference model can be learned from synthetic data and applied to real camera data. We further show that lightweight LoRA fine-tuning enables efficient adaptation when ISP components or pipeline configurations are changed. The proposed method outperforms direct metric regression in estimating metric values and achieves higher agreement with full-reference rankings than conventional blind IQA methods. These results demonstrate the feasibility of reference-free estimation of full-reference metrics for practical camera-pipeline evaluation.

Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.

While global localization using spinning radar has gained attention for its robustness to adverse weather and challenging environments, many studies have focused on individual components such as place recognition or pose estimation. In this paper, we take a holistic view of radar sensor-based global localization and present RadLoc, a fast, robust, and lightweight end-to-end pipeline from place recognition to 3-DoF pose estimation. RadLoc accelerates pre-processing using 1D CA-CFAR filtering and leverages the near-range dominance in spinning radar images to design a compact descriptor and an efficient hierarchical coarse-to-fine retrieval strategy. Moreover, coupled with phase correlation-based 3-DoF pose estimation, it forms a versatile global localization module applicable to SLAM and multi-session SLAM systems. Extensive experiments on 15 sequences across 5 datasets demonstrate that RadLoc achieves robust performance while maintaining the smallest descriptor size and fastest retrieval time among state-of-the-art approaches. The supplementary materials are available at https://sparolab.github.io/research/radloc/.

We introduce Directed Hypergraph Signal Processing (DHGSP), a unified framework that extends graph signal processing to accommodate both higher-order (polyadic) and asymmetric (directional) relationships simultaneously. Using the tensor singular value decomposition (t-SVD) within the t-product algebra, we define a novel adjacency tensor for directed hypergraphs, a topologically faithful shift operator, and a lossless Directed Hypergraph Fourier Transform (t-DHGFT). Experiments on real traffic networks demonstrate that DHGSP outperforms matrix-based (graph and digraph) and undirected tensor-based (hypergraph) baselines in denoising tasks.

The increasing interest in data-driven methods for power system protection is accompanied by a lack of standardized, publicly available high-voltage waveform datasets that enable transparent and reproducible evaluation. To address this gap, this paper introduces the PROTECT-90 dataset, an open electromagnetic transient (EMT)-simulated reference benchmark for high-voltage fault studies with consistent digital-fault-recorder-like measurements, publicly released with this work. The dataset comprises 9,022 physically consistent short-circuit simulation episodes generated on a standardized 90 kV double-line topology with systematically documented domain randomization of grid operating points, line parameters, and fault conditions. For each episode, synchronized three-phase voltage and current waveforms are recorded at eight measurement locations and released together with structured, machine-readable metadata describing fault type, fault location, inception time, and operating conditions. All modeling assumptions, parameter ranges, and data-generation procedures are explicitly documented to ensure transparency and cross-study comparability. By combining physically grounded EMT simulation, balanced scenario coverage, and open accessibility, PROTECT-90 establishes a standardized foundation for reproducible benchmarking of protection-oriented signal processing and learning-based methods.

A crucial assumption in graph signal processing (GSP) is the existence of an underlying graph that captures the pairwise similarities between nodes, allowing filters to be designed based on this graph for tasks such as denoising. For spatial-temporal data in which node-to-node similarities evolve over time, a static spatial graph is insufficient. In this paper, to represent slowly time-varying pairwise relationships, we model the graph changes in two consecutive adjacency matrices $P = W^{(2)} - W^{(1)}$ across time as a low-rank matrix. % Specifically, given an initial adjacency matrix $W^{(1)}$ at time $t=1$, we jointly interpolate a signal $x_2$ and estimate $W^{(2)}$ at $t=2$ using both a graph signal smoothness prior for $x_2$ and a low-rank prior on $¶$. We alternate optimization steps. With $W^{(2)}$ fixed, $x_2$ is interpolated by solving a linear system. Alternatively, holding $x_2$ fixed, $W^{(2)}$ is updated via proximal gradient descent (PGD). The proximal mapping of the rank term $Gamma(W^{(2)} - W^{(1)})$ is approximated in linear time using a fast orthogonal matching pursuit (OMP) algorithm that selects a sparse combination of atoms from a dictionary $cR$ formed by the outer products of $W^{(1)}$'s eigenvectors. We unroll iterations of our algorithm into layers to build a lightweight neural network for limited data-driven parameter tuning. Experiments show that our joint optimization achieves better signal interpolation compared to existing time-varying graph models.

As Large Language Models asymptotically approach human-level fluency in natural language generation, solely relying on surface-level semantic artifacts for detecting LLM-generated texts has become increasingly precarious. Existing detectors often falter when facing three critical challenges: adversarial perturbations, cross-domain shifts, and the rapid temporal evolution of the foundation model. To address these issues, we propose \wavedetect, a novel framework that reformulates text detection as a signal processing task within the time-frequency domain. Unlike previous methods that analyze static token probability distributions, \wavedetect models the generated output as a probability signal, upon which a differentiable Continuous Wavelet Transform is applied to convert them into learnable spectral representations. This process reveals the intrinsic ``spectral fingerprints'' in machine-generated texts--patterns that remain invisible in time domain. Comprehensive evaluations on three well-curated datasets (RAID, EvoBench, and Domain-Shift) show that our method achieves a new state-of-the-art. It not only achieves superior accuracy but also exhibits remarkable robustness against sophisticated attacks, generalization across out-of-distribution topics and unseen evolving LLMs. Our results validate the efficacy of spectral analysis as a promising paradigm for LLM-generated texts detection.

Adaptive room equalization remains challenging under time-varying acoustic conditions and complex excitation signals, such as music. In these scenarios, classical filtered-x least mean squares (Fx-LMS) methods falter due to their rigid formulation. We present a modular differentiable digital signal processing (DDSP) framework for closed-loop adaptive room equalization that recovers Fx-LMS as a special case through automatic differentiation. The framework supports interchangeable EQ structures, response estimation methods, loss functions, and optimizers. Experiments with time-varying measured room impulse responses show that frequency-domain objectives provide more stable adaptation than time-domain objectives in the considered scenarios. Relative to the non-equalized response, system distance is reduced by 70% and mel-spectral distance by 13% (worst-case scenario). We further examine how online room response estimation accuracy and frame length affect the trade-off between responsiveness and convergence stability. Overall, the framework provides a unified open-source basis for exploring synergies between classical adaptive filtering and DDSP-based optimization.

Differentiable audio processors are habitually designed and optimised in machine-learning frameworks, but deploying them as real-time audio effects still often requires non-automatic implementation in a dedicated digital signal processing language. The translation is error-prone, demands an onerous verification process, and detaches research prototypes from usable production tools. That being so, we present ADAC, a compiler that lowers a trained model to a framework-agnostic intermediate representation and emits efficient FAUST code whose impulse response matches the source model to within floating-point arithmetic noise, direct paths included. The optimisation loop is made audible by replacing the model in a running plugin after each gradient step. The exported processor carries a small set of macro-controls that leave its stability intact. A stability certificate computed from the shipped parameters is checked before the plugin is built. At the demonstration, a feedback delay network is trained and exported to a working plugin.

We describe faust2clap, a framework establishing the first officially maintained compilation pathway from Faust DSP specifications to the CLAP format. The system operates in two different modes. A static mode employs ahead-of-time compilation to yield native binaries of optimal efficiency, while a dynamic mode uses runtime interpretation to permit DSP code modification without interrupting the host application. This latter capability addresses a persistent friction in audio software development, namely the cumulative overhead of the edit, compile, and reload cycle. We detail the algorithmic machinery underlying both modes, focusing specifically on the problem of parameter identity. To preserve both parameter values and their bindings to host automation across structural DSP mutations, we introduce an address-based identity matching algorithm and a stable slot allocation scheme. The implementation, comprising approximately 2,400 lines of C++ architecture and Python tooling code, has been integrated into the main Faust distribution.

In recent years, graph signal processing has emerged as a powerful framework at the intersection of signal processing and graph theory, providing tools for the analysis of signals defined on nodes while accounting for their relationships represented by edges. These tools have been successfully applied to various settings, including statistical hypothesis testing. In particular, non-parametric approaches based on surrogate generation have been proposed for signals on undirected graphs. However, they are yet to be extended to directed graphs. In this work, we first revisit the notion of stationary graph signals on directed graphs. Specifically, and through the eigendecomposition of the graph shift operator, we define directed graph wide-sense stationary signals. Then, we propose a new framework to generate surrogate graph signals that preserve covariance structure under stationarity assumptions. Null distributions of the test metric can then be constructed from these surrogates and serve as a reference for the empirical data. Finally, we provide guiding examples and an application on real data, in which we compare the performance of our framework with existing techniques for undirected graphs or based on naive permutation, demonstrating feasibility and superiority of the proposed approach.

This study presents a comprehensive approach for the clustering and classification of upper-limb surface electromyography (sEMG) signals during functional reach and grasp movements. The methodology was applied to the NINAPRO DB4 dataset, which provides multichannel EMG recordings of 52 gestures. A four-stage pipeline was designed, including signal preprocessing, fea-ture extraction, gesture selection via hierarchical clustering, and comparative model evaluation. Preprocessing involved a fourth-order low-pass filter (0.6 Hz) and Hilbert envelope transformation, effectively reducing noise and enhancing signal clarity. Feature extraction yielded 26 temporal and frequency-domain met-rics, which were later refined using visual analysis, mutual information, principal component analysis, and decision tree importance scores. A final subset of five key features was selected for classification tasks. Gesture selection was per-formed through hierarchical clustering using Mahalanobis distance, resulting in six representative movements that balanced biomechanical diversity and compu-tational efficiency. A 200 ms window was identified as optimal for temporal seg-mentation based on stability and physiological plausibility. Classifier models were evaluated in two stages. Automated comparison using PyCaret identified Extra Trees (ET) and Artificial Neural Networks (ANN) as top performers. Sub-sequent independent training confirmed their stability and generalization capac-ity, with ANN showing progressive learning and ET maintaining robust, con-sistent results. The findings support the implementation of adaptive, low-latency control strategies for myoelectric prostheses and provide a scalable pipeline for future real-time applications.

Distributed acoustic sensing (DAS) systems generate continuous, ultra-high-channel-count data streams at rates that exceed the capabilities of conventional batch-oriented analysis frameworks. As a result, essential tasks such as interactive exploration of long-duration recordings, scalable event annotation, and real-time algorithm-in-the-loop monitoring remain inadequately supported by workflows built around manually selected data segments and offline processing. This paper presents FiLark (Fiber Lark), a Python framework that applies a \emph{streaming-first} principle uniformly across data access, signal processing, visualization and monitoring for DAS. Instead of operating on manually selected data segments, FiLark presents any DAS sources-including continuous multi-file recordings-as a unified stream and builds all system components around that abstraction. An OpenGL-based ring-buffer renderer enables interactive browsing and visualization of arbitrarily long recordings with constant memory usage. An integrated annotation interface supports event labeling directly within continuous data streams, facilitating the creation of reproducible machine-learning-ready labeled datasets without offline preprocessing. The signal processing library includes temporal, spatial, spectral, and decomposition-based operators, with both CPU implementations and GPU-accelerated variants via PyTorch, alongside stateful chunked execution that preserves processing continuity and application semantics across segment boundaries. A standardized monitor interface further integrates streaming detectors and learning-based models into the visualization workflow. By sharing a common streaming abstraction across all layers, FiLark allows processing configurations and workflows developed interactively to transfer directly to scalable production pipelines without modification.

Many modern datasets are large and carry complex structural relationships. Graph-based methods have traditionally been used to represent networked data, modeling individual elements as nodes and pairwise interactions as edges. Furthermore, Graph Signal Processing (GSP) has been developed to analyze signals on graph nodes, such as temperature measurements (node signals) across different regions of a country represented as a graph. Topological Signal Processing (TSP) is an emerging field that generalizes GSP, enabling the analysis of signals defined not only on nodes but also on edges, triangles, and higher-dimensional network elements, modeled as simplicial complexes and related topological structures. This makes TSP naturally well-suited for studying higher-order interactions in complex systems by extending classical signal processing concepts, such as filtering and Fourier transforms, to the topological level. Despite its versatility, TSP remains challenging for many practitioners. Therefore, we present an accessible overview of TSP foundations while drawing connections with application-oriented settings. We focus on processing techniques based on the combinatorial Hodge Laplacian, which generalizes the graph Laplacian to simplicial complexes. In particular, we review key TSP concepts, relate them to real-world examples, and discuss how higher-order structures and signals can be derived from datasets. For instance, we introduce an edge-level signal capturing lagged interactions between nodal signals, and demonstrate its use in a case study on TSP-based analysis of brain imaging data, revealing nontrivial interactions between sets of brain regions. Overall, we aim to promote a broader adoption of TSP by bridging methodological developments with applications, fostering its use among a wide community of theoretical and applied researchers.

The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization, adaptive and active sensing, and sequential detection and decision-making. By organizing these advances from a signal-processing perspective, we intend to equip practitioners with practical tools and a coherent roadmap for deploying sequential GP models in real-world systems.

Symbolic regression aims to recover closed-form expressions from numerical data, but in differentiable symbolic regression the recovered expression depends not only on the grammar but also on the fixed architecture through which variables are routed during training. This is relevant to signal-processing settings in which closed-form models and interpretable nonlinear structure are useful. This architecture-specific effect has rarely been isolated directly, because existing comparisons often vary architecture together with operator family, grammar, or search procedure. Three depth-3 architectures are compared across twenty-four operator--shape--leaf combinations, holding operator family, grammar, and training protocol fixed as far as possible while varying the variable-routing architecture. Recovery changes from $0/64$ to $64/64$ trials on the same target under an architecture-plus-native-training-protocol comparison. The best architecture on one target is the worst on another, and trees with two equal-depth subtrees fail in every configuration tested ($0/3{,}776$). As a proof-of-concept mitigation, a small architecture set is trained and the hardened expression with the lowest held-out RMSE is selected. On the jointly-run subset, this improves recovery from $34.4\%$ for the only architecture present in all three configurations to $50.1\%$. On a Shockley diode target, the validation selector recovers cases missed by that baseline architecture, which by itself recovers $0/32$ seeds. Since the jointly-run subset contains only three configurations, the selector result is evidence that validation-based architecture selection is promising, not a complete benchmark. These results support treating architecture as a measurable design variable that should be reported, stress-tested, and selected using held-out validation rather than fixed a priori.

Low-rank adaptation (LoRA) has emerged as the de facto standard for parameter-efficient fine-tuning (PEFT) of foundation models, enabling the adaptation of billion-parameter networks with minimal computational and memory overhead. Despite its empirical success and rapid proliferation of variants, it remains elusive which architectural choices, optimization techniques, and deployment constraints should guide practical method selection. This overview revisits LoRA through the lens of signal processing (SP), bridging modern adapter designs with classical low-rank modeling tools and inverse problems, as well as highlighting how SP principles can inform principled advances of fine-tuning approaches. Rather than providing a comprehensive enumeration and empirical comparisons of LoRA variants, emphasis is placed on the technical mechanisms underpinning these approaches to justify their effectiveness. These advances are categorized into three complementary axes: architectural design, efficient optimization, and pertinent applications. The first axis builds on singular value decomposition (SVD)-based factorization, rank-augmentation constructions, and cross-layer tensorization, while the second axis deals with initialization, alternating solvers, gauge-invariant optimization, and parameterization-aware methods. Beyond fine-tuning, emerging applications of LoRA are accounted across the entire lifecycle of large models, ranging from pre- and post-training to serving/deployment. Finally, open research directions are outlined at the confluence of SP and deep learning to catalyze a bidirectional frontier: classical SP tools provide a principled vocabulary for designing principled PEFT methods, while the unique challenges facing modern deep learning, especially the overwhelming scale and prohibitive overhead, also offer new research lines benefiting the SP community in return.

Denoising of periodic signals and accurate waveform estimation are core tasks across many signal processing domains, including speech, music, medical diagnostics, radio, and sonar. Although deep learning methods have recently shown performance improvements over classical approaches, they require substantial computational resources and are usually trained separately for each signal observation. This study proposes a computationally efficient method based on DCNN and Re-sampling, termed R-DCNN, designed for operation under strict power and resource constraints. The approach targets signals with varying fundamental frequencies and requires only a single observation for training. It generalizes to additional signals via a lightweight resampling step that aligns time scales in signals with different frequencies to re-use the same network weights. Despite its low computational complexity, R-DCNN achieves performance comparable to state-of-the-art classical methods, such as autoregressive (AR)-based techniques, as well as conventional DCNNs trained individually for each observation. This combination of efficiency and performance makes the proposed method particularly well suited for deployment in resource-constrained environments without sacrificing denoising or estimation accuracy.

Implicit neural representations (INRs) mark a fundamental shift in signal modeling, moving from discrete sampled data to continuous functional representations. By parameterizing signals as neural networks, INRs provide a unified framework for representing images, audio, video, 3D geometry, and beyond as continuous functions of their coordinates. This functional viewpoint enables signal operations such as differentiation to be carried out analytically through automatic differentiation rather than through discrete approximations. In this article, we examine the evolution of INRs from a signal processing perspective, emphasizing spectral behavior, sampling theory, and multiscale representation. We trace the progression from standard coordinate based networks, which exhibit a spectral bias toward low frequency components, to more advanced designs that reshape the approximation space through specialized activations, including periodic, localized, and adaptive functions. We also discuss structured representations, such as hierarchical decompositions and hash grid encodings, that improve spatial adaptivity and computational efficiency. We further highlight the utility of INRs across a broad range of applications, including inverse problems in medical and radar imaging, compression, and 3D scene representation. By interpreting INRs as learned signal models whose approximation spaces adapt to the underlying data, this article clarifies the field's core conceptual developments and outlines open challenges in theoretical stability, weight space interpretability, and large scale generalization.

We establish that temporal averaging over multiple observations is the degenerate case of algebraic group action with the trivial group $G=\{e\}$. A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation. The Trivial Group Embedding Theorem proves that the sample covariance is the accumulation of trivial-group estimates, with variance governed by a $(G,L)$ continuum as $1/(|G|\cdot L)$. The processing gain $10\log_{10}(M)$ dB equals the classical beamforming gain, establishing that this gain is a property of group order, not sensor count. The DFT, DCT, and KLT are unified as group-matched special cases. We conjecture a General Algebraic Averaging Theorem extending these results to arbitrary statistics, with variance governed by the effective group order $d_{\mathrm{eff}}$. Monte Carlo experiments on the first four sample moments across five group types confirm the conjecture to four-digit precision. The framework exploits the $structure$ of information (representation-theoretic symmetry of the data object) rather than the content, complementing Shannon's theory. Five applications are demonstrated: single-snapshot MUSIC, massive MIMO, single-pulse waveform classification, graph signal processing, and analysis of transformer LLMs. Techniques for blind group matching are described.

This paper reformulates Transformer/Attention mechanisms in Large Language Models (LLMs) through measure theory and frequency analysis, theoretically demonstrating that hallucination is an inevitable structural limitation. The embedding space functions as a conditional expectation over a σ-algebra, and its failure to be isomorphic to the semantic truth set fundamentally causes logical consistency breakdown. WavePhaseNet Method The authors propose WavePhaseNet, which explicitly constructs a Semantic Conceptual Hierarchy Structure (SCHS) using Discrete Fourier Transform (DFT). By applying DFT along the sequence dimension, semantic information is decomposed into frequency bands: low-frequency components capture global meaning and intent, while high-frequency components represent local syntax and expression. This staged separation enables precise semantic manipulation in diagonalized space. Dimensionality Reduction GPT-4's 24,576-dimensional embedding space exhibits a 1/f spectral structure based on language self-similarity and Zipf's law. Through cumulative energy analysis, the authors derive that approximately 3,000 dimensions constitute the lower bound for "complete representation." This demonstrates that reduction from 24,576 to 3,000 dimensions preserves meaning and intent while enabling rigorous reasoning and suppressing hallucination. Cohomological Consistency Control The reduced embedding space, constructed via cohomological regularization over overlapping local windows, allows defining a graph structure and cochain complex. This quantifies inconsistencies among local inferences as coboundary-based losses. Applying harmonic projection based on Hodge theory positions cohomology as a computable regularization principle for controlling semantic consistency, extracting maximally consistent global representations.

Over-the-Air Federated Learning (AirFL) is an emerging paradigm that tightly integrates wireless signal processing and distributed machine learning to enable scalable AI at the network edge. By exploiting wireless superposition over a shared multiple-access channel, AirFL turns simultaneous transmissions of local model updates into an analog aggregate at the receiver, thereby reducing communication latency, bandwidth usage, and energy consumption in wireless aggregation domains. This article develops a design-oriented tutorial view of analog AirFL. We organize existing schemes according to the signal-processing mechanism used to enable AirFL aggregation: transmitter-side channel compensation and power control, receiver-side equalization and high-dimensional processing, or learning-aware aggregation weighting. This viewpoint leads to three representative classes -- CSIT-aware, blind, and weighted AirFL -- and clarifies their assumptions, performance tradeoffs, complexity, and deployment limitations. We further discuss synchronization, digital and hybrid analog-digital realization, and open research directions for integrating AirFL into practical wireless edge-AI systems.

Virtual analog (VA) modeling methods seek to emulate analog audio hardware using digital signal processing (DSP). Modeling approaches fall into three broad categories: white-box methods, which use detailed device knowledge for accurate simulation; gray-box methods that use generic DSP blocks to model the system; and black-box methods, which rely solely on opaque models learned from input-output data. A category of architectures used widely in black-box modeling are recurrent neural networks (RNNs). To model device controls, the control values can be provided as conditioning input to the network. However, when the conditioning is time-varied, the models are susceptible to producing noise artifacts. Regularization of the RNN dynamics significantly reduces these artifacts, though at a loss in modeling accuracy. This paper closes the dynamics regularization quality gap by introducing deep control-conditioned LSTMs and a gammatone filterband (GFB) loss. Experiments indicate that the proposed method achieves comparable modeling performance as unregularized baselines while avoiding the noise artifacts caused by time-varying control inputs.

Monitoring respiration parameters such as respiratory rate could be beneficial to understand the impact of training on equine health and performance and ultimately improve equine welfare. In this work, we compare deep learning-based methods to an adapted signal processing method to automatically detect cyclic respiratory events and extract the dynamic respiratory rate from microphone recordings during high intensity exercise in Standardbred trotters. Our deep learning models are able to detect exhalation sounds (median F1 score of 0.94) in noisy microphone signals and show promising results on unlabelled signals at lower exercising intensity, where the exhalation sounds are less recognisable. Temporal convolutional networks were better at detecting exhalation events and estimating dynamic respiratory rates (median F1: 0.94, Mean Absolute Error (MAE) $\pm$ Confidence Intervals (CI): 1.44$\pm$1.04 bpm, Limits Of Agreements (LOA): 0.63$\pm$7.06 bpm) than long short-term memory networks (median F1: 0.90, MAE$\pm$CI: 3.11$\pm$1.58 bpm) and signal processing methods (MAE$\pm$CI: 2.36$\pm$1.11 bpm). This work is the first to automatically detect equine respiratory sounds and automatically compute dynamic respiratory rates in exercising horses. In the future, our models will be validated on lower exercising intensity sounds and different microphone placements will be evaluated in order to find the best combination for regular monitoring.

Model-free and reinforcement learning-based adaptive filtering methods are gaining traction for denoising in dynamic, non-stationary environments such as wireless signal channels, biomedical monitoring, and sensor networks. Traditional filters such as LMS, RLS, Wiener, and Kalman are often limited by assumptions of stationarity, the need for exact noise statistics, or fragile parameter tuning. This paper proposes an adaptive filtering framework using Proximal Policy Optimization (PPO), guided by a composite reward that balances SNR improvement, MSE reduction, and residual smoothness. We frame adaptive filtering as a Markov decision process and train a PPO agent to adjust filter coefficients directly in response to changing noise. Experiments on synthetic nonstationary signals with diverse noise types show that the PPO agent generalizes beyond its training distribution. Moreover, real-world analysis is made and evaluated on ECG recordings from the MIT-BIH Noise Stress Test Database corrupted by baseline wander, electrode motion, and muscle artifacts. The learned PPO policy achieves real-time inference and slightly outperforms strong classical baselines on ECG denoising. These results demonstrate the viability of policy-gradient reinforcement learning as a computationally efficient and flexible tool for adaptive filtering in nonlinear, time-varying dynamical systems.

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure. Here, conceptually beyond spectral analysis, we demonstrate that topological methods not only effectively capture intrinsic and complex structural information but can also enhance neural networks. We provide a transparent methodology, TopCap, to capture topological features inherent in time series for basic machine learning. Compared to prior approaches, we obtain descriptors that probe finer information such as the vibration of a time series. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy consistently standing in comparison with neural network models. Moreover, by integrating TopCap features into those neural networks, our approach improves upon state-of-the-art methods in terms of robustness against noise, as well as accuracy, stability, convergence of loss function, and interpretability.