Unified Signal Processing Framework

• Unified signal processing framework is an integrative approach consolidating methodologies, mathematical structures, and design principles across diverse signal domains.
• It employs operator theory, convex optimization, and spectral decompositions to address challenges in communications, sensor processing, machine learning, and quantum computing.
• The framework enables cross-domain algorithm transfer, robust performance, and practical applications in biomedical imaging, distributed sensing, and advanced signal reconstruction.

A unified signal processing framework is an integrative approach that emphasizes common principles, architectures, and mathematical structures underlying a wide variety of signal processing domains. Across time, frequency, and spatial domains—and including both classical and emerging applications such as sparse recovery, adaptive filtering, dimensionality reduction, graph/simplicial signal processing, machine learning, and even quantum and distributed computation—a unified framework systematically consolidates design methodologies, algorithmic primitives, and analytical tools. Central to this approach is the revelation that many seemingly disparate problems can be reformulated in terms of structured optimization, spectral analysis, or operator theory, enabling the transfer of methods and theoretical guarantees across domains.

1. Underlying Mathematical Principles and Common Structures

Unified signal processing frameworks are characterized by the identification and exploitation of common mathematical backbones shared by multiple subfields:

• Sparsity as a Unifying Paradigm: Sparse signal processing reveals that many acquisition, error-correction, spectral estimation, source separation, and channel estimation problems reduce to recovery of sparse representations in an appropriate basis or dictionary. This makes possible drastic reductions in sampling rates (often far below the Nyquist limit), robust recovery from corruption or erasure, and links iterative reconstruction in sampling theory to error location methods in coding via the annihilating filter or error locator polynomials (0902.1853).
• Operator-Theoretic Linear Algebra: The equation $Ax = b$ serves as a backbone unifying adaptive signal processing algorithms—from the steepest descent, LMS, NLMS, and Kaczmarz to RLS and Kalman filters. Iterative algorithms are essentially different strategies to solve or approximate this core equation, with extensions (reduced-rank, multigrid, preconditioning, Krylov subspace) derived by varying the solution representation, update strategy, or regularization (Anjum, 2015).
• Spectral Representations and Transforms: Frameworks such as Xampling (0911.0519), graph signal processing (Ji et al., 2020, Marques et al., 2020, Ji et al., 2020), and even signal processing on cell complexes (Roddenberry et al., 2021) exploit spectral decompositions and Laplacians or their higher-dimensional analogs. The structure of the Laplacian or operator set determines filtering, sampling, and denoising properties. In cyclostationary analysis, the cyclic spectrum organizes correlations to bridge classical Fourier analysis with a KL (Karhunen–Loève) basis tailored to periodic statistics (Vilà-Insa et al., 27 Mar 2024).

2. Domain-General Algorithms and Workflow Integration

A unified framework supports the transposition of algorithms across traditionally disparate areas:

• Sampling, Coding, and Estimation: Union-of-subspaces models (as in Xampling) encode signals that can be acquired and reconstructed using low-rate analog compression, followed by subspace detection via sparse recovery. Coding techniques such as Reed–Solomon and LDPC link to annihilating filters used in finite rate-of-innovation sampling by exploiting the shared structure of error locator polynomials (0902.1853, 0911.0519).
• Iterative Optimization and Convexity: Many recovery, learning, or estimation problems are cast as convex or structured optimization tasks. Algorithms such as IMAT, MIMAT, FOCUSS, and convex relaxations (e.g., l1 minimization in sparse component analysis) are adaptively employed across compressed sensing, source separation, and channel estimation (0902.1853). Similarly, in support vector machine-based DSP estimation, explicit signal structural constraints are embedded in the learning objective (Rojo-Álvarez et al., 2013).
• Information-Theoretic Bounds and Mutual Information Metrics: Unified frameworks apply capacity-like bounds (sample complexity lower limits, mutual information per measurement) to both linear and nonlinear problems—ranging from sparse regression and group testing to problems with missing features—drawing a deep analogy to channel coding theorems (Aksoylar et al., 2013).
• Low-Rank and Dimensionality Reduction: Eigen-decomposition, Krylov subspace, and iterative joint optimization strategies provide reduced-rank filtering and dimensionality reduction for high-dimensional data in wireless, sensor, array, speech, image, and video processing, with the transformation and estimator jointly optimized (Lamare, 2015).

3. Extensions Beyond Classical Domains

By generalizing signal models and processing architectures, unified frameworks adapt to new and more expressive domains:

• Higher-Order and Non-Euclidean Domains: Signal processing on simplicial complexes and regular cell complexes extends analysis and filtering to signals supported on generic topological spaces—encompassing graphs, meshes, and even cubical complexes. The introduction of generalized Laplacians and Hodge theory enables the decomposition of complex flows or fields into gradient, curl, and harmonic components, supporting linear (polynomial filters) and non-linear (convolutional neural network) processing (Ji et al., 2020, Roddenberry et al., 2021).
• Distributional and Uncertain Topologies: When network structure is not precisely known, frameworks operating over distributions of graph operators allow robust transform, filtering, and sampling theory aggregated over possible topologies, generalizing classical graph signal processing (special case: delta operator distribution), and enhancing performance in practical scenarios such as sensor network anomaly detection or incomplete measurement recovery (Ji et al., 2020).
• Generalized Stationarity and Joint Structure: By associating each vertex with a Hilbert-space-valued signal, generalized GSP allows the definition of joint wide-sense stationarity. Covariance operators that commute with joint shift operators admit joint spectral decompositions, permitting optimal Wiener filtering for denoising and completion in multidomain settings (e.g., time, frequency, multi-channel) (Jian et al., 2021).
• Quantum Signal Processing: Generalized frameworks for quantum signal processing (GQSP) and quantum power iteration integrate polynomial approximation, block encoding, and controlled rotations to realize various quantum algorithms—eigenstate preparation, spectral projections, and quantum simulation—within a single architectural template (Khinevich et al., 15 Jul 2025). In quantum sensing, polynomial transformations of continuous-variable observables, achieved using sequences of qubit-controlled gates, enable optimal, adaptively engineered discrimination between quantum states with nearly Heisenberg-limited scaling (Sinanan-Singh et al., 2023).

4. Impact on Application Domains

Unified signal processing frameworks have facilitated advances across a broad range of technical applications:

• Communications: Compressed sensing-based channel estimation (e.g., in OFDM) leverages sparse recovery methods to outperform traditional interpolation in multipath environments, particularly exploiting MIMAT and MMSE estimation in iterative thresholding approaches (0902.1853). Cyclostationary analysis in synchronous communications quantifies the MMSE gain achievable by matching processing to underlying periodicity, with explicit formulas generalized from the stationary (WSS) case (Vilà-Insa et al., 27 Mar 2024).
• Sensor and Array Processing: Unified dimensionality reduction leads to reduced-rank beamformers and high-dimensional sensor algorithms that maintain performance while dramatically decreasing computational burden, with switching architectures (JIOS) further improving adaptability (Lamare, 2015).
• Biomedical and Multimodal Signal Analysis: Tensor-based multi-dimensional frameworks are constructed for EEG analysis and denoising, employing HOSVD and multi-dimensional Fourier transforms to simultaneously exploit time, spatial, and frequency structures for advanced real-time tasks (e.g., seizure detection, BCI) (Govil et al., 10 Jan 2024).
• Distributed and Edge Learning: In federated learning, a signal-processing-centric abstraction exposes the bottlenecks of uplink transmission and global aggregation. Encoding, quantization, scheduling, and OTA computation strategies derived from source and channel coding theory yield scalable, robust, and privacy-aware learning at the network edge (Gafni et al., 2021).
• Industrial and Fault Monitoring: Unified signal representations, such as knowledge-enhanced vibration signal embeddings, support large-scale health diagnostics across diverse datasets. Integration of signal features with LLMs via embedding alignment leverages a unified input for state-of-the-art prediction and interpretability in multi-modal industrial monitoring (Peng et al., 21 Aug 2024).
• Embedded and Accelerator Hardware: Hardware frameworks such as SigDLA introduce unified on-chip computation for both deep learning and classical signal processing by regularizing irregular dataflows (e.g., butterfly FFTs) into tensor operations, supporting energy-efficient inference and DSP within a shared accelerator architecture (Fu et al., 17 Jul 2024).

5. Analytical Performance and Theoretical Guarantees

Unified frameworks systematically generalize and extend established theoretical results:

• MMSE and Spectral Optimality: The asymptotic MMSE formulas for filtering, smoothing, and prediction in cyclostationary environments hinge on eigen-decomposition of the cyclic PSD matrix, generalizing the Kolmogorov–Szegö formula beyond stationary cases (Vilà-Insa et al., 27 Mar 2024).
• Sample Complexity and Information Limits: Exact bounds, derived from mutual information and entropy characterizations, yield universal necessary and sufficient conditions on the number of samples for correct support recovery (linear regression, 1-bit regression, group testing, and structured sparsity), encapsulating both parametric and combinatorial uncertainty (Aksoylar et al., 2013).
• Robustness to Model Mismatch and Uncertainty: Analyses demonstrate that frameworks accommodating distributions of operators or support multi-resolution transforms maintain improved recovery and denoising performance under topology or hardware uncertainty, compared to those based on fixed models (Ji et al., 2020, 0911.0519).

6. Future Directions and Open Challenges

Emerging research aims to further generalize unified signal processing frameworks along multiple axes:

• Algorithmic Integrations with Machine Learning: The embedding of signal processing models into end-to-end learnable architectures (e.g., neural networks on cell complexes, kernel-augmented SVM-based DSP) blurs the line between classical signal design and data-driven inference, suggesting richer architectures for hybrid and domain-adaptive processing (Rojo-Álvarez et al., 2013, Roddenberry et al., 2021, Peng et al., 21 Aug 2024).
• Abstract Algebraic and Topological Extensions: Generalizing Laplacians, shift operators, and filter constructions to even more general domains (higher-order complexes, noncommutative operator settings) is an area of active expansion, involving both mathematical generalization and practical implementation (e.g., in physics-inspired or quantum systems).
• Quantum and Multivariable Algorithm Theory: As algorithmic primitives such as multivariable quantum signal processing (M-QSP) and generalized QSP are formalized, efficient constructive algorithms, operational resource analysis, and concise algebraic descriptions are open targets (Ito et al., 3 Oct 2024, Khinevich et al., 15 Jul 2025).
• High-Scalability and Hardware-Software Co-Design: Developing unified platforms that horizontally scale across IoT, sensor, and cloud-edge continuum, with reconfigurable accelerators capable of handling both legacy DSP and AI tasks, remains a key engineering driver (Fu et al., 17 Jul 2024).

A plausible implication is that as unified frameworks deepen cross-domain connections, innovation in one domain (e.g., sparse recovery, spectral graph theory, quantum polynomial algorithms) rapidly propagates to others, further accelerating both theoretical understanding and practical impact.