signal processing

Traditional time-series analysis has predominantly focused on time- or state-domain approaches, while frequency-domain methods have long lacked a systematic, cross-disciplinary survey. Method: This paper presents the first comprehensive review of Fourier, Laplace, and wavelet transforms in time-series analysis—covering theoretical foundations, applicability boundaries, strengths, and limitations—and evaluates their applications across finance, meteorology, molecular dynamics, and other domains. We propose a unified evaluation framework, a reproducible technical pipeline, and open-source an integrated toolchain (hosted on GitHub). Contribution/Results: The work fills a critical gap in the literature by delivering the first systematic, domain-agnostic survey of frequency-domain techniques for time-series modeling. It provides both authoritative scholarly reference and practical engineering support for cross-domain temporal modeling, enabling rigorous method selection, benchmarking, and deployment.

This study addresses the limitations of conventional image enhancement, filtering, and pattern recognition—namely, heavy reliance on manual feature engineering and insufficient real-time performance—by proposing a theory-driven, end-to-end machine learning framework. Methodologically, it is the first to systematically integrate discrete Fourier transform (DFT), Z-transform, and continuous Fourier analysis into deep learning pipelines, synergistically coupling them with convolutional neural networks (CNNs) and classical digital filtering algorithms to enable frequency-domain-guided automated feature extraction and real-time joint signal–image processing. The key contributions include: (i) development of an extensible Python framework; (ii) average PSNR improvement of 3.2 dB in image enhancement and noise suppression tasks; and (iii) 40% acceleration in feature extraction efficiency. This work establishes a novel paradigm for AI-powered real-time computer vision that simultaneously ensures high performance and interpretability.

This work addresses the statistical modeling and application of spectrogram zeros of noisy signals. Specifically, it investigates the random point process formed by spectrogram zeros in the complex plane—a fundamental object in time-frequency analysis—and establishes, for the first time, a rigorous theoretical connection between these zeros and those of Gaussian analytic functions, thereby bridging time-frequency analysis, random analytic function theory, and spatial point process theory. Building upon this foundation, we develop a statistically principled model for zero-point distributions and design novel signal detection and adaptive denoising algorithms grounded in spatial statistical inference. The proposed methods enjoy strong theoretical guarantees—including consistency and asymptotic optimality—and demonstrate robustness and interpretability even at low signal-to-noise ratios. By recasting time-frequency signal processing through the lens of stochastic geometry and random zero sets, this work introduces a new paradigm for analyzing and processing nonstationary signals in the time-frequency domain.

This paper addresses the spectral analysis of non-decaying, unbounded continuous-time signals—those not belonging to the L² space—for which no rigorous spectral representation theory previously existed. Method: We develop the first mathematically rigorous spectral representation framework by integrating generalized Fourier analysis, distribution theory, and spectral operator methods, enabling precise definitions of key concepts—including transfer functions, spectral degeneracy, spectral gaps, and bandlimitedness—for such signals. Contributions/Results: (1) We introduce novel, rigorously formulated definitions of spectral degeneracy and spectral gaps, extending classical L²-based spectral theory beyond its traditional domain of applicability; (2) we design low-pass and high-pass filters for unbounded signals with explicit, analytically tractable transfer functions; (3) we prove that sublinearly growing signals exhibiting single-point spectral degeneracy are predictable, and we construct an explicit predictor. Collectively, these results establish necessary and sufficient criteria for bandlimitedness in non-L² signals and significantly broaden the theoretical foundations of linear systems and signal processing.

Traditional discrete Fourier transform (DFT) is constrained by uniform sampling and fixed-length sequences, rendering it inadequate for non-uniformly sampled, missing-data, or ultra-long signals. To address this, we propose the Extended Discrete Fourier Transform (EDFT), which formulates spectral estimation as an optimization problem minimizing the Fourier integral residual. EDFT adaptively constructs frequency-domain basis functions without requiring equispaced time-domain sampling or identical sequence lengths. Our method integrates iterative optimization, explicit Fourier integral constraints, and adaptive inverse DFT-based signal reconstruction. It enables high-resolution spectral estimation, time-domain extrapolation, missing-data imputation, and direct processing of non-uniformly sampled signals. Compared to DFT, EDFT substantially broadens the applicability of Fourier analysis while preserving theoretical rigor and computational feasibility.

This study systematically investigates the frequency-domain encoding capabilities of the Chronos foundation model, addressing a critical gap in understanding how such models represent fundamental signal properties. Through controlled experiments using discrete sinusoidal signals and a lightweight online Minimum Description Length (MDL) probing framework, the work examines the existence, separability, and cross-spectral fidelity of internal frequency representations within the Chronos decoder. The research reveals, for the first time, a degradation in representation quality in high-frequency regions, thereby delineating both the strengths and limitations of Chronos’s frequency encoding mechanism. These findings offer novel insights into the interpretability of time-series foundation models and provide practical guidance for applications in signal processing and multimodal fusion.

本文探讨了使用时频表示法来规范CPS需求，提出了一种新的逻辑语言S2TL，并比较了时域和时频域规格在适用性、表达保真度及噪声容忍度上的差异。

This study addresses a fundamental challenge in system identification: distinguishing spurious eigenvalues arising from limited data from those genuinely reflecting the underlying system dynamics. To this end, the paper introduces—for the first time—the probabilistic sampling pseudospectrum \( P(\lambda) \) and its computationally efficient estimator \( \hat{P}(\lambda) \). By leveraging resampling and statistical inference, this framework quantifies the uncertainty of eigenvalues across the complex plane. The proposed approach provides a general and rigorous statistical criterion for data-driven methods such as Dynamic Mode Decomposition and subspace identification, substantially enhancing the reliability of identifying true dynamical modes from noisy, finite-length observations.

Traditional relational databases struggle to process online signal data streams in medical monitoring efficiently and deterministically. To address this challenge, this work proposes the first deterministic stream processing framework tailored for medical monitoring scenarios. The framework introduces a formal algebraic system for data sequences grounded in rigorous mathematical foundations and designs stream processing operators with precise semantics. This algebraic structure is theoretically linked to Beatty’s and Fraenkel’s theorems, ensuring predictability and correctness throughout the processing pipeline. By providing formal guarantees alongside high performance and reliability, the proposed model significantly enhances the capability of medical monitoring systems to handle time-critical, continuous physiological data streams.

针对封闭架构音频处理器导致内部过程不透明的问题，通过建立分析评估工具及实践实施方法来解决。