higher-order spectral convolution

This work addresses the need for independent post-processing of spectral coefficients across two channels in signal processing applications. We propose a novel real-imaginary separated complex discrete Fourier transform (DFT) algorithm, departing from conventional approaches that treat complex inputs as atomic entities. Our method establishes the first fully decoupled computational framework for real and imaginary components, formulated via vector-matrix representation and integrated with a divide-and-conquer strategy leveraging real-domain-optimized FFT structures. By eliminating redundant complex arithmetic, the algorithm retains the asymptotic O(N log N) complexity while significantly reducing memory access overhead and computational latency on hardware platforms—thereby enhancing throughput for dual-channel spectral processing. The core contribution lies in the physical separation of real and imaginary components at both input and output stages, coupled with complete decoupling of their computational paths. This enables an efficient new paradigm for resource-constrained or channel-isolated signal processing systems.

To address the dual challenges of stringent computational constraints and hard real-time requirements in embedded real-time spectrum analysis, this work presents the first complete fixed-point 36-point FFT implementation on the Nuvoton NUC140V2 microcontroller (ARM Cortex-M0, 72 MHz), based on the Prime Factor Algorithm (PFA). Unlike conventional Cooley–Tukey FFTs, PFA exploits prime-factor decomposition and divide-and-conquer DFT restructuring to eliminate both twiddle-factor look-up tables and complex multiplications, thereby drastically reducing arithmetic overhead. Leveraging C-language fixed-point arithmetic and assembly-level optimization of critical execution paths, the implementation achieves a runtime of 127 μs and consumes only 288 bytes of memory. This work establishes a new paradigm for high-throughput, low-overhead spectral analysis on resource-constrained MCUs, satisfying deterministic latency requirements in diverse real-time audio processing applications.

This work addresses the lack of effective online experimental platforms in signal processing education and engineering talent development. To bridge this gap, the authors developed and have continuously refined J-DSP, a web-based simulation environment that pioneered the migration of the original Java-based DSP toolkit to an HTML5 architecture, enabling cross-platform— including mobile—accessibility. The platform integrates advanced topics such as digital filter design, FFT-based spectral analysis, machine learning for signal classification, and quantum Fourier transform. Having operated reliably for 25 years, J-DSP has been widely adopted in university courses and National Science Foundation–funded programs, including REU, IRES, and RET initiatives, significantly advancing the modernization of signal processing pedagogy and fostering STEM workforce development.

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 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 lack of a unified theoretical foundation for classical and modern signal transforms, which has hindered systematic understanding and automated selection. By leveraging group representation theory, the authors unify a broad class of transforms—including the DFT, DCT, Walsh–Hadamard, Haar wavelets, KLT, spherical harmonics, and fractional Fourier transform—as eigenbases of covariance matrices that are covariant under specific group actions. Central to this framework is the identification of a “matching group” that leaves the signal covariance invariant, combined with the Peter–Weyl theorem and the Algebraic Diversity (AD) formalism. The study further introduces a novel, data-driven polynomial-time algorithm to automatically discover the optimal matching group without expert intervention, enabling automatic transform selection. This approach naturally extends to cutting-edge domains such as massive MIMO systems, graph neural networks, and Transformer attention mechanisms.

sp-DBA框架通过动态激活变换域计算块来加速科学计算和信号处理，无需重构建现有求解器，实现高效并行执行。

This work proposes a novel mixture-of-experts (MoE) adapter that overcomes the limitations of existing parameter-efficient fine-tuning methods, which are confined to fixed spatial or frequency domains and struggle to adapt to task-, layer-, or token-specific optimal representations. By introducing the fractional Fourier transform (FrFT) into the MoE architecture for the first time, each expert is equipped with a learnable FrFT order, enabling continuous interpolation between spatial and frequency domains and dynamic selection of the most compact low-rank update space. This design naturally induces expert decorrelation through learnable domain selection, substantially enhancing multi-task compositionality with minimal computational overhead. Experiments on LLaMA-3.1-8B and Qwen2.5-7B demonstrate consistent superiority over strong baselines such as FlyLoRA and FourierMoE across commonsense, mathematical, coding, and knowledge-intensive tasks, while maintaining low active parameter counts and revealing interpretable patterns of order specialization at both task and layer levels.

本文针对水下图像因光线折射和吸收导致的颜色失真、雾霾及可见度低的问题，提出了一种基于频域变换的Dynamic SpectraFormer方法，有效提升了图像质量。

本文针对大规模稀疏矩阵的频谱分析难题，提出了一种基于GPU的二进制稀疏快速傅里叶变换方法及两种压缩技术，有效减少了内存使用和计算时间。