The fractional Fourier transform is critical in signal processing and supports many applications. Signal processing is one notable application. Implementing the fractional Fourier transform requires discrete versions. As a result, defining a discrete coupled fractional Fourier transform (DCFrFT) is essential. This paper presents a discrete version of the continuous, two-dimensional coupled fractional Fourier transform, which is not a tensor product of one-dimensional transforms. We examine the main characteristics of the operator and illustrate its relationship with the existing two-dimensional discrete fractional Fourier transforms. Examples help clarify the approach.

Random sequences are critical to cryptographic technologies and applications. Randomness testing typically employs probabilistic statistical techniques for evaluating the randomness properties of such sequences. Both the National Institute of Standards and Technology (NIST, Gaithersburg, MD, U.S.) and the State Cryptography Administration (SCA, China) have issued guidelines for randomness testing, each of which includes the Discrete Fourier Transform (DFT) test as one of the mandatory assessments. This paper focuses on the efficient implementation of the DFT test and proposes a fast implementation approach that leverages FFTW (Fastest Fourier Transform in the West). Comprehensive experimental tests and performance evaluations were performed both before and after optimization of the algorithm. The results show that the optimized algorithm increases the speed of the DFT test for a single sample by a factor of 3.37.

Abstract In this article, we formalize and analyze qualia motion, i.e., the process by which a composition transitions across distinct harmonic qualities through the Fourier qualia space (FQS)—a multidimensional and transposition-independent space based on the discrete Fourier transform (DFT) coefficients’ magnitude. In the FQS, the plot of set classes relies on their harmonic qualities—such as diatonicity and octatonicity—enabling us to (1) identify the pitch-class set in a musical phrase that best represents its qualia—a reference sonority; (2) define a harmonic progression using all sequential reference sonorities in a piece; (3) visualize trajectory in space; and (4) establish a statistical metric for the ambiguity of harmonic qualia. Finally, we discuss Schoenberg's Op. 19, No. 1, analyzing the sense of its harmonic path. The proposed space leverages a bipartite, symmetrical, and consequential structure and unveils ambiguity as an element of nondirected linearity and counterpoint.

Epilepsy is a chronic neurological disorder that significantly affects the quality of life (QoL), often causing irreversible brain damage and physical impairment. Electroencephalography (EEG) signal analysis is crucial for monitoring epilepsy, enabling early seizure detection and timely intervention. Effective seizure detection requires the identification of interpretable features from the EEG signal to improve clinical outcomes. This study proposes a novel interpretable multi-view feature learning approach (IMV-FL), in which the time-domain signals and Discrete Fourier Transform (DFT) are applied to convert the time-domain EEG signal into frequency-domain representations. To develop initial multiview feature extraction and compression, spatial and temporal morphological features are extracted from optimal layers of ResNet and Long Short-Term Memory (LSTM) models, with feature compression performed using a Deep Neural Network (DNN). To construct an interpretable multi-view feature fusion, linear and nonlinear properties are calculated for the feature and with fusion strategies. The selected features are processed using the Mutual Information-Based Feature (MIBF) selection algorithm, and a Stacking Ensemble Classifier (SAEC) is adopted for unified view classification. To enhance clinical interpretability, SHapley Additive exPlanations (SHAP) is applied. The proposed framework outperforms single-view feature learning methods by 3% on average and state-of-the-art techniques by 2% in classification accuracy, sensitivity, specificity, and F1-score using the CHB-MIT Scalp and Bonn EEG datasets. This approach offers an effective tool for EEG-based seizure detection (ESD) in clinical and healthcare settings.

Abstract This paper develops a constructive numerical scheme for Fourier-Bessel approximations on disks compatible with convolutions supported on disks. We address accurate finite Fourier-Bessel transforms (FFBT) and inverse finite Fourier-Bessel transforms (iFFBT) of functions on disks using the discrete Fourier Transform (DFT) on Cartesian grids. Whereas the DFT and its fast implementation (FFT) are ubiquitous and are powerful for computing convolutions, they are not exactly steerable under rotations. In contrast, Fourier-Bessel expansions are steerable, but lose both this property and the preservation of band limits under convolution. This work captures the best features of both as the band limit is allowed to increase. The convergence/error analysis and asymptotic steerability of FFBT/iFFBT are investigated. Conditions are established for the FFBT to converge to the Fourier-Bessel coefficient and for the iFFBT to uniformly approximate the Fourier-Bessel partial sums. The matrix form of the finite transforms is discussed. The implementation of the discrete method to compute numerical approximation of convolutions of compactly supported functions on disks is considered as well.

Abstract Fluid flow exhibits strong nonlinearity, and traditional solutions based on computational fluid dynamics (CFD) are limited by high computational costs. To efficiently model complex fluid dynamics, this study proposes a Fourier neural operator network (FNONet) to approximate solutions for 2D incompressible non-uniform steady laminar flows around obstacles. FNONet transforms flow geometry into Fourier space via discrete Fourier transform (DFT), parameterizes integral kernels using a neural network, and reconstructs velocity and pressure fields through hybrid frequency-spatial computations. A pressure gradient-constrained data-driven loss function is introduced to incorporate spatial-domain errors and frequency-domain pressure gradient constraints, thereby enhancing prediction accuracy near fluid-solid interfaces. Experiments demonstrate that the total mean squared error (MSE) of FNONet is over 20 times lower than that of convolutional neural network (CNN) based framework and 1.5 times lower than that of CNN-Transformer based framework. Compared with traditional CFD solvers, FNONet achieves a prediction speedup of over 3 orders of magnitude on central processing units (CPUs) and 4 orders of magnitude on graphics processing units (GPUs). The robustness and generalization of the model to unseen geometries are validated through qualitative visual analysis and quantitative error metrics on test datasets, providing a promising data-driven approach for CFD.

Long-term time series forecasting often relies on frequency- or time-domain decomposition to capture trends and seasonalities. Frequency based methods, such as the Discrete Fourier Transform, isolate dominant periodic components and suppress noise, but they operate in the complex domain. Time-domain models apply trend–seasonality decomposition but often capture only a single seasonality entangled with residual noise. To address these limitations, we propose the \ac{FragFM}-Fragmented Fourier Matrix--a time domain framework that incorporates spectral insights through fixed Fourier fragments. FragFM removes trends via moving averages, extracts multiple seasonal components with fragmented Fourier bases, and corrects residual noise. Operating fully in the real domain, FragFM achieves forecasting accuracy comparable to leading frequency-domain models while consistently outperforming traditional time-domain approaches. Its lightweight, interpretable design offers an efficient solution for long-term forecasting, bridging time- and frequency-domain methods. The code repository is: https://github.com/K0DX-DV1NUX/FragFM

Parameter-efficient fine-tuning (PEFT) methods like Low-Rank Adaptation (LoRA) are widely used to adapt large pre-trained models under limited resources, yet they often underperform full fine-tuning in low-resource automatic speech recognition (ASR). This gap stems partly from initialization strategies that ignore speech signals’ inherent spectral-phase structure. Unlike SVD/QR-based approaches (PiSSA, OLoRA) that construct mathematically optimal but signal-agnostic subspaces, we propose SPaRLoRA (Spectral-Phase Residual LoRA), which leverages Discrete Fourier Transform (DFT) bases to create speech-aware low-rank adapters. SPaRLoRA explicitly incorporates both magnitude and phase information by concatenating real and imaginary parts of DFT basis vectors, and applies residual correction to focus learning exclusively on components unexplained by the spectral subspace. Evaluated on a 200-h Sichuan dialect ASR benchmark, SPaRLoRA achieves a 2.1% relative character error rate reduction over standard LoRA, outperforming variants including DoRA, PiSSA, and OLoRA. Ablation studies confirm the individual and complementary benefits of spectral basis, phase awareness, and residual correction. Our work demonstrates that signal-structure-aware initialization significantly enhances parameter-efficient fine-tuning for low-resource ASR without architectural changes or added inference cost.

Human daily physical activity (PA), monitored via wearable devices, provides valuable information for real-time health assessment and disease prevention. However, analyzing time-domain PA data is challenging due to large data volumes and high inter- and intra-individual heterogeneity. Traditional PA analyses often rely on demographics, while advanced methods utilize time-domain summary statistics (e.g., L5, M10) or functional principal component analysis (FPCA). This study presents a data-efficient approach utilizing the Discrete Fourier Transform (DFT) to convert time-domain data into a compact set of frequency-domain variables. Our research suggests that adding these DFT variables can significantly enhance model performance. We demonstrate that incorporating DFT-derived variables substantially improves model performance. Specifically, (1) a small subset of DFT variables effectively captures major PA levels with effective dimensionality reduction; (2) these variables retain known associations with factors like age, sex, and weekday/weekend status; (3) they enhance the performance of classifiers. Mathematical and empirical analyses further confirm the reliability and interpretability of DFT-based features in dimension reduction. Across three mental health studies, these DFT-derived variables successfully capture key PA characteristics while retaining known associations and strengthening model performance. Overall, the proposed DFT-based framework offers a robust and scalable tool for analyzing accelerometer data, with broad applicability in health and behavioral research.

Advanced concept for identifying chemico-biological interactions associated with programmed cell death using a multi-scale attention residual convolutional neural network.

Abstract Sensor fusion algorithms are of fundamental importance for the odometric pose estimation of Differential Drive Wheeled Robots (DDWR) as it can be integrated into other algorithms to localize and map the environment. Different sensors are used for odometric pose estimation and, in some cases, Virtual Sensors (VS) are also used. The purpose of this work is to present a new VS can be integrated into sensor fusion algorithms for estimating the odometric pose of the robot. For this reason, a new VS for real-time estimation of the angular speed of the wheels connected to a DC of a DDWR for indoor applications is presented. The estimated wheel angular speed is obtained by processing the vibrational signals sampled from an onboard Inertial Measurement Unit (IMU) through the Discrete Fourier Transform (DFT). This technique is named Fast Fourier Transform as Wheel Angular Speed Estimator (FFT-WASE). The estimation process, integrated with the IMU, enables the realization of a new VS. The estimated angular speed obtained through the VS-based approach is adopted for making the speed control of the DDWR, as an application to validate the proposed technique. The comparison between the estimated and the measured speed coming from the encoders, the study of the absolute percentage errors and RMSE, the reconstruction of DDWR linear speed and yaw rate, and the stability analysis of the controlled DDWR have proved the validity of VS. Further tests are provided to show the applicability of the same acquired vibrational signals for real-time wheel slip monitoring.

Eye-tracking technology enables communication for individuals with muscle control difficulties, making it a valuable assistive tool. Traditional systems rely on electrooculography (EOG) or infrared devices, which are accurate but costly and invasive. While vision-based systems offer a more accessible alternative, they have not been extensively explored for eye-writing recognition. Additionally, the natural instability of eye movements and variations in writing styles result in inconsistent signal lengths, which reduces recognition accuracy and limits the practical use of eye-writing systems. To address these challenges, we propose a novel vision-based eye-writing recognition approach that utilizes a webcam-captured dataset. A key contribution of our approach is the introduction of a Discrete Fourier Transform (DFT)-based length normalization method that standardizes the length of each eye-writing sample while preserving essential spectral characteristics. This ensures uniformity in input lengths and improves both efficiency and robustness. Moreover, we integrate a hybrid deep learning model that combines 1D Convolutional Neural Networks (CNN) and Temporal Convolutional Networks (TCN) to jointly capture spatial and temporal features of eye-writing. To further improve model robustness, we incorporate data augmentation and initial-point normalization techniques. The proposed system was evaluated using our new webcam-captured Arabic numbers dataset and two existing benchmark datasets, with leave-one-subject-out (LOSO) cross-validation. The model achieved accuracies of 97.68% on the new dataset, 94.48% on the Japanese Katakana dataset, and 98.70% on the EOG-captured Arabic numbers dataset—outperforming existing systems. This work provides an efficient eye-writing recognition system, featuring robust preprocessing techniques, a hybrid deep learning model, and a new webcam-captured dataset.

This paper presents a modified Orthogonal Frequency Division Multiplexing (OFDM) system that combines Discrete Wavelet Transform (DWT) with Discrete Sine Transform (DST) to enhance data rate capacity over traditional Discrete Fourier Transform (DFT)-based OFDM systems. By applying Inverse Discrete Wavelet Transform (IDWT) to the modulated Binary Phase Shift Keying (BPSK) bits, the constellation diagram reveals that half of the time-domain samples after single-level Haar IDWT are zeros, while the other half are real. The proposed system utilizes these 0.5N zero values, modulating them with the DST (IDST) and assigning them as the imaginary part of the signal. Performance comparisons demonstrate that the Bit-Error-Rate (BER) of this hybrid DWT-DST configuration lies between that of BPSK and Quadrature Phase Shift Keying (QPSK) in a DWT-based system, while also achieving data rate improvement of 0.5N. Additionally, simulation results indicate that the proposed approach demonstrates stable performance even in the presence of estimation errors, with less than 3.4% BER degradation for moderate errors, and consistently better robustness than QPSK-based systems while offering improved data rate efficiency over BPSK. This novel configuration highlights the potential for more efficient and reliable data transmission in OFDM systems, making it a promising alternative to conventional DWT or DFT-based methods.

The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella [Linear Algebra Appl. 493 (2016), pp. 281–300] developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and convex polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevič arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of four-by-four stochastic matrices. Similar to the Karpelevič region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices. Implications for further inquiry are also given.

Development and validation of a new simulator of running impacts.

The Kirkwood-Dirac (KD) distribution is a quantum state representation that relies on two chosen fixed orthonormal bases, or alternatively, on the transition matrix of these two bases. In recent years, it has been discovered that the KD distribution has numerous applications in quantum information science. The presence of negative or nonreal KD distributions may indicate certain quantum features or advantages. If the KD distribution of a quantum state consists solely of positive or zero elements, the state is called a KD-positive state. Consequently, a crucial inquiry arises regarding the determination of whether a quantum state is KD-positive when subjected to various physically relevant transition matrices. When the transition matrix is discrete Fourier transform (DFT) matrix of dimension p [Langrenez et al., J. Math. Phys. 65, 072201 (2024)] or p2 [Yang et al., J. Phys. A: Math. Theor. 57, 435303 (2024)] with p being prime, it is proved that any KD-positive state can be expressed as a convex combination of pure KD-positive states. In this work, we prove that when the transition matrix is the DFT matrix of any finite dimension, any KD-positive state can be expressed as a real linear combination of pure KD-positive states.

Expectation Propagation (EP) has emerged as a promising detection algorithm for large-scale multiple-input multiple-output (MIMO) systems owing to its excellent performance and practical complexity. However, transmit antenna correlation significantly degrades the performance of EP detection, especially when the number of transmit and receive antennas is equal and high-order modulation is adopted. Based on the fact that the eigenvector matrix of the channel transmit correlation matrix approaches asymptotically to a discrete Fourier transform (DFT) matrix, a DFT precoder is proposed to effectively eliminate transmit antenna correlation. Simulation results demonstrate that for high-order, high-dimensional massive MIMO systems with strong transmit antenna correlation, employing the proposed DFT precoding can significantly accelerate the convergence of the EP algorithm and reduce the detection error rate.

Voltage sags are a frequent power quality issue that causes the root mean square (RMS) voltage to decrease from 10 milliseconds to 1 second. Even a slight drop can lead to serious problems, including equipment malfunctions, process interruptions, and financial losses. Therefore, timely and accurate detection of voltage sags is crucial to ensure system reliability and minimize potential damage. In this paper, we will review various voltage sag detection methods, including peak voltage detection, conventional d-q transform, discrete Fourier transform (DFT), wavelet transform (WT), Monte Carlo simulation, digital prolate spheroidal window (DPSW), and virtual positive sequence sag detection (VPS²D). Each method's implementation complexity, detection time, accuracy, and strength will be evaluated, even when harmonics and noise are present.

Discrete Fourier Transform and Minimum Variance Distortionless Response Beamforming Algorithms for Searching for Transients in Radio Interferometers that Experience Radio-frequency Interference

Entangled measurements are an indispensable tool for quantum information processing, such as Bell-state measurements in quantum teleportation and entanglement swapping. However, to date, the realization of entangled measurements has mainly focused on bipartite systems or Greenberger-Horne-Zeilinger (GHZ) states. Here, we demonstrate a practical scheme to realize entangled measurements for states. Thanks to the cyclic shift symmetry in the discrete Fourier transformation (DFT) of bosonic modes, the DFT measurement outcomes can be used to deterministically project multiqubit states onto states. Experimentally, we show that three-qubit state discrimination can be achieved by detecting the cyclic shift symmetry with a three-mode DFT optical circuit, yielding a measurement discrimination fidelity of 0.871 ± 0.039. Our experimental demonstration opens the door for the development of new quantum network protocols between multipartite systems.