Fractional Fourier Transform Research Articles

New Wigner Distribution and Ambiguity Function for the Offset Fractional Fourier Transform and Their Applications

ABSTRACTThis paper introduces the new offset fractional Fourier Wigner distribution and offset fractional Fourier ambiguity function (OFrWD and OFrAF). Moreover, many various useful properties of them are also derived. Besides, convolutions via offset fractional Fourier transform are also introduced. Furthermore, the relationships between proposed convolutions and the OFrWD as well as OFrAF are also obtained. In addition, with the help of simulation, applications of OFrWD and OFrAF are established, such as designing multiplicative filters in the offset fractional Fourier transform (OFrFT) domain and detecting the parameters of single‐component and multicomponent linear frequency‐modulated (LFM) signals.

Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems

In indoor visible light communication (VLC) and visible light positioning (VLP) systems, the performance of conventional orthogonal frequency-division multiplexing (OFDM) schemes is often compromised due to the nonlinear characteristics and limited modulation bandwidth of light-emitting diodes, the multipath effect in enclosed indoor environments, and the relative positions of transmitters and receivers. This paper proposes an OFDM scheme based on the fractional Fourier transform (FRFT) to address these issues, demonstrating promising results when applied to indoor VLC and VLP systems. The FRFT, a generalization of the conventional Fourier transform (FT) in the fractional domain, captures information in both the time and frequency domains, offering greater flexibility than the FT. In this paper, we first introduce the computation method of the reality-preserving FRFT for an intensity modulation/direct detection VLC system and integrate it with OFDM to optimize system performance. By adopting FRFT-OFDM under the optimal fractional order, we enhance both the bit error ratio (BER) performance and positioning accuracy. Simulation results reveal that the FRFT-OFDM scheme with an optimized fractional order significantly improves the BER and positioning accuracy compared to the FT-OFDM scheme across most receiver positions within the indoor observation plane. For communication, the FRFT-OFDM scheme achieves over 6 dB Eb/N0 gain compared to the FT-OFDM scheme at a BER of 3×10−4 when the receiver is positioned at most locations in the room. For positioning, the FRFT-OFDM scheme enhances positioning accuracy by more than 1 cm relative to the FT-OFDM scheme at most locations in the room. Notably, both systems maintain the same computational complexity and spectral efficiency.

Fast Parameter Estimation of Linear Frequency Modulation Signals in Marine Environments Based on Gradient Optimization Strategy

Multi-buoy sonar systems achieve target localization by receiving broadband Linear Frequency Modulation signals emitted from the transmitter. Accurate estimations of the parameters of Linear Frequency Modulation signals significantly enhance the localization accuracy. Linear Frequency Modulation signals can be focused into the fractional domain through Fractional Fourier Transform, but this increases the computational complexity. In marine environments, the low signal-to-noise ratio and multipath effects degrade the parameter estimation accuracy further. To address these issues, this paper proposes a fast estimation algorithm based on the Fractional Fourier Transform and a Gradient Subtraction-Average-Based Optimizer. This algorithm leverages the impulsive characteristics of Linear Frequency Modulation signals after Fractional Fourier Transform transformation, using the Fractional Fourier Transform as the fitness function. The Gradient Subtraction-Average-Based Optimizer algorithm includes three enhancement strategies: chaotic mapping initialization, a Golden Sine Algorithm, and an adaptive t-distribution variational operator. The simulation results demonstrate that the Gradient Subtraction-Average-Based Optimizer algorithm improves the issues of low diversity in the search agents, imbalanced global and local search capabilities, and susceptibility to local optima. A comparative analysis and statistical testing confirm that under a low signal-to-noise ratio and multipath effect conditions, the Gradient Subtraction-Average-Based Optimizer algorithm not only ensures real-time parameter estimation but also improves the estimation accuracy. The results of the parameter estimation provide reliable data support for subsequent target localization.

Hyperspectral anomaly detection via low-rank and sparse decomposition with cluster subspace accumulation

Anomaly detection (AD) has emerged as a prominent area of research in hyperspectral imagery (HSI) processing. Traditional algorithms, such as low-rank and sparse matrix decomposition (LRaSMD), often struggle to effectively address challenges related to background interference, anomaly targets, and noise. To overcome these limitations, we propose a novel method that leverages both spatial and spectral features in HSI. Initially, the original HSI is segmented into several subspaces using the k-means method, which reduces redundancy among HSI bands. Subsequently, the fractional Fourier transform (FrFT) is applied within each subspace, enhancing the distinction between background and anomaly target information while simultaneously suppressing noise. To further improve the stability and discriminative power of the HSI, LRaSMD is employed. Finally, the modified Reed–Xiaoli (RX) detector is utilized to identify anomalies within each subspace. The results from these detections are then aggregated to produce a comprehensive final outcome. Experiments conducted on five real HSI data sets yield an average area under the curve (AUC) of 0.9761 with a standard deviation of 0.0156 for the proposed algorithm. These results indicate that our method is highly competitive in the field of anomaly detection.

Fitting the Seven-Parameter Generalized Tempered Stable Distribution to Financial Data

This paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes maximum-likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavy-tailed data (Bitcoin and Ethereum returns) and two peaked data (S&P 500 and SPY ETF returns). For each historical data, the estimation results show that six-parameter estimations are statistically significant except for the local parameter, μ. The goodness of fit was assessed through Kolmogorov–Smirnov, Anderson–Darling, and Pearson’s chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value and outperforms the Kobol, Carr–Geman–Madan–Yor, and bilateral Gamma distributions.

Review on application of fractional Fourier transform in LinearFrequencyModulation signal and communication system

Traditional Fourier transform often apply to analyze and process stationary signals, however, it is weak for time-varying non-stationary signals, and fractional Fourier transform (FRFT) can better solve such problems. The FRFT can be comprehended as the expressive methods on the fractional Fourier domain constituted by the spinning coordinate axis of the signal anticlockwise about the origin at arbitrarily Angle in the time-frequency plane. In this paper, the improved fractional Fourier transform is combined with other calculation methods to achieve high precision estimation of chirp signal parameters. And the communication system built on weighted fractional Fourier transform and discrete fractional Fourier transform is studied and simulated respectively, which verifies the feasibility and improves the anti-jamming and anti-interception ability of the communication system.

Enhanced SAR Compression through Multi-Look Doppler Compensation and Auto-Focusing Technique.

This paper presents a simple and streamlined compensation technique for improving the quality of synthetic aperture radar (SAR) images based on the Range Doppler Algorithm (RDA). Incorrect Doppler estimation in the space orbit, caused by unexpected radar motion errors, orbit mismatches, and other factors, can significantly degrade SAR image quality. These inaccuracies result in mismatches between the azimuth-matched filter and the received Doppler chirp signal. To address this issue, we propose a Doppler estimation method that leverages the Fractional Fourier Transform (FrFT) and cross-correlation techniques. The received signals are compared with the azimuth-matched filter based on the rotation angle in the FrFT domain, and the Doppler centroid is adjusted to achieve the optimal alignment. This process ensures high correlation values and enhanced resolution in the final SAR image. The efficacy of the proposed technique is validated through experiments using real spaceborne SAR data from the practical satellite. The results demonstrate significant improvements in image quality and resolution compared to conventional algorithms, highlighting the advantages of our approach for various remote sensing applications.

On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces

On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces

Quantifying the frequency modulation in electrograms during simulated atrial fibrillation in 2D domains

Atrial fibrillation (AF) affects millions of people in the world, causing increased morbidity and mortality. Treatment involves antiarrhythmic drugs and catheter ablation, showing high success for paroxysmal AF but challenges for persistent AF. Experimental evidence suggests reentrant waves and rotors contribute to AF substrates. Ablation procedures rely on electroanatomical maps and electrogram (EGM) signals; however, current methods used in clinical practice lack consideration for time–frequency varying EGM components. The fractional Fourier transform (FrFT) can be adopted to capture time-varying frequency components, thereby enhancing the comprehension of arrhythmogenic substrates during AF for improved ablation strategies. To this end, a FrFT-based algorithm is developed to characterize non-stationary components in EGM signals from simulated AF episodes. The proposed algorithm comprises a pre-processing step to enhance the coarser features of the EGM waveform, a windowing process for dynamic assessment of the EGM, and a FrFT order optimization stage that seeks compact signal representations in fractional Fourier domains. The resulting order is related to the rate of frequency change in the signal, making it a useful indicator for frequency-modulated components. The FrFT-based algorithm is implemented on EGM signals from AF simulations in 2D domains representing a region of the atrial tissue. Consequently, the computed optimum FrFT orders are used to build maps that are spatially correlated to the underlying propagation dynamics of the simulated AF episode. The results evince that the extreme values in the optimum orders map pinpoint the localization of fibrillatory mechanisms, generating EGM activation waveforms with varying frequency content over time.

Matched power-frequency-modulated signal transform and its application in bat call signal analysis.

Bat call signal analysis is an important research topic, which is meaningful for bat species identification, and the design of various biomimetic systems. In addition to the commonly used methods in the time-frequency domain, the fractional Fourier transform (FRFT) is a valuable signal processing tool, as it is a generalization of the Fourier transform. However, the FRFT is constrained to the analysis of the linear frequency modulated-like bat call signal, while the modulation of the harmonics in a bat call is often nonlinear. For this reason, this paper proposes an integral transform, named matched power-frequency-modulated (PFM) signal transform (MPST), which is also the generalization of the Fourier transform, more precisely, a time-warping Fourier transform. As with the limitation of FRFT, the MPST is constrained to the analysis of the PFM-like bat call with the instantaneous frequency defined as an approximate power function abut time, in which the power can be an arbitrary positive integer or a fraction. The applications of MPST on the PFM-modeled bat call analysis are mainly parameter estimation and harmonic separation, and the performance is fully validated using the recordings of the feeding buzzes, social calls, and distress calls from the European bats.

A GNSS Anti-Interference Method Based on Fractional Fourier Transform

A GNSS Anti-Interference Method Based on Fractional Fourier Transform

AbFTNet: An Efficient Transformer Network with Alignment before Fusion for Multimodal Automatic Modulation Recognition

Multimodal automatic modulation recognition (MAMR) has emerged as a prominent research area. The effective fusion of features from different modalities is crucial for MAMR tasks. An effective multimodal fusion mechanism should maximize the extraction and integration of complementary information. Recently, fusion methods based on cross-modal attention have shown high performance. However, they overlook the differences in information intensity between different modalities, suffering from quadratic complexity. To this end, we propose an efficient Alignment before Fusion Transformer Network (AbFTNet) based on an in-phase quadrature (I/Q) and Fractional Fourier Transform (FRFT). Specifically, we first align and correlate the feature representations of different single modalities to achieve mutual information maximization. The single modality feature representations are obtained using the self-attention mechanism of the Transformer. Then, we design an efficient cross-modal aggregation promoting (CAP) module. By designing the aggregation center, we integrate two modalities to achieve the adaptive complementary learning of modal features. This operation bridges the gap in information intensity between different modalities, enabling fair interaction. To verify the effectiveness of the proposed methods, we conduct experiments on the RML2016.10a dataset. The experimental results show that multimodal fusion features significantly outperform single-modal features in classification accuracy across different signal-to-noise ratios (SNRs). Compared to other methods, AbFTNet achieves an average accuracy of 64.59%, with a 1.36% improvement over the TLDNN method, reaching the state of the art.

High-resolution multicomponent LFM parameter estimation based on deep learning

This paper addresses the complex challenge of parameter estimation in multi-component Linear Frequency Modulation (LFM) signals by introducing an innovative approach to high-resolution Fractional Fourier Transform (FrFT) parameter estimation, facilitated by convolutional neural networks. Initially, it analyzes the issues of peak shifts and the masking of weaker components due to spectral overlap in the FrFT domain of multi-component LFM signals. Convolutional neural networks are then employed to train and achieve high-resolution representations of FrFT parameters. Specifically, convolutional modules with residual structures are utilized to learn coarse features, while a weighted attention mechanism refines independent features across both channel and spatial dimensions. This approach effectively addresses the challenges posed by spectral peak overlap and frequency shifts in multi-component LFM signals, thereby enhancing the quality of high-resolution parameter estimation. Experimental results demonstrate that the proposed method significantly outperforms traditional methods in processing multi-component LFM signals. Moreover, it exhibits robust detection capabilities for both weak and compact components, thereby underscoring its potential applicability in the field of complex signal processing.

MIMO Signal Detection Based on IM-LSTMNet Model

Signal detection is crucial in multi-input multi-output orthogonal frequency division multiplexing (MIMO-OFDM) systems, yet classical detection methods often struggle with nonlinear issues in wireless channels. To handle this challenge, we propose a novel signal detection method for MIMO-OFDM system based on the fractional Fourier transform (FrFT), leveraging the robust time series processing capabilities of long short-term memory (LSTM) networks. Our innovative approach, termed IM-LSTMNet, integrates LSTM with convolutional neural networks (CNNs) and incorporates a Squeeze and Excitation Network to emphasize critical information, enhancing neural network performance. The proposed IM-LSTMNet is applied to the FrFT-based MIMO-OFDM system to improve signal detection performance. We compare the detection results of IM-LSTMNet with zero forcing (ZF), minimum mean square error (MMSE), simple LSTM neural network, and CNN–LSTM network by evaluating the bit error rate. Experimental results demonstrate that IM-LSTMNet outperforms ZF, MMSE, LSTM, and other methods, significantly enhancing system signal detection performance. This work offers a promising advancement in MIMO-OFDM signal detection, presenting a deep learning-based solution that effectively improves the system signal detection performance.

Discussion on Weighted Fractional Fourier Transform and Its Extended Definitions

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions.

Dual-chirp-based photonic THz-ISAC system with adaptive frequency synchronization.

Recent advancements have brought significant attention to photonic terahertz (THz)-integrated sensing and communication (ISAC) systems. In this work, we present an adaptive frequency offset (FO) compensation method for dual-chirp-based ISAC waveforms, using the fractional Fourier transform (FrFT) method. The proposed scheme can enable frequency synchronization without a need for training preambles and exhibit robustness against system noise. We validate this approach through an experimental demonstration in a 300 GHz photonic THz-ISAC system with 20 Gbps quadrature-phase shift keying (QPSK) data transmission and 1.5 cm range resolution. The experiment successfully compensates for frequency offsets ranging from -5 to 5 GHz, achieving an estimation error of less than 0.08% and a chirp-pilot power overhead of 0.5%.

Extending the Tight‐Binding Model by Discrete Fractional Fourier Transform

An extension of the tight‐binding approximation model using the discrete fractional Fourier transform is proposed. The quantum state between localized and delocalized states is formulated, where the intermediate state is continuously parameterized. The mixed features of the localized molecular‐like state and delocalized wave‐like state are confirmed when the wavefunction and band diagram of the intermediate state are represented. The proposed model is expected to be used to represent the quantum state with localized/delocalized features in, for example, organic semiconductors.

Pseudopolar Format Matrix Description of Near-Range Radar Imaging and Fractional Fourier Transform

Near-range radar imaging (NRRI) has evolved into a vital technology with diverse applications spanning fields such as remote sensing, surveillance, medical imaging and non-destructive testing. The Pseudopolar Format Matrix (PFM) has emerged as a promising technique for representing radar data in a compact and efficient manner. In this paper, we present a comprehensive PFM description of near-range radar imaging. Furthermore, this paper also explores the integration of the Fractional Fourier Transform (FrFT) with PFM for enhanced radar signal analysis. The FrFT—a powerful mathematical tool for signal processing—offers unique capabilities in analysing signals with time-frequency localization properties. By combining FrFT with PFM, we have achieved significant advancements in radar imaging, particularly in dealing with complex clutter environments and improving target detection accuracy. Meanwhile, this paper highlights the imaging matrix form of FrFT under the PFM, emphasizing the potential for addressing challenges encountered in near-range radar imaging. Finally, numerical simulation and real-world scenario measurement imaging results verify optimized accuracy and computational efficiency with the fusion of PFM and FrFT techniques, paving the way for further innovations in near-range radar imaging applications.

Fractional fourier transform and its application

The Fourier Transform (FT) is a linear transformation for the primitive function. It takes some set of functions to be an orthogonal basis. Its physical meaning is to transfer the primitive function onto each set of base functions. Because it can convert functions between the time and frequency domains, the FT is widely employed in many fields. The Fractional Fourier Transform (FrFT) is an improvement and progress based on the FT. This paper will define the FT and FrFT. Then the distinction between FrFT and FT is discussed. Finally, specific examples of its application in processing digital image are provided. FrFT is the process of transforming an image function into a series of periodic functions. The FrFT is used as a powerful mathematical tool to understand non- smooth signals, nonlinear systems and complicated phenomena, which is significant and has broad possibilities in the fields of signal processing, communication, image processing, optical imaging and quantum information processing.

Multi‐shaft stirred reactors mixing efficiency: Rapid characterization strategy based on chaotic attractors

AbstractThe complex fluid dynamics mechanism of multi‐shaft stirring systems has constrained their application in industrial production. In this work, we have extracted torque signals to correlate power consumption with rotational speeds across single, dual, and triple shaft stirred reactors. Remarkably, the reasons behind the unique linear power consumption‐rotational speed relationship exhibited in triple‐shaft reactor, through the application of Fractional Fourier Transform. Our comparative analysis, through mixing energy, reveals multi‐shaft systems' superiority in energy‐efficient mixing. More importantly, the interaction between impellers and fluid within a reactor is reflected in the evolving of torque chaotic attractors. We have established a novel characterization strategy by fractal dimension of attractors which reflect the reactors' mixing efficiency. By mathematically modeling these attractors, we have demonstrated their potential as a tool for establishing the internal dynamics equations of stirred reactors. This approach offers a new method for evaluating mixing effectiveness and automated control of stirring machinery.