Fractional Fourier Transform Research Articles

Fractional Fourier transforms on L and applications

This paper is devoted to the L p ( R ) theory of the fractional Fourier transform (FRFT) for 1 ≤ p < 2 . In view of the special structure of the FRFT, we study FRFT properties of L 1 functions, via the introduction of a suitable chirp operator. However, in the L 1 ( R ) setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss L p multiplier results and a Littlewood-Paley theorem associated with FRFT.

Gamma variance model: Fractional Fourier Transform (FRFT)

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the probability density function and its derivatives; and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed, and the distributional proprieties are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY ETF historical prices. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the cumulative distribution of the sample data than the CLM. The best VG model comes from the FRFT estimation.

Fractional Fourier transforms of vortex Hermite-cosh-Gaussian beams

In this paper, we investigate the propagation properties of a vortex Hermite-cosh-Gaussian beam (vHChGB) passing through a fractional Fourier transform (FRFT) optical system. The analytical expression of a vHChGB in FRFT plane is derived based on the Huygens–Fresnel diffraction integral. From the obtained expression, the evolution of the intensity and phase distributions of output beam are illustrated numerically as a function of the order of the FRFT and under different beam parameters conditions. The results show that a vHChGB in FRFT plane evolves gradually and periodically versus the order of the FRFT optical system. The beam shape depends strongly on the fractional order, and it is affected by the beam order, the decentered parameter and the topological vortex charge. The results obtained may be beneficial to applications in optical trapping, micro-particle manipulation, and beam shaping.

An optimized framework for digital watermarking based on multi-parameterized 2D-FrFT using PSO

In this paper, we propose a multi-parameter image watermarking algorithm using the fractional Fourier transform (FrFT). The proposed algorithm employs a double transform mechanism to increase the robustness of the algorithm against attacks. We also couple the application of the transform with the Particle Swarm Optimization (PSO) algorithm to decide on the ideal fractional domain for the embedding process. The PSO finds the optimal fractional domain by minimizing the RMSE between the input image and the watermarked image. The results obtained are tested against standard parameters, e.g., robustness, imperceptibility, payload capacity, to evaluate the performance of the algorithm.

Fetal heart rate estimation using fractional Fourier transform and wavelet analysis

ObjectiveMonitoring fetal cardiac activity during pregnancy is a critical part of assessing the fetus's health. Non-invasive fetal electrocardiogram (NIFECG) is a safe emerging fetal cardiac monitoring approach receiving considerable interest. This paper proposes an effective way to separate the fetal ECG signal from the single-channel abdominal ECG signals. MethodsThe paper proposes a novel algorithm based on time–frequency analysis combining fractional Fourier transform (FrFT) and wavelet analysis to extract fetal ECG from abdominal signals at higher accuracy. The abdominal signals acquired from pregnant women are preprocessed and subjected to suppressing maternal ECG using fractional Fourier transform and maximum likelihood estimate. The estimated maternal signal is removed from the abdominal ECG. The residue is processed using wavelet decomposition to obtain a clean fetal ECG and calculate fetal heart rate. ResultsThe proposed algorithm's performance is validated using signals from the Daisy database and Physionet challenge 2013 set-a dataset. Real-time signals acquired using Powerlab data acquisition hardware are also included for validation. The obtained results show that the proposed algorithm can effectively extract the fetal ECG and accurately estimate the fetal heart rate. ConclusionThe proposed method is a promising and straightforward algorithm for FECG extraction. Fractional Fourier transform maps the time domain abdominal signal into the fractional frequency domain, distinguishing the fetal and maternal ECG. The Wavelet transform can efficiently denoise the residue abdominal signal and provides a clean fetal ECG. The proposed approach achieves 98.12% of accuracy, 98.85% of sensitivity, 99.16% of positive predictive value, and 99.42% of F1 measure.

Unsymmetric Image Encryption Using Lower-Upper Decomposition and Structured Phase Mask in the Fractional Fourier Domain

Background: An asymmetric cryptosystem using Structured Phase Mask (SPM) and Random Phase Mask (RPM) in fractional Fourier transform (FrFT) using lower-Upper decomposition with partial pivoting is proposed to provide extra security to the system. The usage of structured phase mask offers additional parameter in encryption. In the encoded process the phase-truncation (PT) part is replaced by the LUDP that is by the decomposition part. Objective: Introducing an asymmetric cryptosystem with LUDP is to avert quick identification of encrypted image in the FrFT domain. Method: Input image is firstly convoluted with SPM, in FrFT and in LUDP then the obtained result is convoluted with RPM in inverse FrFT and in LUDP. Finally, the encoded image is attained. Results: The strength and legitimacy of the proposed scheme have been verified by using numerical analysis on MATLAB R2018a (9.4.0.813654). For checking the viability of the proposed scheme mathematical simulations have been carried out which determines the performance and better quality of an image based on key sensitivity, occlusion attack, noise attacks and histograms. Conclusion: A novel asymmetric cryptosystem is proposed using two phase masks one is SPM and another is RPM. LUDP is proposed in which the encoded procedure is different from the decoded procedure. Security is enhanced by increasing the number of keys and the scheme is also robust against attacks. Statistical simulations are also carried out for inspection the strength and viability of the algorithm.

Detection of T-wave Alternans in ECG Signals Using FRFT and Tensor Decomposition

T-wave alternans (TWA) refers to the periodic beat-to-beat variation in the amplitude of T-wave in the electrocardiogram (ECG) signal in an ABAB-pattern. TWA has been proven to be a very important indicator of malignant arrhythmia risk stratification. A new method to detect TWA by combining fractional Fourier transform (FRFT) and tensor decomposition is proposed. First, the T-wave vector is extracted from the ECG of each heartbeat, and its FRFT amplitudes at multiple orders are arranged to form a T-wave matrix. Then, a third-order tensor is composed of T-wave matrices of several consecutive heart beats. After tensor decomposition, projection matrices are obtained in three dimensions. The complexity of the projection matrix is measured by Shannon entropy to obtain feature vector to detect the presence of TWA. Results show that the sensitivity, specificity, and accuracy of the algorithm on the MIT-BIH database are 91.16%, 94.25%, and 92.68%, respectively. This method effectively utilizes the fractional domain information of ECG, and shows the promising potential of the FRFT in ECG signal processing.

Research Progress of the Sampling Theorem Associated with the Fractional Fourier Transform

Sampling is a bridge between continuous-time and discrete-time signals, which is important to digital signal processing. The fractional Fourier transform (FrFT) that serves as a generalization of the FT can characterize signals in multiple fractional Fourier domains, and therefore can provide new perspectives for signal sampling and reconstruction. In this paper, we review recent developments of the sampling theorem associated with the FrFT, including signal reconstruction and fractional spectral analysis of uniform sampling, nonuniform samplings due to various factors, and sub-Nyquist sampling, where bandlimited signals in the fractional Fourier domain are mainly taken into consideration. Moreover, we provide several future research topics of the sampling theorem associated with the FrFT.

A New Tensor Factorization Based on the Discrete Simplified Fractional Fourier Transform

Tensor analysis approaches are of great importance in various fields such as computation vision and signal processing. Thereinto, the definitions of tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) are significant in practice. This work presents new t-product and t-SVD definitions based on the discrete simplified fractional Fourier transform (DSFRFT). The proposed definitions can effectively deal with special complex tenors, which further motivates the transform based tensor analysis approaches. Then, we define a new tensor nuclear norm induced by the DSFRFT based t-SVD. In addition, we analyze the computational complexity of the proposed t-SVD, which indicates that the proposed t-SVD can improve the computational efficiency.

Key Validity Using the Multiple-Parameter Fractional Fourier Transform for Image Encryption

As a symmetric encryption algorithm, multiple-parameter fractional Fourier transform (MPFRFT) is proposed and applied to image encryption. The MPFRFT with two vector parameters has better security, which becomes the main technical means to protect information security. However, our study found that many keys of the MPFRFT are invalid, which greatly reduces its security. In this paper, we propose a new reformulation of MPFRFT and analyze it using eigen-decomposition-type fractional Fourier transform (FRFT) and weighted-type FRFT as basis functions, respectively. The results show that the effective keys are extremely limited. Furthermore, we analyze the extended encryption methods based on MPFRFT, which also have the security risk of key invalidation. Theoretical analysis and numerical simulation verify our point of view. Our discovery has important reference value for a class of generalized FRFT image encryption methods.

A refocusing iterative optimization method based on the quad-beam mode for accurate estimation of the azimuth velocity of slow-moving targets using SAR

ABSTRACT SAR slow-moving target detection and velocity estimation in ultra-high speed platform and strong clutter are a serious problem. The existing slow-moving target velocity estimation methods including fractional Fourier transform (FrFT), bidirectional imaging mode (DiBi) and the dual-beam interferometry mode (DBI) cannot suppress the clutter, which reduces the accuracy of velocity estimation. To solve the problem, we propose an accurate azimuth velocity estimating method refocusing iterative optimization (RIO) based on the quad-beam mode. Firstly, we use the back projection (BP) algorithm for the quad-beam imaging, and perform accurate registration and clutter suppression for the two forward (fore-) and two afterward (aft-) beams respectively. After that, the azimuth offset in fore- and aft-beam images is utilized to achieve the coarse azimuth velocity estimation. Then we put forward the RIO algorithm to compensate the error in velocity estimation. Through iterative search and compensation the azimuth velocity, the azimuth offset is gradually reduced to disappear, thus a more accurate estimation of azimuth velocity is realized. The experiments validate the accuracy of azimuth velocity estimation of the proposed method and the error is less than 0.01 m , which is much better than the other three typical methods (FrFT, DiBi, DBI).

A Novel MuLoRa Modulation Based on Fractional Fourier Transform

LoRa has been widely used in the Internet of Things (IoT). However, since LoRa only supports single symbol transmission in its bandwidth, the bit rate of LoRa is limited. This restricts the future development of LoRa, such as Smart Home, Internet of Vehicles, Satellite IoT, etc. Toward this end, based on the energy convergence characteristic of LoRa symbol in the fractional domain, we propose a multiplexed LoRa (MuLoRa) which transmits multiple LoRa symbols simultaneously in the same bandwidth. By adding address bits, the collision between multiple LoRa symbols of MuLoRa is avoided. Considering the hardware consumption of MuLoRa modulation in the time domain, a fractional domain MuLoRa modulation method based on fractional Fourier transform (FRFT) is designed. Simulation results show that MuLoRa can significantly improve the transmission bit rate than conventional LoRa under the same bandwidth. And MuLoRa can be configured more flexibly to meet the performance requirements of rate, power, collision avoidance and BER in different application scenarios in the future high data rate IoT era.

Multipath Interference Removal in Receivers of Linear Frequency Modulated Radar Pulses

The article describes algorithms for the retrieval of linear frequency modulated (LFM) pulses from a received signal which is corrupted by noise and multipath reflections from ground and stationary objects. Such a problem appears, for example, in electronic intelligence (ELINT) or spectrum sensors that receive and analyze signals originating from an active radar emitting LFM signals. Two computationally efficient methods to solve this problem are presented in this paper. The first is a version of time-frequency filtering based on spectrogram zeros. The second method is based on the chirp rate estimation, fractional Fourier Transform (FrFT) and band-pass filtering to retrieve the original pulse signal. Both of the techniques presented were tested using synthetic signals to measure their statistical properties and then positively verified on real-life recorded signals originating from a non-cooperative radar.

Parametric Decomposition of Pulsed Lidar Signals with Noise Corruption Using FRFT Spectrum Analysis

The parametric decomposition of full-waveform Lidar data is challenging when faced with heavy noise scenarios. In this paper, we report a fractional Fourier transform (FRFT)-based approach for accurate parametric decomposition of pulsed Lidar signals with noise corruption. In comparison with other joint time-frequency analysis (JTFA) techniques, FRFT is found to present a one-dimensional Lidar signal by a particular two-dimensional spectrum, which can exhibit the mathematical distribution of the multiple components in Lidar signals even with a heavy noise interference. A FRFT spectrum-processing solution with histogram clustering and moving LSM fitting is designed to extract the amplitude, time offset, and pulse width contained in the mathematical distribution. Extensive experimental results demonstrate that the proposed FRFT spectrum analysis method can remarkably outperform the conventional Levenberg–Marquardt-based method. In particular, it can accurately decompose the amplitudes, time offsets, and pulse widths of the pulsed Lidar signal with a −10-dB signal-to-noise-ratio by mean deviation ratios of 4.885%, 0.531%, and 7.802%, respectively.

Improved VMD-FRFT based on initial center frequency for early fault diagnosis of rolling element bearing

The extraction of weak fault signatures with periodic impulse characteristics is a crucial aspect of detecting an early fault of bearing. In order to suppress the interference of background noise to the weak fault signatures extraction by variational mode decomposition (VMD), some scholars recently proposed to use fractional Fourier transform (FRFT) to make the weak fault signatures concentrate more energy in the narrowband, which is called VMD-FRFT. Nevertheless, VMD-FRFT is more sensitive to the initial center frequency (ICF) than VMD according to our study, which will make it derive undesirable mode components due to unreasonable ICFs. Inspired by the resonant demodulation method to find the appropriate frequency band for demodulation, an improved VMD-FRFT with the guidance of ICF is proposed, which is to take the center frequencies corresponding to K selected frequency bands as the initial values of VMD-FRFT. Hence, the modal components hidden in multiple resonant frequency bands can be obtained in terms of performing iterative optimization. In this paper, a novel indicator named reciprocal of spectral autocorrelation smoothness index being sensitive to the periodic impulses is introduced to detect resonance frequency bands. Both simulated and experimental examples are used to verify the proposed method. The results show that the proposed method can effectively identify weak faults and detect bearing faults at an earlier time than other methods such as Infogram, reciprocal of spectral smoothness index and improved VMD.

Orbital-angular-momentum-based super-resolution ISAR imaging for maneuvering targets: Modeling and performance analysis

Vortex electromagnetic (EM) waves carrying orbital angular momentum (OAM) wave, whose infinitely helical wavefront and orthogonal eigenvalues of unique physical properties introduce additional rotational degrees of freedom, which has demonstrated superior performance in target detection and azimuthal imaging. In order to enhance the imaging performance for maneuvering targets, this paper offers a solution to develop the vortex inverse synthetic aperture radar (ISAR) technique based on OAM beams design. Firstly, by the space geometry of radar and maneuvering targets, the vortex echo signal model is derived, and the characteristics are analyzed as well. Secondly, pre-processing approach is developed for initial phase compensation by the design of the transmitted OAM beams. Thirdly, the imaging method, based on fractional Fourier transform (FrFT) and Doppler centroid tracking (DCT), is proposed to achieve high-precision range compression and motion compensation to accomplish the range alignment while removing the phases caused by residual azimuth displacements. Finally, after the slow-time pulse compression, the time-invariant Doppler shift is generated, and the well-focused ISAR image can be obtained. The simulation results demonstrate that the proposed vortex ISAR imaging can achieve a higher azimuth resolution than the conventional planar EM ISAR paradigm with the same synthetic aperture length. This discovery provides an approach for OAM-based microwave imaging in a rather efficient manner, which has no harsh requirements for complicated and heavily computational post-processing.

Mittag–Leffler–Hyers–Ulam Stability of Delay Fractional Differential Equation via Fractional Fourier Transform

Mittag–Leffler–Hyers–Ulam Stability of Delay Fractional Differential Equation via Fractional Fourier Transform

Fractional Fourier single-pixel imaging.

Single-pixel imaging technology has a number of advantages over conventional imaging approaches, such as wide operation wavelength region, compressive sampling, low light radiation dose and insensitivity to distortion. Here, we report on a novel single-pixel imaging based on fractional Fourier transform (FRFT), which captures images by acquiring the fractional-domain information of targets. With the use of structured illumination of two-dimensional FRFT base patterns, FRFT coefficients of the object could be measured by single-pixel detection. Then, the object image is achieved by performing inverse FRFT on the measurements. Furthermore, the proposed method can reconstruct the object image from sub-Nyquist measurements because of the sparsity of image data in fractional domain. In comparison with traditional single-pixel imaging, it provides a new degree of freedom, namely fractional order, and therefore has more flexibility and new features for practical applications. In experiments, the proposed method has been applied for edge detection of object, with an adjustable parameter as a new degree of freedom.

Hyperspectral Anomaly Detection via Integration of Feature Extraction and Background Purification

Anomaly detection (AD) has become a hotspot in hyperspectral imagery (HSI) processing due to its advantage in detecting potential targets without prior knowledge, and a variety of algorithms are proposed for a better performance. However, they usually either fail to extract intrinsic features underlying HSIs, or suffer from the contamination of noise and anomalies. To address these problems, we propose a new anomaly detector by integrating fractional Fourier transform (FrFT) with low rank and sparse matrix decomposition (LRaSMD). First, distinctive features of HSI data are extracted via FrFT. Then, row-constrained LRaSMD (RC-LRaSMD), which is more practical and stable than the traditional LRaSMD, is employed to separate background from noise and anomalies. Finally, we implement an atom-selection strategy to construct the background covariance matrix for detection. The experimental results with several HSI data sets demonstrate satisfying detection performance compared with other state-of-the-art detectors.

Asymmetric multiparameter encryption of hyperspectral images based on hybrid chaotic mapping and an equal modulus decomposition tree.

We present an asymmetric encryption scheme for hyperspectral images using hybrid chaotic maps (HCMs) and an equal modulus decomposition tree (EMDT) structure in a discrete multiple-parameter fractional Fourier transform (dmpFrFT) domain. The original hyperspectral image was scrambled by an HCM and then encrypted into asymmetric ciphertext using the EMDT. In the EMDT, each pair of the band images of the scrambled hyperspectral image were regarded as leaf nodes, while the encryption modules using chaotic random phase mask, dmpFrFT, and improved equal modulus decomposition were regarded as branch nodes, and the encryption process was implemented along the paths from the leaf nodes to the topmost branch node. The EMDT structure could provide multiparameter encryption, real-valued output, and different pairs of band images with different secret keys and encryption/decryption paths. Compared with the previous optical encryption approaches for hyperspectral images, our asymmetric cryptosystem had larger key space, less data amount of storage and transmission, and stronger resistance to statistical attacks. Various numerical simulations verified the performance of our proposed asymmetric cryptosystem.