Discrete Fractional Fourier Research Articles

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.

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.

Integrated photonic fractional convolution accelerator

An integrated photonic circuit architecture to perform a modified-convolution operation based on the discrete fractional Fourier transform (DFrFT) is introduced. This is accomplished by utilizing two nonuniformly coupled waveguide lattices with equally spaced eigenmode spectra, the lengths of which are chosen so that the DFrFT and its inverse operations are achieved. A programmable modulator array is interlaced so that the required fractional convolution operation is performed. Numerical simulations demonstrate that the proposed architecture can effectively perform smoothing and edge detection tasks even for noisy input signals, which is further verified by electromagnetic wave simulations. Notably, mild lattice defects do not jeopardize the architecture performance, showing its resilience to manufacturing errors.

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.

Optical Color Image Encryption Algorithm Based on Two-Dimensional Quantum Walking

The double random phase encoding (DRPE) image encryption method has garnered significant attention in color image processing and optical encryption thanks to its R, G, and B parallel encryption. However, DRPE-based color image encryption faces two challenges. Firstly, it disregards the correlation of R, G, and B, compromising the encrypted image’s robustness. Secondly, DRPE schemes relying on Discrete Fourier Transform (DFT) and Discrete Fractional Fourier Transform (DFRFT) are vulnerable to linear attacks, such as Known Plaintext Attack (KPA) and Chosen Plaintext Attack (CPA). Quantum walk is a powerful tool for modern cryptography, offering robust resistance to classical and quantum attacks. Therefore, this study presents an optical color image encryption algorithm that combines two-dimensional quantum walking (TDQW) with 24-bit plane permutation, dubbed OCT. This approach employs pseudo-random numbers generated by TDQW for phase modulation in DRPE and scrambles the encrypted image’s real and imaginary parts using the generalized Arnold transform. The 24-bit plane permutation helps reduce the R, G, and B correlation, while the generalized Arnold transform bolsters DRPE’s resistance to linear attacks. By incorporating TDQW, the key space is significantly expanded. The experimental results validate the effectiveness and security of the proposed method.

Breast ultrasound image despeckling using multi-filtering DFrFT and adaptive fast BM3D

Improving the quality of breast ultrasound images is of great significance for clinical diagnosis which can greatly boost the diagnostic accuracy of ultrasonography. However, due to the influence of ultrasound imaging principles and acquisition equipment, the collected ultrasound images naturally contain a large amount of speckle noise, which leads to a decrease in image quality and affects clinical diagnosis. To overcome this problem, we propose an improved denoising algorithm combining multi-filter DFrFT (Discrete Fractional Fourier Transform) and the adaptive fast BM3D (Block Matching and 3D collaborative filtering) method. Firstly, we provide the multi-filtering DFrFT method for preprocessing the original breast ultrasound image so as to remove some speckle noise early in fractional transformation domain. Based on the fractional frequency spectrum characteristics of breast ultrasound images, three types of filters are designed correspondingly in low, medium, and high frequency domains. And by integrating filtered images, the enhanced images are obtained which not only remove some speckle noise in background but also preserve the details of breast lesions. Secondly, for further enhancing the image quality on the basis of multi-filter DFrFT, we propose the adaptive fast BM3D method by introducing the DBSCAN-based super pixel segmentation to block matching process, which utilizes super pixel segmentation labels to provide a reference on how similar it is between target block and retrieval blocks. It reduces the number of blocks to be retrieved and make the matched blocks with more similar features. At last, the local noise parameter estimation is also adopted in the hard threshold filtering process of traditional BM3D algorithm to achieve local adaptive filtering and further improving the denoising effect. The synthetic data and real breast ultrasound data examples show that this combined method can improve the speckle suppression level and keep the fidelity of structure effectively without increasing time cost.

The fast computation of multi-angle discrete fractional Fourier transform

The discrete fractional Fourier transform (DFRFT) plays an important role in processing time-varying signals. Nevertheless, directly computing the DFRFT involves high complexity, particularly when addressing multi-angle DFRFT scenarios. This article presents a method to compute multi-angle DFRFTs through the execution of a single complex DFRFT, exploiting several properties of the DFRFT. We first calculate the DFRFTs with rotation angles 0 and α. Subsequently, when dealing with the DFRFTs of two real signals with rotation angles α and β, which is so-called the multi-angle DFRFTs, our method only need one complex DFRFT and some additional manipulations with complexity O(M), which reduce the computational complexity efficiently. Furthermore, the proposed method is also applicable to the processing of two dimensional (2D) signals. Additionally, as a generalized form of the DFRFT, the fast computation of the multi-angle discrete affine Fourier transform (DAFT) is also considered. Finally, the simulation results confirm that the proposed methods can effectively reduce the computational complexity without compromising precision.

Auto-calibrating universal programmable photonic circuits: hardware error-correction and defect resilience.

It is recently shown that discrete N × N linear unitary operators can be represented by interlacing N + 1 phase shift layers with a fixed intervening operator such as discrete fractional Fourier transform (DFrFT). Here, we show that introducing perturbations to the intervening operations does not compromise the universality of this architecture. Furthermore, we show that this architecture is resilient to defects in the phase shifters as long as no more than one faulty phase shifter is present in each layer. These properties enable post-fabrication auto-calibration of such universal photonic circuits, effectively compensating for fabrication errors and defects in phase components.

Circuit of Quantum Fractional Fourier Transform

In this paper, we first use the quantum Fourier transform (QFT) and quantum phase estimation (QPE) to realize the quantum fractional Fourier transform (QFrFT). As diverse definitions of the discrete fractional Fourier transform (DFrFT) exist, the relationship between the QFrFT and a classical algorithm is then established; that is, we determine the classical algorithm corresponding to the QFrFT. Second, we observe that many definitions of the multi-fractional Fourier transform (mFrFT) are flawed: when we attempt to propose a design scheme for the quantum mFrFT, we find that there are many invalid weighting terms in the definition of the mFrFT. This flaw may have very significant impacts on relevant algorithms for signal processing and image encryption. Finally, we analyze the circuit of the QFrFT and the reasons for the observed defects.

Estimation of LFM signal parameters using RD compressed sampling and the DFRFT dictionary

In this paper, a method combining random demodulator (RD) and discrete fractional Fourier transform (DFRFT) dictionary is suggested to directly estimate the parameters of linear frequency modulation (LFM) signals from compressed sampling data. First, the RD system parameters are modified in accordance with the characteristics of the LFM signal to produce effective compressed sampling data. Next, a DFRFT dictionary is built using the fractional Fourier transform theory, and sparse representation coefficients are obtained by reconstructing the compressed sampling data using the recovery algorithm and DFRFT dictionary. The signal exhibits characteristics that make it pulse under the best fractional transform order, so the problem of signal parameter estimation can be reduced to searching for the location of the maximum value of sparse representation coefficients. The location is determined by a parameter optimization algorithm, and from there, the initial frequency and Chirp rate of the LFM signal can be estimated. Lastly, simulation and real data tests are performed to confirm that the suggested method can directly be utilized to estimate the parameter of LFM signals using compressed sampling data in addition to having high sparse representation ability for LFM signals.

Discrete combined fractional Fourier transform and its application to image enhancement

Discrete combined fractional Fourier transform and its application to image enhancement

Hardware Architectures for Computing Eigendecomposition-Based Discrete Fractional Fourier Transforms with Reduced Arithmetic Complexity

Hardware Architectures for Computing Eigendecomposition-Based Discrete Fractional Fourier Transforms with Reduced Arithmetic Complexity

On refinements to QMFD based chirp parameter estimation

Commuting matrix methods furnish a full basis of orthogonal eigenvectors for the discrete Fourier transform or its centered version needed for computing the discrete fractional Fourier transform and multicomponent chirp signal analysis. However, these approaches suffer from ill-conditioning issues at higher matrix sizes, and require a computationally expensive eigenvalue decomposition.In this paper, ill-conditioning issues associated with the QMFD approach developed previously by the authors are addressed via diagonal modification. Further symmetries of the eigenvectors are used to reduce the size of the underlying eigenvalue problem. These modifications are then incorporated into the real-arithmetic implementation of the QMFD approach that is shown to be significantly superior to the conventional implementation and the corresponding MSE of the chirp parameter estimates are shown to approach their Cramer–Rao lower bounds.

Real-Time Discrete Fractional Fourier Transform Using Metamaterial Coupled Lines Network

Discrete fractional Fourier transforms (DFrFTs) are universal mathematical tools in signal processing, communications, and microwave sensing. Despite the excessive applications of DFrFT, the implementation of corresponding fractional orders in the baseband signal often leads to bulky, power-hungry, and high-latency systems. In this article, we present a passive metamaterial coupled lines network (MCLN) that performs the analog DFrFT in real-time at microwave frequencies. The proposed MCLN consists of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$</tex-math> </inline-formula> parallel microstrip transmission lines (TLs) in which adjacent TLs are loaded with interdigital capacitors to enhance the coupling level. We show that with the proper design of the coupling coefficients between adjacent channels, the MCLN can perform an M-point DFrFT of an arbitrary fractional order that can be designed through the length of the network. In the context of real-time signal processing for the realization of DFrFT, we design, model, simulate, and implement a 16 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$</tex-math> </inline-formula> 16 MCLN and experimentally demonstrate the performance of the proposed structure. The proposed innovative approach is versatile and is capable to be used in various applications where DFrFT is an essential tool. The proposed design scheme based on MCLN is scalable across the frequency spectrum and can be applied to the millimeter and submillimeter wave systems.

A use-case efficient DFrFT-based index modulated-OFDM system in the presence of CFO

Index modulated orthogonal frequency division multiplexing (OFDM-IM) only utilizes partial resources to transmit signals according to use case requirement, it leads to higher energy efficiency and spectral efficiency than the classical OFDM. However, in the presence of inter-carrier interference (ICI) due to carrier frequency offset (CFO), the performance of index modulated OFDM system severely deteriorates. To mitigate this issue, a novel discrete fractional Fourier transform (DFrFT) based OFDM-IM scheme over Rayleigh frequency selective fading channel is proposed in this paper. The optimum DFrFT angle (αopt) for this new scheme at various CFO values has been calculated theoretically. Incorporating these estimated αopt, the proposed system achieves a gain of 8 dB over conventional FFT-based OFDM-IM at bit error rate (BER) = 10−3 in a high normalized CFO (ϵ) environment (ϵ=0.2). The system model is also verified for different user applications, i.e., high data rate and low data rate applications. Simulation and analytical results show that the proposed model has a robust and superior performance compared to the conventional FFT-based OFDM-IM, DFrFT-based OFDM, and classical OFDM. Moreover, the proposed DFrFT-OFDM-IM model achieves all these superiorities with the same complexity as the conventional FFT-based OFDM-IM system.

Double image encryption based on a hybrid phase-truncation scheme

We propose a double image encryption approach based on a hybrid phase-truncation scheme. This approach is realized by spatial and Fresnel encoding, where two plaintexts are encrypted by a random amplitude mask into two private key masks and one ciphertext mask. The random amplitude mask is introduced to break the functional relation of the plaintexts to the ciphertext. Compared with other reported phase-truncation-based techniques, the proposed hybrid scheme is not only free from information leakage but also resistant to several different types of attacks. In addition, the parameters of the Fresnel transform and discrete multiple parameter fractional Fourier transform serve as additional keys to strengthen security significantly. Numerical simulations and a performance analysis are conducted to demonstrate the reliability and validity of the proposed approach.

Local discrete fractional fourier transform: An algorithm for calculating partial points of DFrFT

Fractional Fourier transform (FrFT) is widely utilized in nonstationary signal processing owing to its linearity and flexibility. Various discrete algorithms focus on the global discrete FrFT (DFrFT) points. However, only a part of the DFrFT points are considered because of a priori knowledge of applications and limited memory space of digital machines. Therefore, a local DFrFT (LDFrFT) was proposed in this study. LDFrFT for the first Q continuous DFrFT points is proposed, and its complexity was reduced from O(Mlog2(M)) to O(Mlog2(Q)) for an M-point sequence with Q being a factor of M. It is further extended to the LDFrFT for any Q continuous DFrFT points having complexity less than that of the DFrFT. These two algorithms were then generalized to local discrete linear canonical transform algorithms. The LDFrFT uses all sampling points, thus maintaining the resolution of the DFrFT. Moreover, LDFrFT is not limited by the starting point and is more flexible than the recently proposed local DFrFT algorithm. Finally, simulated frequency-modulated continuous-wave radar signals, gravitational waves, and real-world bat echolocation chirp data were utilized to demonstrate the effectiveness of the LDFrFT, which works well when the SNR is over -30 dB.

Discrete Quadratic-Phase Fourier Transform: Theory and Convolution Structures

The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval’s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.

Machine learning-based fault diagnosis for research nuclear reactor medium voltage power cables in fraction Fourier domain

Fault diagnosis of Medium Voltage power Cables (MVCs) research nuclear reactor, incredibly inaccessible/remote ones, has to be carefully identified, located, and fixed within a short time. Therefore, this paper proposes a perfect simultaneous fault diagnosis scheme based on Multiclass Support-Vector Machine (MCSVM) in the fractional Fourier domain. First, the three-phase sending currents are simulated under different conditions then their features are extracted using Discrete Fractional Fourier Transform (DFRFT). Afterward, the features reduction process occurs via the Singular Value Decomposition (SVD) approach. MCSVM scheme is used to diagnose faults (i.e., discover, categorize, and trace) using reduced features outcome from DFRFT and SVD stages. Alternating Transient Program/Electromagnetic Transient Program (ATP/EMTP) simulations have been carried out for 22 kV unreachable MVC. Different kernels of SVM, i.e., linear, quadratic, or polynomial, and diverse factors of DFRFT, i.e., α, are investigated in simulations to obtain the optimum performance (i.e., best α and kernel pair). Hence, performance analysis of the proposed diagnosis method under different conditions (i.e., various fault resistances, locations, and inception angles) concluded two highest accuracy and lowest time settings, which were found at α = 0.5 (for both) quadratic kernel, and linear kernel, respectively. Moreover, the linear kernel achieves 99.8% accuracy rate, the lowest execution time (10 ms), and fault tracing error rate of 0.525789%, which is proper for real-time applications. Besides, our proposed method is more reliable and accurate against variable operating conditions (fault resistances, distances, and inception angles), leading to more reliable power production systems.

Double-image compression and encryption scheme based on chaotic system and real-preserving multi-order discrete fractional Fourier transform

In order to reduce the amount of data transmission and key consumption, a double-image compression and encryption scheme based on chaotic system and real-preserving multi-order discrete fractional Fourier transform (RPMODFrFT) is proposed in this paper. It can compress and encrypt two images simultaneously, and the measurement matrix of compression sensing is controlled by 2D Logistic-Sine-Coupling map (2D-LSCM). After consolidating two compressed images into one image, the image is further encrypted by scrambling and 2D-LSCM controlled RPMODFrFT operation. At last, the bitwise XOR operation using 2D-LSCM with the initial value related to the plain image is carried out. The simulation results show that the proposed encryption-compression scheme has high security and robustness through a lot of attack analysis, i.e. differential attack analysis (NPCR > 99.6022% and UACI >33.5658%), entropy values (average 7.99710), statistical attack (correlation coefficient values are almost 0), noise attack and cropping attack.