Discrete Fractional Fourier Transform Research Articles

Graph Neural Networks optimized with Gazelle Optimization Algorithm for Urban Plantscape Design Based on Large-scale Street View

The center of human activity and a key point of contact for social interaction between city people and the built environment is the street. With more street-level photos becoming available, urban landscape studies have more chances to examine and evaluate streetscapes closely and from various angles. This manuscript present the graph neural networks (GNN) optimized with gazelle optimization algorithm (GOA) for urban plants cape design based on large-scale street view (UPLS-GNN-GOA). Initially data is taken from ADE20K dataset. Afterward the data is fed to General Robust Subband Adaptive Filtering (GRSAF) based pre-processing process. The pre-processing output is given to Discrete Fractional Fourier Transforms (DFrFT) to extract the street greenery and openness of street canyons. After that, the extracted features are fed to Graph Neural Networks (GNN) to classify the public health, urban microclimate, human perception. The weight parameters of the GNN are optimized using Gazelle Optimization Algorithm (GOA).The UPLS-GNN-GOA method is implemented in Python, and its efficiency is determined using a series of performance evaluation metrics analysis, including accuracy, precision, recall, f1-score, and sensitivity. The proposed UPLS-GNN-GOA method shows the highest accuracy of 98%, precision of 98%, and F1-score of 97% while comparing other existing methods such as urban plantscape design based on convolutional neural network (UPLS-CNN), urban plantscape design based on deep convolutional neural network (UPLS-DCNN) and urban plantscape design based on deep learning and convolutional neural network (UPLS-DL-CNN) respectively.

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.

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.

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.

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.

A Novel OTFS System Based on DFrFT-OFDM

Orthogonal Time Frequency Space modulation (OTFS) is a promising modulation technique expected to counter the severe Doppler effects encountered in a doubly dispersive channel. OTFS scheme is developed on the basis of Orthogonal Frequency Division Multiplexing (OFDM) systems to support communication between high-speed vehicles, whose superiority lies in the domain of the equivalent channel. OTFS converts the Time-Frequency (TF) domain channel in the OFDM system to Delay-Doppler (DD) domain channel that becomes a handy tool to overcome the difficulties faced in a frequency selective channel. This letter presents an OTFS system design that is developed on the Discrete Fractional Fourier Transform (DFrFT) based OFDM system, which is designed to perform better compared to the conventional OTFS system with the same design complexity. The simulation results evidentially show 1 dB gain in power at a Bit-Error-Rate (BER) of 10^&#x2212;5 and a significant 3 dB decrement in the PAPR when a high power pilot with a 25 dB is inserted in the OTFS data frame.

Multiweighted-Type Fractional Fourier Transform: Unitarity

The definition of the discrete fractional Fourier transform (DFRFT) varies, and the multiweighted-type fractional Fourier transform (M-WFRFT) is its extended definition. It is not easy to prove its unitarity. We use the weighted-type fractional Fourier transform, fractional-order matrix and eigendecomposition-type fractional Fourier transform as basic functions to prove and discuss the unitarity. Thanks to the growing body of research, we found that the effective weighting term of the M-WFRFT is only four terms, none of which are extended to M terms, as described in the definition. Furthermore, the program code is analyzed, and the result shows that the previous work (Digit Signal Process 2020: 104: 18) based on MATLAB for unitary verification is inaccurate.

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.

Light field image encryption based on spatial-angular characteristic

The 4D light field (LF) can be sliced into the spatial domain and the angular domain. Different from traditional images, LF image can simultaneously capture the direction and position of light propagation. With the development of image encryption technology, the security protection of LF images need to be paid further attention. Existing LF image encryption methods only explore the spatial information of the LF image, but ignore the angular information of LF image. In this paper, we propose a novel scheme for LF image encryption based on the internal spatial-angular characteristic of LF image. Firstly, considering the characteristic representation of the LF, we use the discrete fractional Fourier transform (DFrFT) to process the spatial domain of the LF image. Afterwards, in the angular domain, we employ the difference in viewing angles extracted from multiple angles to compose an epipolar plane image (EPI). The ciphertext EPI obtained by Arnold transform is displayed as an angular domain ciphertext image after reverse reconstruction. Furthermore, weighted linear blending and Chen’s chaotic system are performed on the ciphertext images in the spatial and angular domains to obtain the final ciphertext image. Experimental results demonstrate the effectiveness and applicability of the proposed scheme for LF image encryption.

Application Specific Integrated Circuit Implementation of Discrete Fractional Fourier Transform

Asian Journal of Organic & Medicinal Chemistry includes communications, full papers (articles), short papers, green chemistry, reviews, and perspectives. Specifically, these include synthesis, synthetic methods and regents, mechanism of reactions, biologically active compounds with their possible mechanism of action, natural products, in vitro and in vivo tests of compounds, computational chemistry, physical-analytical-inorganic chemistry, nanotechnology, drug carriers, proteomics and genomics. Rapid synthesis of new compounds following eco-friendly methods and their functions as medicinally active agents is certainly the central part of chemical and medicinal research in modern times. On the basis of this fact, significant emphasis is placed to rapid publication and disseminates original discoveries in this new journal.

Low complexity joint equalization and interference compensation in fractional Fourier domain for DCT-OFDM system with CFO

Orthogonal frequency division multiplexing (OFDM) systems based on discrete fractional Fourier transform (DFRFT) chirp basis replacing the complex exponential basis of discrete Fourier transform (DFT) is extensively studied in the literature. Authors of this paper have recently proposed fractional Fourier domain equalization (FFDE) for an OFDM system with a discrete cosine transform (DCT) multicarrier basis. The proposed DCT-OFDM system with FFDE nearly achieves the performance of the DFRFT-OFDM system without requiring any chirp rate feedback to the transmitter. This is enabled by separately performing the tasks of multicarrier modulation/demodulation with IDCT/DCT kernels at transmitter/receiver and FFDE using DFRFT/IDFRFT kernels both at the receiver. However, in the presence of a high value of CFO and also with higher modulation order, the residual interference power existing after channel equalization in fractional Fourier domain (FFD) using the channel decomposed with DFRFT significantly degrades the error rate performance. In this paper, the equalization and interference compensation is treated as a joint and linear problem in FFD after DFRFT at the receiver. Full compensation by performing direct matrix inverse can ensure the performance free from CFO for the DCT-OFDM system with both FDE and FFDE, but will drastically increase the computational cost to the O(N3) for implementing the joint equalizer at the receiver for N subcarriers. Therefore, low complexity joint equalization and interference compensation (JEIC) schemes are proposed for the DCT-OFDM system with FFD decomposition utilizing the banded implementation of joint ICI matrix after DFRFT at the receiver. The banded JEIC can closely preserve the interference-free performance and only requires operations in the O(N) to implement the joint equalizer.

The hopping discrete fractional Fourier transform

The discrete fractional Fourier transform (DFrFT) is a powerful signal processing tool for non-stationary signals. Many types of DFrFT have been derived and successful used in different areas. However, for real-time applications that require recalculating the DFrFT at each or several samples, the existing discrete algorithms aren’t the optimal. In this paper, the sliding window algoritm is used to resolve this problem. First, the sliding DFrFT (SDFrFT) algorithm with sliding step p is proposed, termed as the hopping DFrFT (HDFrFT) algorithm. Two different windowing methods which can realize windowing in the sliding process are also proposed to reduce fractional spectral leakage. Second, we apply the sliding window algorithm in computing the discrete time fractional Fourier transform (DTFrFT) and propose the hopping DTFrFT (HDTFrFT) algorithm to obtain a continuous fractional spectrum. Third, the sliding algorithm is further extended to compute the discrete fractional cosine/sine/Hartley transform (DFrCT/DFrST/DFrHT), respectively. Finally, the simulations results confirm that in a sliding process, our proposed sliding algorithms can greatly reduce the computation complexity without degrading the precision.

Low complexity interference compensation for DFRFT‐based OFDM system with CFO

Discrete fractional Fourier transform (DFRFT) based orthogonal frequency division multiplexing (OFDM) system significantly outperforms the conventional discrete Fourier transform (DFT) based OFDM system in the presence of carrier frequency offset (CFO). However, to achieve the interference-free performance, the residual interference remaining after the channel equalisation in the fractional Fourier domain needs to be compensated. To enable this residual interference compensation with low complexity after DFRFT at the receiver, an accurate and generalized expression for inter-carrier interference (ICI) coefficient from interfering to the desired subcarrier is derived over a multipath fading channel with CFO. The derived ICI coefficient is validated at the DFRFT angle equal to π / 2 (i.e., for DFT-OFDM system) and over AWGN and flat channel. Monte Carlo simulation of symbol error rate (SER) over the Rayleigh fading channel is also carried out to verify the derived theoretical closed-form ICI coefficient for different values of the DFRFT angle. At the receiver, a low complexity residual interference cancellation stage is introduced by utilizing the banded matrix implementation of the derived closed-form residual ICI matrix for the DFRFT-OFDM system which only needs to perform O ( N ) operations for closely preserving the interference-free performance.

Efficient equalisers for OFDM and DFrFT-OCDM multicarrier systems in mobile E-health video broadcasting with machine learning perspectives

Recently, the orthogonal frequency-division multiplexing (OFDM) system has become an appropriate technique to be applied on the physical layer in various requests, mainly in wireless communication standards, which is the reason to use OFDM within mobile wireless medical applications. The OFDM with cyclic prefix (CP) can compensate lacks for the time-invariant multi-path channel effects using a single tap equaliser. However, for mobile wireless communication, such as the use of OFDM in ambulances, the Doppler shift is expected, which produces a doubly dispersive communication channel where a complex equaliser is needed. This paper proposes a low-complexity band LDLH factorisation equaliser to be applied in mobile medical communication systems. Moreover, the discrete fractional Fourier transform (DFrFT) is used to improve the communication system’s performance over the OFDM. The proposed low-complexity equaliser could improve the OFDM, and the DFrFT-orthogonal chirp-division multiplexing (DFrFT-OCDM) system’s performance, as illustrated in the simulation results. This proves that the recommended system outperforms the standard benchmark system, which is an essential factor as it is to be applied within mobile medical systems.

Pulse shape discrimination of LSO and LuYAP scintillators based on Discrete Fractional Fourier Transform

Parallax error decreases the accuracy of the Positron Emission Tomography (PET) scanner. One of suitable solutions to reduce this error is to gain the depth of interaction information in a PET which uses the phoswich detectors. The pulse shape of the scintillator material can identify the corresponding layer of interaction within the phoswich detector by using Pulse Shape Discrimination (PSD) methods. In this work, we propose the PSD based on a Discrete Fractional Fourier Transform (DFRFT) to extract the features of the scintillation pulses. Then, we use the Support Vector Machine (SVM) to classify these features. A data set consists of 100 000 scintillation pulses for LSO and LuYAP crystals are discriminated using the proposed method. Different fraction factors of the DFRFT are studied to select the optimum one. Also, the SVM is applied using linear, radial basis function, or quadratic kernel. The highest efficiency of the proposed PSD is 96.3% when the applied fraction factor is 0.8 and a quadratic SVM kernel is used.