This paper introduces for the first time the generalized concept of the mutual inductance in the fractional-order domain where the symmetrical and unsymmetrical behaviors of the fractional-order mutual inductance are studied. To use the fractional mutual inductance in circuit design and simulation, an equivalent circuit is presented with its different conditions of operation. Also, simulations for the impedance matrix parameters of the fractional mutual inductance equivalent circuit using Advanced Design System and MATLAB are illustrated. The Advanced Design System and MATLAB simulations of the double-tuned filter based on the fractional mutual inductance are discussed. A great matching between the numerical analysis and the circuit simulation appears, which confirms the reliability of the concept of the fractional mutual inductance. Also, the analysis of the impedance matching using the fractional-order mutual inductance is introduced. © 2015 John Wiley & Sons, Ltd.
Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts
This paper discusses the diffusion and reaction behaviors of catalyst pellets in the fractional-order domain as well as the case of nth-order reactions. Two generic models are studied to calculate the concentration of reactant in a porous catalyst in the case of a spherical geometric pellet and a flat-plate particle with different examples. A controlled Picard analytical method is introduced to obtain an approximated solution for these systems in both linear and nonlinear cases. This method can cover a wider range of problems due to the extra auxiliary parameter, which enhances the convergence and is suitable for higher-order differential equations. Moreover, the exact solution in the linear fractional-order system is obtained using the Mittag–Leffler function where the conventional solution is a special case. For nonlinear models, the proposed method gives matched responses with the homotopy analysis method (HAM) solutions for different fractional orders. The effect of fractional-order parameter on the dimensionless concentration of the reactant in a porous catalyst is analyzed graphically for different cases of order reactions and Thiele moduli. Moreover, the proposed method has been applied numerically for different cases to predict and calculate the dual solutions of a nonlinear fractional model when the reaction order n = ?1. © 2017, Copyright © Taylor & Francis Group, LLC.
Three Fractional-Order-Capacitors-Based Oscillators with Controllable Phase and Frequency
This paper presents a generalization of six well-known quadrature third-order oscillators into the fractional-order domain. The generalization process involves replacement of three integer-order capacitors with fractional-order ones. The employment of fractional-order capacitors allows a complete tunability of oscillator frequency and phase. The presented oscillators are implemented with three active building blocks which are op-Amp, current feedback operational amplifier (CFOA) and second generation current conveyor (CCII). The general state matrix, oscillation frequency and condition are deduced in terms of the fractional-order parameters. The extra degree of freedom provided by the fractional-order elements increases the design flexibility. Eight special cases including the integer case are illustrated with their numerical discussions. Three different phases are produced with fixed sum of 2p which can be completely controlled by fractional-order elements. A general design procedure is introduced to design an oscillator with a specific phase and frequency. Two general design cases are discussed based on exploiting the degrees of freedom introduced by the fractional order to obtain the required design. Spice circuit simulations with experimental results for some special cases are presented to validate the theoretical findings. © 2017 World Scientific Publishing Company.
A Study on Fractional Power-Law Applications and Approximations
The frequency response of the fractional-order power-law filter can be approximated by different techniques, which eventually affect the expected performance. Fractional-order control systems introduce many benefits for applications like compensators to achieve robust frequency and additional degrees of freedom in the tuning process. This paper is a comparative study of five of these approximation techniques. The comparison focuses on their magnitude error, phase error, and implementation complexity. The techniques under study are the Carlson, continued fraction expansion (CFE), Padé, Charef, and MATLAB curve-fitting tool approximations. Based on this comparison, the recommended approximation techniques are the curve-fitting MATLAB tool and the continued fraction expansion (CFE). As an application, a low-pass power-law filter is realized on a field-programmable analog array (FPAA) using two techniques, namely the curve-fitting tool and the CFE. The experiment aligns with and validates the numerical results. © 2024 by the authors.
Secure blind watermarking using Fractional-Order Lorenz system in the frequency domain
This paper investigates two different blind watermarking systems in the frequency domain with the development of a Pseudo Random Number Generator (PRNG), based on a fractional-order chaotic system, for watermark encryption. The methodology is based on converting the cover image to the YCbCr color domain and applying two different techniques of frequency transforms, Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT), to the Y channel. Then, the encrypted watermark is embedded in the middle-frequency band and HH band coefficients for the DCT and DWT, respectively. For more security and long encryption key size, the fractional-order Lorenz system is used to double the encryption key size and make it secure against brute-force attacks. The proposed algorithms successfully detect the hidden watermark by using the statistical properties of the embedding media, where the PRNG is examined using statistical tests and the watermarking systems are evaluated using standard imperceptibility and robustness measures. Common attacks such as noise-adding attacks, image enhancement attacks and geometric transformation attacks are discussed. Results of the PRNG demonstrate sensitivity to the system parameters, and results of the watermarking systems show good imperceptibility while keeping the robustness measures in a good range. © 2023 Elsevier GmbH
Analysis and Guidelines for Different Designs of Pseudo Random Number Generators
The design of an efficient Pseudo Random Number Generator (PRNG) with good randomness properties is an important research topic because it is a core component in many applications. Based on an extensive study of most PRNGs in the past few decades, this paper categorizes six distinct design scenarios under two primary groups: non-chaotic and chaotic generators. The non-chaotic group comprises Linear Feedback Shift Registers (LFSR) with S-Boxes, primitive roots, and elliptic curves, whereas the chaotic group encompasses discrete, continuous, and fractional-order chaotic generators. This paper delves into the related scientific summaries, equations, flowcharts, and designs with necessary recommendations for each PRNG scenario. Even though the focus is on the basic design characteristics that provide simple, functional and secure PRNGs, it is possible to enhance those designs for additional features and improved efficiency. Simulation outcomes and system key configurations, which produce long random sequences, are also presented and evaluated using leading criteria. The evaluation criteria include the National Institute of Standards and Technology (NIST) SP-800-22 test suite, TestU01 randomness tests, histogram, entropy, autocorrelation, and cross-correlation. Furthermore, key space, key sensitivity, and bit rate indicate that all designed examples meet international standards with high quality. The presented PRNGs are compared and integrated into an image encryption system. Although each PRNG design scenario can have a different key space, simple designs with fixed-length system keys are chosen for the sake of proper comparisons. Statistical and security assessments of the encryption system demonstrate that the PRNGs are cryptographically secure. © 2013 IEEE.
Hardware Accelerator of Fractional-Order Operator Based on Phase Optimized Filters With Applications
Hardware accelerators outperform CPUs in terms of performance by parallelizing the algorithm architecture and using the device’s programmable resources. FPGA is a type of hardware accelerator that excels not only in performance but also in energy efficiency. So, it provides a suitable platform for implementing complicated fractional-order systems. This paper proposes a novel phase-based optimization method to implement fractional operators using FIR and IIR filters. We also compare five fractional operator implementation methods on FPGA regarding resource utilization, execution time, power, and accuracy. These methods and the proposed one are evaluated in terms of power consumption, delay, and resources to assist the designer in determining the most suitable implementation method for the given application. The proposed method has a lower phase error of 14.7% in the case of derivative operation and a lower phase error of 18.83% in the case of integration compared to the literature. In addition, the proposed methods decreased the consumed power and area by more than three times compared to the fixed-window GL fractional operator. The proposed approach implements Heaviside’s inductor-terminated lossy line. In addition, it is employed as an edge detection kernel to demonstrate its effectiveness in image processing applications. © 2023 IEEE.
Optimization of Double fractional-order Image Enhancement System
Image enhancement is a vital process that serves as a tool for improving the quality of a lot of real-life applications. Fractional calculus can be utilized in enhancing images using fractional order kernels, adding more controllability to the system, due to the flexible choice of the fractional order parameter, which adds extra degrees of freedom. The proposed system merges two fractional order kernels which helps in image enhancement techniques, and the contribution of this work is based on the study of how to optimize this process. The optimization of the two fractional kernels was done using the neural network optimization algorithm (NNA) to utilize the best order for the two kernels. In this paper, three fractional kernels are studied to highlight the performance of image enhancement using fractional kernels against different metrics. Furthermore, three different combinations of two kernels are combined and studied to enhance the metrics score by utilizing two different fractional orders for each kernel. Various optimization algorithms are used to obtain the optimum fractional order for both single and combined kernels. Using the constrained NNA, the evaluation metrics of the image enhancement show a 33% increase in measure of enhancement metric (EME), 21% increase in contrast, and 4% increase in average gradient compared to the best-achieved metrics by the literature while keeping the similarity metric above 0.75. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Circuit realization and FPGA-based implementation of a fractional-order chaotic system for cancellable face recognition
Biometric security has been developed in recent years with the emergence of cancellable biometric concepts. The idea of the cancellable biometric traits is concerned with creating encrypted or distorted traits of the original ones to protect them from hacking techniques. So, encrypted or distorted biometric traits are stored in databases instead of the original ones. This can be accomplished through non-invertible transforms or encryption schemes. In this paper, a cancellable face recognition algorithm is introduced based on face image encryption through a fractional-order multi-scroll chaotic system. The fundamental concept is to create random keys that will be XORed with the three components of color face images (red, green, and blue) to obtain encrypted face images. These random keys are generated from the Least Significant Bits of all state variables of a proposed fractional-order multi-scroll chaotic system. Lastly, the encrypted color components of face images are combined to produce a single cancellable trait for each color face image. The results of encryption with the proposed system are full-encrypted face images that are suitable for cancellable biometric applications. The strength of the proposed system is that it is extremely sensitive to the user’s selected initial conditions. The numerical simulation of the proposed chaotic system is done with MATLAB. Phase and bifurcation diagrams are used to analyze the dynamic performance of the proposed fractional-order multi-scroll chaotic system. Furthermore, we realized the hardware circuit of the proposed chaotic system on the PSpice simulator. The proposed chaotic system can be implemented on Field Programmable Gate Arrays (FPGAs). To model our generator, we can use Verilog Hardware Description Language HDL, Xilinx ISE 14.7 and Xilinx FPGA Artix-7 XC7A100T based on Grunwald-Letnikov algorithms for mathematical analysis. The numerical simulation, the circuit simulation and the hardware experimental results confirm each other. Cancellable face recognition based on the proposed fractional-order chaotic system has been implemented on FERET, LFW, and ORL datasets, and the results are compared with those of other schemes. Some evaluation metrics containing Equal Error Rate (EER), and Area under the Receiver Operating Characteristic (AROC) curve are used to assess the cancellable biometric system. The numerical results of these metrics show EER levels close to zero and AROC values of 100%. In addition, the encryption scheme is highly efficient. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Mathematical techniques of fractional order systems : A volume in advances in nonlinear dynamics and chaos (ANDC)
Mathematical Techniques of Fractional Order Systems illustrates advances in linear and nonlinear fractional-order systems relating to many interdisciplinary applications, including biomedical, control, circuits, electromagnetics and security