Fractional Domain Research Articles

Bounded $$H^\infty $$-calculus for a class of nonlocal operators: the bidomain operator in the $$L_q$$-setting

The bidomain operator \({{\mathbb {A}}}\), a nonlocal operator, is studied in a bounded domain \(\Omega \subset {{\mathbb {R}}}^d\) with boundary \(\partial \Omega \) of class \(C^{2-}\) within the \(L_q\)-setting for \(1<q<\infty \). Assuming a fairly general framework, it is shown that this operator is sectorial, invertible on functions with mean zero, and admits a bounded \({H}^\infty \)-calculus with \({H}^\infty \)-angle 0. In particular, it has the property of maximal \(L_p-L_q\)-regularity and the fractional power domains \({{\mathcal {D}}}({{\mathbb {A}}}^\alpha )\) are identified. Furthermore, it is shown that \(-{{\mathbb {A}}}\) generates a strongly continuous, compact, analytic and exponentially stable semigroup on \(L_{q,0}(\Omega )\) with q-independent, purely countable point spectrum \(\sigma ({{\mathbb {A}}})\).

Smart noise jamming suppression method based on fast fractional filtering

Smart noise jamming based on digital radio frequency memory (DRFM) has both oppressive jamming effect and deceptive jamming effect, and it is difficult to be effectively suppressed by traditional anti-jamming methods. In order to solve this problem, an anti-jamming method for linear frequency modulation (LFM) pulse compression radar system is proposed. The proposed method is combined with fractional Fourier transform (FrFT) and fractional domain filtering strategy. Here, first, the FrFT transform is performed on the received signal, and the optimal transform angle is calculated by calculating the fractional fourth-order origin moment. Then, the narrow-band filter is performed in the fractional domain to extract the target signal and suppress the jamming signal. Finally, fractional Fourier inverse transform is done to restore the target signal. The simulation results show that the method not only can effectively suppress smart noise jamming but also can accurately detect the position of the target.

The application of fractional wavelet transform in image enhancement

The fractional wavelet transform is a generalized extension of the wavelet multi-resolution analysis in time-fractional frequency domain. It can characterize signal features in time domain and fractional frequency domain. Based on the multi-resolution analysis of fractional wavelet, the process of decomposition and reconstruction of discrete fractional wavelet transform is derived, and it is extended to two dimensions. According to the characteristic that the subband energy of fractional wavelet transform varies with the order P, a novel image enhancement method is proposed. All the fractional wavelet coefficients at large scale and low resolution are retained, and a nonlinear adaptive gain method is used for the high-resolution subband coefficients. The experimental results show that the performance of this method is better than the traditional wavelet enhancement method both in vision effect and contrast improvement index.

Fractional X-shape controllable multi-scroll attractor with parameter effect and FPGA automatic design tool software

Abstract This paper proposes a new fractional-order multi-scrolls chaotic system. More complex systems and flexible ranges of the chaotic behavior are obtained due to the extra parameters added by the fractional-order. The proposed system has novel complex chaotic behaviors. The effect of changing the system parameters on the system behavior is investigated and their bifurcation diagrams have been provided. The MLE for the proposed system in integer and fractional domain has been discussed. It shows that the proposed chaotic system is richer in the case of fractional-order. A novel FPGA design automation tool for the proposed system is also proposed. It provides an optimized FPGA implementation of the system according to different system parameters. Additionally, it provides a facility and fast way to accelerate the design process for the developer to design their own module while applying the system in different applications. The proposed tool has been tested using Xilinx's Virtex-5 XC5VLX50T FPGA and verified using MATLAB. The experimental results are provided for different design peripherals options. Finally, a procedure to generalize the proposed tool to involve any other chaotic system is presented.

Fractional Order High Pass Filter Based on Operational Transresistance Amplifier with Three Fractional Capacitors of Different Order

Design, realization and performance evaluation of fractional order high pass filter based on Operational Transresistance Amplifier (OTRA) using three fractional elements of different order α, β, γ are presented in this paper. The projected circuit uses a single active current mode device i.e. OTRA and utilizes fractional capacitors of varying orders to boost the design flexibility. The design equations for fractional order high pass filter are computed from the transfer function. The paper elaborates the impact of fractional order elements of varying order on the frequency response of the filter. Subsequently, the sensitivity and stability of the transfer function of the proposed filter are also analyzed. The potential usefulness of the proposed filter is additionally incontestable through the Total Harmonic Distortion (THD), Power Supply Rejection Ratio (PSRR), Temperature sweep, Corner, Supply Voltage variation and Noise Analysis. It has been observed that the proposed pass filter in fractional domain supported OTRA provides greater flexibility in controlling the magnitude characteristics.

The Time Dimensional Measurability Aware FDE Based Analysis of Active Circuit in The Fractional Domain

In this research, the analysis of the active circuit in the fractional domain has been performed by using the fractional differential equation approach where themeasurability in time dimension of the derivative term has also been concerned unlike the previous work. The OTA-C filter has been adopted as the candidate active circuit due to its compactness and renown. The derivative term of the fractional differential equation which includes the fractional time component parameters for obtaining such time dimensional measurability, has been interpreted in Caputo sense and the analytical solution of such equation has been determined with the aid of Laplace transformation. With the obtained solution, the time dimensional measurability aware fractional derivative based circuit responses to various inputs have been determined, the circuit fractional time constant and other crucial time parameters has been determined and thetemporal behaviour of the circuit has been analysed in the fractional domain. The loci of the pole on W-plane has been determined and the stability analysis has also been presented. Moreover, we also mathematically proof that the OTA-C filter in the fractional domain can be electronically realized with time dimensionalmeasurability awareness by using the state of the art fractional capacitor.

An asymmetric cryptosystem based on the random weighted singular value decomposition and fractional Hartley domain

A new asymmetric encryption system for double random phase encoding based on random weighted singular value decomposition and fractional Hartley transform domain has been proposed. Random weighted singular value decomposition is purely based upon random weights, isometric matrix and orthogonal triangular decomposition and all these fragments enhances the security of double random phase encoding cryptosystem. Random weights and orthogonal triangular decomposition are considered as heart of this cryptosystem. This system is carried out in fractional Hartley domain, where fractional orders play a vital role. On the receiver side, it is only possible to decrypt the image if anyone knows all the three components, its multiplication order, fractional order of fractional Hartley transform. Proposed cryptosystem is efficiently compared with singular value decomposition and truncated singular value decomposition. Similar to singular value decomposition and truncated singular value decomposition, proposed cryptosystem also yields three components. Because of random weights, these three components are highly differing from traditional singular value decomposition and truncated singular value decomposition components. Some analysis is offered to authenticate the opportunity.

Reliability enhancement in multi-numerology-based 5G new radio using INI-aware scheduling

Multi-numerology waveform-based 5G new radio (NR) systems offer great flexibility for different requirements of users and services. However, there is a new type of problem that is defined as inter-numerology interference (INI) between multiple numerologies. This paper proposes novel scheduling and resource allocation techniques to enhance the overall reliability and also provide extra protection for ultra-reliable and low-latency communications (uRLLC) users and cell edge users against INI. Proposed methods are useful for Internet of Things (IoT) communications, and they do not cause additional spectral usage, computational complexity, and latency. Practical INI-aware schemes in this paper include fractional numerology domain (FND) scheduling, power difference-based (PDB) scheduling, and machine learning-based (MLB) scheduling algorithms. INI and signal-to-interference ratio (SIR) results for multi-numerology systems are obtained through computer simulations to show trade-offs between different scenarios and success of the proposed algorithms.

Triple-level cryptosystem using deterministic masks and modified gerchberg-saxton iterative algorithm in fractional Hartley domain by positioning singular value decomposition

We present a novel method for image encryption by Modified Gerchberg–Saxton Iterative Algorithm (MGSIA) using deterministic masks (DMKs) and singular value decomposition (SVD) in fractional Hartley transform (FrHT) domain. This proposal delivers triple-level encryption (L1, L2 and L3) and all these three levels can be iterated number of times. First and third level of encryption routines MGSIA, DMK and FrHT, whereas mid-level encryption uses SVD. Since L1 delivers single cipher image and L3 yields double cipher images, this technique is also known as multiple image encryption system. Each MGSIA generates private keys and these keys are engendered in such a way that decryption is to be expected with traditional DRPE system. Because of the nature of these two asymmetric keys, the proposed system is non-linear and by the way, it boosts security. In the place of traditional random phase masks, the proposed cryptosystem uses statistically independent deterministic masks in L1 and L3. DMKs are nothing but the class of structured phase masks. DMKs are generated and used as the phase for MGSIA in L1 and L3. These DMKs are formed by the deviation from conventional rectangular function and limited range values which delivers key components with reduced size, better performance and lower complexity. Deployment of SVD on L2 decomposes the cipher image1 into three components and out of these three components, due to the nature of diagonality, S components plays a major role in proposed work. Numerical simulations have been achieved to prove the strength and feasibility of the proposed scheme.

A Comparison of ECG Signal Pre-processing Using FrFT, FrWT and IPCA for Improved Analysis

ObjectiveElectrocardiogram (ECG) is a diagnostic tool for recording electrical activities of the human heart non-invasively. It is detected by electrodes placed on the surface of the skin in a conductive medium. In medical applications, ECG is used by cardiologists to observe heart anomalies (cardiovascular diseases) such as abnormal heart rhythms, heart attacks, effects of drug dosage on subject's heart and knowledge of previous heart attacks. Recorded ECG signal is generally corrupted by various types of noise/distortion such as cardiac (isoelectric interval, prolonged depolarization and atrial flutter) or extra cardiac (respiration, changes in electrode position, muscle contraction and power line noise). These factors hide the useful information and alter the signal characteristic due to low Signal-to-Noise Ratio (SNR). In such situations, any failure to judge the ECG signal correctly may result in a delay in the treatment and harm a subject (patient) health. Therefore, appropriate pre-processing technique is necessary to improve SNR to facilitate better treatment to the subject. Effects of different pre-processing techniques on ECG signal analysis (based on R-peaks detection) are compared using various Figures of Merit (FoM) such as sensitivity (Se), accuracy (Acc) and detection error rate (DER) along with SNR. MethodsIn this research article, a new fractional wavelet transform (FrWT) has been proposed as a pre-processing technique in order to overcome the disadvantages of other existing commonly used techniques viz. wavelet transform (WT) and the fractional Fourier transform (FrFT). The proposed FrWT technique possesses the properties of multiresolution analysis and represents signal in the fractional domain which consists of representation in terms of rotation of signals in the time–frequency plane. In the literature, ECG signal analysis has been improvised using statistical pre-processing techniques such as principal component analysis (PCA), and independent component analysis (ICA). However, both PCA and ICA are prone to suffer from slight alterations in either signal or noise, unless the basis functions are prepared with a worldwide set of ECG. Independent Principal Component Analysis (IPCA) has been used to overcome this shortcoming of PCA and ICA. Therefore, in this paper three techniques viz. FrFT, FrWT and IPCA are selected for comparison in pre-processing of ECG signals. ResultsThe selected methods have been evaluated on the basis of SNR, Se, Acc and DER of the detected ECG beats. FrWT yields the best results among all the methods considered in this paper; 34.37dB output SNR, 99.98% Se, 99.96% Acc, and 0.036% DER. These results indicate the quality of biology-related information retained from the pre-processed ECG signals for identifying different heart abnormalities. ConclusionCorrect analysis of the acquired ECG signal is the main challenge for cardiologist due to involvement of various types of noises (high and low frequency). Twenty two real time ECG records have been evaluated based on various FoM such as SNR, Se, Acc and DER for the proposed FrWT and existing FrFT and IPCA preprocessing techniques. Acquired real-time ECG database in normal and disease situations is used for the purpose. The values of FoMs indicate high SNR and better detection of R-peaks in a ECG signal which is important for the diagnosis of cardiovascular disease. The proposed FrWT outperforms all other techniques and holds both analytical attributes of the actual ECG signal and alterations in the amplitudes of various ECG waveforms adequately. It also provides signal portrayals in the time-fractional-frequency plane with low computational complexity enabling their use practically for versatile applications.

The search for fractional order in heavy quarkonia spectra

Using the concept of conformable fractional derivative, we study the properties of fractional [Formula: see text]-dimensional Schrödinger equation for the potential [Formula: see text]. The extended Nikiforov–Uvarov method is generalized to the fractional domain and then employed to obtain the analytic exact energy eigenvalues and eigenfunctions and their dependence on the fractional order [Formula: see text] and the dimension [Formula: see text]. To test its applicability, we apply the method on heavy quarkonia systems, and reproduce their mass spectra and fractional radial probabilities at different values of [Formula: see text] and [Formula: see text]. Comparing the mass spectra with the experimental data, we discuss to what extent fractional models can account for some features in the description of heavy quarkonia at certain dimensional space.

A novel in-band OSNR monitoring technique based on fractional Fourier transform of LFM signal

We propose a novel in-band optical signal to noise ratio (OSNR) monitoring technique in digital coherent receiver based on fractional Fourier transform (FrFT), which employs the power aggregation property of linear frequency modulation (LFM) signal in fractional domain. In our method, the LFM signal with extremely low power is added on the data signal. Thus, the received signal exists a peak in fractional domain, which can be used to indirectly differentiate the data signal power from noise thereby calculating the OSNR. Meanwhile, the proposed technique does not occupy additional spectrum resource and avoids any additional hardware on top of the standard digital coherent receiver. Experimental results of a 10 Gbaud quadrature phase-shift keying (QPSK) system show that the proposed technique is highly accurate, with the monitoring error less than 0.55 dB in the OSNR range of 9–29 dB for back-to-back transmission and 0.9 dB in the OSNR range of 11–25 dB for 960 km standard single mode fiber (SSMF) transmission.

Moving Ground Target Detection With Main-Lobe Jamming Based on Fractional Fourier Transform and Differential Cancellation

If the active interference enters the main lobe of a radar antenna, the detection performance of the radar system will be severely degraded. Many traditional techniques for suppressing this type of interference are not sufficiently effective. Through the analysis of the characteristics of linear frequency modulated echoes and jamming in the fractional Fourier transform (FrFT) domain, this paper proposes a novel algorithm for moving ground target detection, which can eliminate multipath signals effectively and suppress the main-lobe jamming significantly. First, by choosing the right pulse number taken into the accumulation, the effect of the active interference and moving target echoes can be suppressed simultaneously in the fractional domain, and then the FrFT of multipath signals can be obtained. Second, the FrFT of moving target echoes can be extracted by differential cancellation, and the second coherent integration is employed to detect the moving targets and construct the FrFT-Doppler plane. By constant false-alarm rate detection, the peaks can be extracted from the FrFT-Doppler plane, and then the target parameters can be estimated properly. It is not necessary to estimate the direction of arrival of each interference or limit the number of interferences in the main lobe. The processing results of simulated and actual datasets verify that for the weak or low-velocity targets, the detection performance of the proposed algorithm is superior to that of conventional match filter processing.

Parameter Estimation of Delay-Doppler Underwater Acoustic Multi-Path Channel Based on Iterative Fractional Fourier Transform

The characterization of wideband underwater acoustic (UWA) multi-path signals is investigated in this paper. The UWA multi-path channel can be considered as a sparse model and it causes changes of the transmitted signal on Doppler factor, time delay, and amplitude, thus the signal component from each path can be differentiated from the set of parameters. This paper proposes an FrFT-based method to estimate the channel parameters, which can decompose and retrieve the multi-path signals from the UWA channels. The proposed algorithm proceeds in an iterative manner and returns a set of estimated parameters at each turn. The estimated time delay is obtained directly without any extra correlation computation. Then, the estimated component is extracted by eliminating its FrFT result from the fractional domain. Compared to the existing methods, the most improvement is that the method can be applied to the wideband LFM signals with a wider frequency range in low signal-to-noise environment. Because the indirect subtractions avoid the bad chain effect caused by the low resolution or estimation error. The effectiveness of the method is confirmed by several simulations with low and high frequency. The performance is compared with a matching pursuit-based algorithm and an FrFT-based algorithm. It is shown that the proposed method outperforms the existing algorithms in low-frequency multi-component separation as well as estimation accuracy.

On the fractional domain generalization of memristive parametric oscillators

AbstractIn this research, we generalize a family of electronic parametric oscillators in the fractional domain by using a state of the art circuit element namely fractional memristor. Such family o...

Approximation of the interface condition for stochastic Stefan-type problems

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

An Indoor Positioning Error Correction Method of Pedestrian Multi-Motions Recognized by Hybrid-Orders Fraction Domain Transformation

Aiming at the inherent problems of heading divergence and error accumulation in indoor inertial positioning of pedestrian field, we proposed a new indoor positioning error correction method for pedestrian multi-motions recognition. This method is aimed at pedestrians' seven common indoor movements. Periodically, divide the motion data of the accelerometer in the sensitive-axis-direction (vertical geodetic direction) of MEMS-IMU worn on the pedestrian's waist. And acquire the feature vector through feature extraction by the hybrid-orders fraction domain transformation. The effective features with high identification degree under the different optimal order transformations are mixed and matched. Then, these are sent to each sub-classifier to complete the classification process by the dichotomy and finish the processes of subsequent machine learning. The error correction of the heading angle and positioning is carried out combining with the improved HDE algorithm, the innovative floor constraints' method, and the motion states' transition correction technique. The final indoor positioning experiment results show that the classification effect of this motion detection method is better than the traditional methods'. The average classification accuracy can reach up to 97%. And the method reduces the computing requirements for hardware. By using the floor constraint method, the vertical height difference can achieve the effect of complete reset of the origin. The trajectory lines of the movement in the same floor are well displayed. At the same time, the best horizontal error positioning result is only 1.72 m during the total travel distance of 412.40 m (TTD ≈ 0.42%).

Off-Grid DOA Estimation for Wideband LFM Signals in FRFT Domain Using the Sensor Arrays

For the direction-of-arrival (DOA) estimators based on sparse signal representation, the feasibility and precision are greatly restricted by the inherent limitation of the predefined spatial discrete grids. In this paper, based on the high aggregation characteristic of wideband linear frequency modulation signals, we derive the modified fractional domain sparse model (MFDSM) in DOA estimation and propose a novel off-grid DOA estimation method via alternating descent iteration (OGDEADI). The simulation results show that our proposed MFDSM-OGDEADI method has enhanced estimation accuracy and better angular resolution in terms of signal-to-noise ratio, angle difference, grid size, and angle bias, and it also has the advantage of being less sensitive to the grid size.

Asymmetric encryption algorithm for colour images based on fractional Hartley transform

ABSTRACTThis paper presents a new scheme for encryption of single-channel colour images. The scheme uses amplitude- and phase-truncation approach to introduce non-linearity for enhanced security. Further, the colour image encryption is performed in the fractional Hartley domain, which is relatively less investigated. The encryption starts with an affine transform of each channel of the input colour image. Thereafter, one of the channels is considered as the input amplitude image while the other two are used as phase masks, one in the spatial and the other in the frequency domain. A detailed analysis of the scheme’s sensitivity to various encryption parameters has been carried out. In addition, security analysis of the scheme against attacks establishes the scheme’s robustness. The combined use of the affine transform and phase-truncation approach for colour image encryption in the fractional Hartley domain is attempted for the first time in this study. It is shown that the proposed scheme resists the special attack.

Volume holographic optical encryption and decryption in photorefractive LiNbO[formula omitted]:Fe crystal

We present general volume holographic encryption and decryption system by means of two wave mixing process that can apply to variety of diffraction domains, including the conventional Fourier hologram domain (using 4f-optical processor), the Fresnel hologram domain, and fractional Fourier hologram domain. A rigorous theory for volume holographic encryption and decryption is developed and presented under the framework of Collins diffraction integral formalism. The ability for volume holographic encryption and decryption are successfully demonstrated in a photorefractive LiNbO3:Fe crystal. We also demonstrate a robustness test of our holographic decryption system with respect to misalignment of the phase key in both longitudinal and transversal directions away from the original position and blind decryption without knowledge of the correct key.