A Physical Security Technology Based upon Doubly Multiple Parameters Weighted Fractional Fourier Transform

Enhancing the security of the wireless communication is necessary to guarantee the reliable of the data transmission, due to the broadcast nature of wireless channels. In this paper, we provide a novel technology referred to as doubly multiple parameters weighted fractional Fourier transform(DMWFRFT), which can strengthen the physical layer security of wireless communication. This paper introduces the concept of DM-WFRFT based on multiple parameters WFRFT(MP-WFRFT), and then presents its four properties. Based on these properties, the parameters decryption probability is analyzed in terms of the number of parameters. The number of parameters for DM-WFRFT is more than that of the MP-WFRFT,which indicates that the proposed scheme can further strengthen the the physical layer security. Lastly, some numerical simulations are carried out to illustrate that the efficiency of proposed DM-WFRFT is related to preventing eavesdropping, and the effect of parameters variety on the system performance is associated with the bit error ratio(BER).

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

A Mathematical Approach for Generating a Highly Non-Linear Substitution Box Using Quadratic Fractional Transformation

Nowadays,one of the most important difficulties is the protection and privacy of confidential data.To address these problems,numerous organizations rely on the use of cryptographic techniques to secure data from illegal activities and assaults.Modern cryptographic ciphers use the non-linear component of block cipher to ensure the robust encryption process and lawful decoding of plain data during the decryption phase.For the designing of a secure substitution box(S-box),non-linearity(NL)which is an algebraic property of the S-box has great importance.Consequently,the main focus of cryptographers is to achieve the S-box with a high value of non-linearity.In this suggested study,an algebraic approach for the construction of 16×16 S-boxes is provided which is based on the fractional transformation Q(z)=1/α(z)^(m)+β(mod257)and finite field.This technique is only applicable for the even number exponent in the range(2-254)that are not multiples of 4.Firstly,we choose a quadratic fractional transformation,swap each missing element with repeating elements,and acquire the initial S-box.In the second stage,a special permutation of the symmetric group S256 is utilized to construct the final S-box,which has a higher NL score of 112.75 than the Advanced Encryption Standard(AES)S-box and a lower linear probability score of 0.1328.In addition,a tabular and graphical comparison of various algebraic features of the created S-box with many other S-boxes from the literature is provided which verifies that the created S-box has the ability and is good enough to withstand linear and differential attacks.From different analyses,it is ensured that the proposed S-boxes are better than as compared to the existing S-boxes.Further these S-boxes can be utilized in the security of the image data and the text data.

SAR target detection based on the optimal fractional Gabor spectrum feature

In this paper,an algorithm based on a fractional time-frequency spectrum feature is proposed to improve the accuracy of synthetic aperture radar(SAR)target detection.By extending the fractional Gabor transform(FrGT)into two dimensions,the fractional time-frequency spectrum feature of an image can be obtained.In the achievement process,we search for the optimal order and design the optimal window function to accomplish the two-dimensional optimal FrGT.Finally,the energy attenuation gradient(EAG)feature of the optimal time-frequency spectrum is extracted for high-frequency detection.The simulation results show the proposed algorithm has a good performance in SAR target detection and lays the foundation for recognition.

Reservoir information extraction using a fractional Fourier transform and a smooth pseudo Wigner-Ville distribution

Currently, it is difficult for people to express signal information simultaneously in the time and frequency domains when analyzing acoustic logging signals using a simple-time or frequency-domain method. It is difficult to use a single type of time-frequency analysis method, which affects the feasibility of acoustic logging signal analysis. In order to solve these problems, in this paper, a fractional Fourier transform and smooth pseudo Wigner Ville distribution （SPWD） were combined and used to analyze array acoustic logging signals. The time-frequency distribution of signals with the variation of orders of fractional Fourier transform was obtained, and the characteristics of the time-frequency distribution of different reservoirs under different orders were summarized. Because of the rotational characteristics of the fractional Fourier transform, the rotation speed of the cross terms was faster than those of primary waves, shear waves, Stoneley waves, and pseudo Rayleigh waves. By choosing different orders for different reservoirs according to the actual circumstances, the cross terms were separated from the four kinds of waves. In this manner, we could extract reservoir information by studying the characteristics of partial waves. Actual logging data showed that the method outlined in this paper greatly weakened cross-term interference and enhanced the ability to identify partial wave signals.

Fractional S transform-Part 1:Theory

The S transform, which is a time-frequency representation known for its local spectral phase properties in signal processing, uniquely combines elements of wavelet transforms and the short-time Fourier transform （STFT）. The fractional Fourier transform is a tool for non-stationary signal analysis. In this paper, we define the concept of the fractional S transform （FRST） of a signal, based on the idea of the fractional Fourier transform （FRFT） and S transform （ST）, extend the S transform to the time-fractional frequency domain from the time- frequency domain to obtain the inverse transform, and study the FRST mathematical properties. The FRST, which has the advantages of FRFT and ST, can enhance the ST flexibility to process signals. Compared to the S transform, the FRST can effectively improve the signal time- frequency resolution capacity. Simulation results show that the proposed method is effective.

High-sensitive state perception method for inverter-fed machine turn insulation based on FrFT-Mel

Amidst the swift advancement of new power systems and electric vehicles,inverter-fed machines have progressively materialized as a pivotal apparatus for efficient energy conversion.Stator winding turn insulation failure is the root cause of inverter-fed machine breakdown.The online monitoring of turn insulation health can detect potential safety risks promptly,but faces the challenge of weak characteristics of turn insulation degradation.This study proposes an innovative method to evaluate the turn insulation state of inverter-fed machines by utilizing the fractional Fourier transform with a Mel filter(FrFT-Mel).First,the sensitivity of the high-frequency(HF)switching oscillation current to variations in turn insulation was analyzed within the fractional domain.Subsequently,an improved Mel filter is introduced,and its structure and parameters are specifically designed based on the features intrinsic to the common-mode impedance resonance point of the electrical machine.Finally,an evaluation index was proposed for the turn insulation state of inverter-fed machines.Experimental results on a 3kW permanent magnet synchronous machine(PMSM)demonstrate that the proposed FrFT-Mel method significantly enhances the sensitivity of turn insulation state perception by approximately five times,compared to the traditional Fourier transform method.

Security-Enhanced Directional Modulation Based on Two-Dimensional M-WFRFT

Directional modulation(DM)is one of the most promising secure communication techniques.However,when the eavesdropper is co-located with the legitimate receiver,the conventional DM has the disadvantages of weak anti-scanning capability,anti-deciphering capability,and low secrecy rate.In response to these problems,we propose a twodimensional multi-term weighted fractional Fourier transform aided DM scheme,in which the legitimate receiver and the transmitter use different transform terms and transform orders to encrypt and decrypt the confidential information.In order to further lower the probability of being deciphered by an eavesdropper,we use the subblock partition method to convert the one-dimensional modulated signal vector into a twodimensional signal matrix,increasing the confusion of the useful information.Numerical results demonstrate that the proposed DM scheme not only provides stronger anti-deciphering and anti-scanning capabilities but also improves the secrecy rate performance of the system.

Aerial target threat assessment based on gated recurrent unit and self-attention mechanism

Aerial threat assessment is a crucial link in modern air combat, whose result counts a great deal for commanders to make decisions. With the consideration that the existing threat assessment methods have difficulties in dealing with high dimensional time series target data, a threat assessment method based on self-attention mechanism and gated recurrent unit(SAGRU) is proposed. Firstly, a threat feature system including air combat situations and capability features is established. Moreover, a data augmentation process based on fractional Fourier transform(FRFT) is applied to extract more valuable information from time series situation features. Furthermore, aiming to capture key characteristics of battlefield evolution, a bidirectional GRU and SA mechanisms are designed for enhanced features.Subsequently, after the concatenation of the processed air combat situation and capability features, the target threat level will be predicted by fully connected neural layers and the softmax classifier. Finally, in order to validate this model, an air combat dataset generated by a combat simulation system is introduced for model training and testing. The comparison experiments show the proposed model has structural rationality and can perform threat assessment faster and more accurately than the other existing models based on deep learning.

An Efficient Radar Detection Method of Maneuvering Small Targets

Detection of maneuvering small targets has always been an important yet challenging task for radar signal processing.One primary reason is that target variable motions within coherent processing interval generate energy migrations across multiple resolution bins,which severely deteriorate the parameter estimation performance.A coarse-to-fine strategy for the detection of maneuvering small targets is proposed.Integration of small points segmented coherently is performed first,and then an optimal inter-segment integration is utilized to derive the coarse estimation of the chirp rate.Sparse fractional Fourier transform(FrFT)is then employed to refine the coarse estimation at a significantly reduced computational complexity.Simulation results verify the proposed scheme that achieves an efficient and reliable maneuvering target detection with-16dB input signal-to-noise ratio(SNR),while requires no exact a priori knowledge on the motion parameters.

PCG-Based Exercise Fatigue Detection Method Using FRFT-Based Fusion Model

Accurate detection of exercise fatigue based on physiological signals is vital for reason-able physical activity.As a non-invasive technology,phonocardiogram(PCG)signals possess arobust capability to reflect cardiovascular information,and their data acquisition devices are quiteconvenient.In this study,a novel hybrid approach of fractional Fourier transform(FRFT)com-bined with linear and discrete wavelet transform(DWT)features extracted from PCG is proposedfor PCG multi-class classification.The proposed system enhances the fatigue detection performanceby combining optimized FRFT features with an effective aggregation of linear features and DWTfeatures.The FRFT technique is employed to convert the 1-D PCG signal into 2-D image which issent to a pre-trained convolutional neural network structure,called VGG-16.The features from theVGG-16 were concatenated with the linear and DWT features to form fused features.The fusedfeatures are sent to support vector machine(SVM)to distinguish six distinct fatigue levels.Experi-mental results demonstrate that the proposed fused features outperform other feature combinationssignificantly.

Fractional Fourier Domain Filter for Image Restoration

The FOURIER transform is one of the most frequently used tools in signal analysis. A generalization of the Fourier transform-the fractional Fourier transform-has become a powerful tool for time-varying signal analysis. The mean square error(MSE) is used as design criteria to estimate signal. Wiener filter, which can be implement in O(NlogN) time, is suited for time-invariant degradation models. For time-variant and non-stationary processes, however, the linear estimate requires O(N 2 ). Filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain filtering while requiring O(NlogN) implementation time. The blurred images that have several degradation models with different SNR are restored in the experiments. The results show that the method presented in this paper is valid and that the effect of restoration is improved as SNR is increased.

Chirp-Rate Resolution of Fractional Fourier Transform in Multi-component LFM Signal

Distinguishing close chirp-rates of different linear frequency modulation （LFM） signals under concentrated and complicated signal environment was studied. Firstly, detection and parameter estimation of multi-component LFM signal were used by discrete fast fractional Fourier transform （FrFT）. Then the expression of chirp-rate resolution in fractional Fourier domain （FrFD） was deduced from discrete normalize time-frequency distribution, when multi-component LFM signal had only one center frequency. Furthermore, the detail influence of the sampling time, sampling frequency and chirp-rate upon the resolution was analyzed by partial differential equation. Simulation results and analysis indicate that increasing the sampling time can enhance the resolution, but the influence of the sampling frequency can he omitted. What＇s more, in multi-component LFM signal, the chirp-rate resolution of FrFT is no less than a minimal value, and it mainly dependent on the biggest value of chirp-rates, with which it has an approximately positive exponential relationship.

Joint Timing and Frequency Synchronization for OFDM based on Fractional Fourier Transform

In this paper a joint timing and frequency synchronization method based on Fractional Fourier Transform （FIFT） is proposed for Orthogonal Frequency-Division Multiplexing （OFDM） system. The combination of two chirp signals with opposite chirp rates are used as the training signal, the received training signal with timing and frequency offset is transformed by FrFT and the two peaks representing two chirps in FrFT domain are detected, then the position coordinates of the two peaks are precisely corrected and substituted into an equation group to calculate timing and frequency offset simultaneously. This method only needs one FrFT calculation to implement synchronization, the computational complexity is equal to that of FFT and less than that of correlation or maximum likelihood calculation of existing methods, and estimation range of frequency offset is Large, greater than half the signal bandwidth, while the simulation results show that even at low SNR it can accurately estimate timing and frequency offset and the estimation error is less than that of existing methods.

Order selection in fractional Fourier transform based beamforming

Traditionally,beamforming using fractional Fourier transform（FrFT） involves a trial-and-error based FrFT order selection which is impractical.A new numerical order selection scheme is presented based on fractional power spectra（FrFT moment） of the linear chirp signal.This method can adaptively determine the optimum FrFT order by maximizing the second-order central FrFT moment.This makes the desired chirp signal substantially concentrated whereas the noise is rejected considerably.This improves the mean square error minimization beamformer by reducing effectively the signal-noise cross terms due to the finite data length de-correlation operation.Simulation results show that the new method works well under a wide range of signal to noise ratio and signal to interference ratio.

Image recovery from double amplitudes in fractional Fourier domain

The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].

New analytical exact solutions of time fractional KdV KZK equation by Kudryashov methods

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

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 import-ant to digital signal processing.The fractional Fourier transform(FrFT)that serves as a generaliz-ation 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 de-velopments 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 as-sociated with the FrFT.

INTERFERENCE MITIGATING BASED ON FRACTIONAL FOURIER TRANSFORM IN TRANSFORM DOMAIN COMMUNICATION SYSTEM

The method of FRactional Fourier Transform (FRFT) is introduced to Transform Domain Communication System (TDCS) for signal transforming in the paper after theoretical analysis. The method yields optimal Basis Function (BF) by FRFT with optimal transform angle. The TDCS using the proposed method has wider usable spectrum, stronger robustness and better ability of anti non-stationary jamming than using usual methods, such as Fourier Transform (FT), Auto Regressive (AR), Wavelet Transform (WT), etc. The main simulation results are as follows. First, the Bit Error Rate (BER) Pb is close to theoretical bound of no jamming no matter in single tone or in linear chirp interference. Second, the interference-to-signal ratio J /E is at least 12dB more than that of Direct Spread Spectrum System (DSSS) under the same BER if the spectrum hopping-to-signal ratio is 1:20 in chirp plus hopping interfering. Third, the Eb /N 0(when estimation difference is 90% between trans- mitter and receiver) is about 3.5dB or about 0.5dB (when estimation difference is 10% between transmitter and receiver) more than that of theoretical result when no estimation difference un-der Pb=10-2.