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Multiplicative noise removal is a challenging image processing problem. In this paper, we study a fractional-order nonlinear cross-diffusion system to denoise images corrupted by multiplicative Gamma noise. The system is coupled by two nonlinear, partial differential equations and gray level information, fractional derivative operator is considered in diffusion process to remove high level noise effectively while preserve structure details for texture images and SAR images. We prove the existence of weak solutions and other theoretical properties of the fractional-order cross-diffusion system under some hypotheses. In the numerical aspect, the model is implemented by virtue of the discrete fractional Fourier transform in the frequency domain. Experimental results show that the new system has a good compromise between multiplicative noise removal and structure preservation when compared to other well-known denoising methods. [ABSTRACT FROM AUTHOR]

Suppose that $ m, k,t $ m , k , t are integers with $ m,k\geq ~1 $ m , k ≥ 1 and $ 0\leq t $ 0 ≤ t and A is the $ k\times k $ k × k matrix with the $ (i,j) $ (i , j) -entry $ \binom {t+(j-1)m}{i-1} $ ( t + (j − 1) m i − 1 ) . When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform and a polynomial formulation for its inverse to the case that the sequence length and the characteristic of the base field are not coprime. At the end, we use this generalized Mattson-Solomon transform to present a lower bound on the length of repeated root cyclic codes, which can be seen as a generalization of the BCH bound. [ABSTRACT FROM AUTHOR]

The Hilbert transform (HT) is an important method for signal demodulation and instantaneous frequency (IF) estimation. The modulus of the analytic signal constructed by the HT is considered as the amplitude, and the derivative of the instantaneous phase of the extracted pure frequency modulation signal is the IF. When the spectra of the amplitude function and the oscillation term overlap each other and fail to satisfy the Bedrosian condition, the instantaneous amplitude and frequency calculated by the HT will contain errors. The recursive Hilbert transform (RHT) is an effective method to overcome this problem. The RHT considers the pure frequency modulation signal obtained by the previous HT as a new signal and recursively computes its HT until convergence. The final pure frequency modulation signal of the recursive procedure has the same zero-crossing points as the original signal, and its corresponding quadrature error signal vanishes. We emphasize the convergence analysis of the algorithm and study the convergent tendency of the quadrature error signal in each recursive step. The key to the proof is that the discrete Fourier transform value of the quadrature error signal is regarded as a vector, and the length/norm of the vector decreases with the recursion process. Finally, we used three examples to demonstrate the effectiveness of this method in signal demodulation, IF identification and damped vibrating signal analysis, which indicate the application potential of the RHT method in mono-component signal processing. [ABSTRACT FROM AUTHOR]

In today's fast-changing power systems, it is vital to have a protection system that can distinguish between power swing and fault conditions. With advanced protective devices (Digital Relays), an appropriate protection strategy can be established to identify power swing and fault conditions individually. The proposed algorithm uses the sliding window concept, which requires very little computational data unless any abnormal conditions exist in the power system. Modified Full Cycle Discrete Fourier Transform (MFCDFT) is more effective in the extraction of the fundamental phasors from the original complex signal by effectively removing the harmonics and DC components. An adaptive impedance characteristic-based MFCDFT is utilized for the discrimination of power swing and fault in this work. Several fault scenarios and power swings are generated on the test system concurrently and independently from the relay station at varied distances. PSCAD/EMTDC is utilized on a single-machine infinite bus system to analyze various test conditions. The simulation is utilized in this case to collect data from various test circumstances. After obtaining the data, the MFCDFT algorithm along with adaptive impedance estimation is performed using MATLAB code. After examining various test results, it is discovered that the suggested technique correctly recognizes power swings and various fault scenarios. It can also identify the difference between stable and unstable power swings. According to the results, the proposed technique is proficient at spotting fault and power swing conditions. In addition, the algorithm ensures dependable operation with quick detection and avoids maloperation under a typical swing condition. [ABSTRACT FROM AUTHOR]

Summary: The non‐orthogonal multiple access (NOMA) is highly capable of serving multiple users at similar times and frequencies. The power allocation (PA) is widely considered as a main factor in NOMA for efficient communication. Here, the application of multiple‐input multiple‐output (MIMO) is added to NOMA to fulfill demands of enriched spectral efficiencies and extra user data. In this research, the Ladybug Beetle Honey Badger Optimization (LBHBO) is proposed for efficient PA in MIMO‐NOMA. Initially, the received signals from the user are modulated for amplitude and frequency. Then, user grouping is conducted by fuzzy local information c‐means (FLICM) clustering followed by using PA done by proposed LBHBO. This power is then moved to the transmitter and then to the channel estimation process. Moreover, cyclic prefix (CP) removal is carried out that tends to discrete Fourier transform (DFT). Finally, quadrature amplitude modulation (QAM) demodulation is performed for reallocated output. Furthermore, LBHBO is formed by combining Ladybug Beetle Optimization (LBO) and Honey Badger Algorithm (HBA). The performance offered by the LBHBO‐PA with maximal values with energy efficiency (EE) of 25.38 Mbits/s, sum rate of 1.29 Mbits/s, and achievable rate of 100.47 Mbits/s. [ABSTRACT FROM AUTHOR]

State-of-the-art smart cities have been calling for economic but efficient energy management over a large-scale network, especially for the electric power system. It is a critical issue to monitor, analyze, and control electric loads of all users in the system. In this study, a non-intrusive load monitoring method was designed for smart power management using computer vision techniques popular in artificial intelligence. First of all, one-dimensional current signals are mapped onto two-dimensional color feature images using signal transforms (including the wavelet transform and discrete Fourier transform) and Gramian Angular Field (GAF) methods. Second, a deep neural network with multi-scale feature extraction and attention mechanism is proposed to recognize all electrical loads from the color feature images. Third, a cloud-based approach was designed for the non-intrusive monitoring of all users, thereby saving energy costs during power system control. Experimental results on both public and private datasets demonstrate that the method achieves superior performances compared to its peers, and thus supports efficient energy management over a large-scale Internet of Things network. [ABSTRACT FROM AUTHOR]

In this article, the main problem under investigation is the detection and diagnosis of short-circuit faults in power transmission lines. The proposed fault detection (FDD) approach is mainly based on principal component analysis (PCA). The proposed fault diagnosis/identification (FAI) approach is mainly based on sliding-window versions of the discrete Fourier transform (DFT) and discrete Hilbert transform (DHT). The main contributions of this article are (a) a fault detection approach based on principal component analysis in the two-dimensional scores space; and (b) a rule-based fault identification approach based on human expert knowledge, combined with a probabilistic decision system, which detects variations in the amplitudes and frequencies of current and voltage signals, using DFT and DHT, respectively. Simulation results of power transmission lines in Portugal are presented in order to show the robust and high performance of the proposed FDD approach for different signal-to-noise ratios. The proposed FDD approach, implemented in Python, that can be executed online or offline, can be used to evaluate the stress to which circuit breakers (CBs) are subjected, providing information to supervision- and condition-based monitoring systems in order to improve predictive and preventive maintenance strategies, and it can be applied to high-/medium-voltage power transmission lines as well as to low-voltage electronic transmission systems. [ABSTRACT FROM AUTHOR]

Due to the limitations of the actual acquisition environment in the field, especially in ocean bottom seismometer (OBS) acquisition, the acquired seismic data are often irregular and incomplete, which affects the subsequent data imaging, interpretation and hydrocarbon‐bearing reservoir prediction. The interpolation reconstruction algorithm based on the compressive sensing theory can reconstruct the data without the limitation of Nyquist sampling interval. However, the reconstruction accuracy and effect are different for different sparse representations of the data. On the basis of compressed sensing theory, we propose a Bregman iterative seismic data reconstruction method based on the sparse decomposition of discrete orthonormal Coiflets and Symlets wavelet transforms. First, the discrete orthonormal matrix is constructed by using the above two wavelet functions to make the original seismic data sparse, then the Bregman iterative algorithm is used to reconstruct the sparse coefficients in the discrete wavelet domain, and finally the recovery matrix is used to reconstruct the seismic data. The discrete orthonormal Coiflets and Symlets wavelet transforms have good sparse representation ability and can compensate for the problem that discrete Fourier transform cannot well sparse representation of the data. After numerical experiments on horizontal‐layered model, Marmousi2 model and actual data, it is verified that the proposed method can reconstruct the missing seismic data under the regular observation system and under the irregular observation system with non‐uniform OBS distribution. Furthermore, this method can hardly bring the interference of random noise, and the reconstruction result is of high accuracy. [ABSTRACT FROM AUTHOR]

We carried out multi-bandpass BVRI CCD photometric observations for the two δ Scuti stars SDSS J231559.32 + 595654.1 (SDSS23 + 59) and ASAS J080529-1357.5 (ASAS08-13) at Kottamia Astronomical Observatory (KAO). The star SDSS23 + 59 has δ Scuti-type with a low amplitude pulsating variable, and the star ASAS08-13 is classified as a high amplitude δ Scuti-type. The time-series light curve data for each star was analyzed using the discrete Fourier transform (DFT), which revealed a dominant fundamental mode (f 0) of oscillation and associated harmonics. The photometric analysis of the two stars suggests SDSS23 + 59 is a star that can pulsate in a short period mode (P = 0.111314d ± 0.000012) with low amplitude (0.101–0.09), and the star ASAS08-13 can pulsate in a short period mode (P = 0.10052247 ± 0.00000281) with high amplitude (0.186–0.168 mag). The nonradial modes for SDSS23 + 59 and radial modes for ASAS08-13 are detected. We calculated new times of maximum light for both stars and combined them with those from other sources (ZTF and AAVSO) to update their corresponding linear ephemerides. The O − C diagram gives us a primary indication that the period of the star is increasing at the rate of (1/P) dP/dt = 3.410(0.006)×10 -10 yr−1 and 2.7321(0.0092)x10-6 yr−1 for SDSS23 + 59 and ASAS08-13 respectively. The physical parameters are calculated, and the positions of the stars on the P-T eff , M−L , M−R , and T eff -log g are investigated. [ABSTRACT FROM AUTHOR]

Long-term coherent integration (CI) can effectively improve the radar detection capability for high-speed targets. However, the range walk (RW) effect caused by high-speed motion significantly degrades the detection performance. To improve detection performance, this study proposes an improved algorithm based on the modified Radon inverse Fourier transform (denoted as IMRIFT). The proposed algorithm uses parameter searching for velocity estimation, designs a compensation function based on the relationship between velocity and distance walk and Doppler ambiguity terms, and performs CI based on the compensated signal. IMRIFT can achieve RW correction, avoid the blind-speed sidelobe (BSSL) effect caused by velocity mismatch, and improve detection performance, while ensuring low computational complexity. In addition, considering the relationship between energy concentration regions and bandwidth in the 2D frequency domain, a fast method based on IMIRFT is proposed, which can balance computational cost and detection capacity. Finally, a series of comparative experiments are conducted to demonstrate the effectiveness of the proposed algorithm and the fast method. [ABSTRACT FROM AUTHOR]

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height n = 0, as well as a certain duality for the En-(co)homology of π-finite spectra, established by Hopkins and Lurie, at heights n ≥ 1. We use this theory to generalize said duality in three different directions. First, we extend it from Z-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of En. Second, we lift it to the telescopic setting by replacing En with T (n)-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg--Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal ∞-categories of local systems of K (n)-local En-modules and relate it to (semiadditive) redshift phenomena. [ABSTRACT FROM AUTHOR]

Aerosol decrease is considered as one of the most important environmental challenges. The current study introduces a novel analytical method for accurate and precise detection and identification of different aerosols. The designed array-based spectroscopic method is based on "discrete fast Fourier transform"-assisted dual resonance ultraviolet-infrared spectroscopy for reliable quantitative/qualitative analysis of aerosols like fug, smog, silica-based micro-/nanoparticles, carbon soot, etc. The detection system is arranged using a slice (5 × 5 cm) of a digital versatile disk as a simple, low-cost, available, and size-controllable wavelength selector (grating) and light reflector with control through a software program. The results show that this method is suitable for real-time detection of different types of chemical agent-modified particles with acceptable sensitivity and selectivity and improved detection limit. [ABSTRACT FROM AUTHOR]

In open networks, the geometric deformation and common image processing are common image manipulation modes, which pose a great challenge in robust watermarking. To improve the robustness of GRHFM‐based watermarking, a watermarking algorithm based on the fast GRHFM calculation method and ant colony optimization (ACO) based fractional parameter selection method is proposed. With the similarity between the discrete Fourier transform and GRHFM moment calculation, the algorithm utilizes fast Fourier transform to improve the GRHFM calculation accuracy and speed. To select the optimal GRHFM fractional parameter for watermarking algorithm, ACO is introduced to the fractional parameter adaptive selection method. Here, the fast Fourier transform‐based calculation method is combined with the adaptive parameter selection method to maximize the invisibility and robustness of watermarking. The experimental results indicate that the algorithm achieves higher robustness with the same payload and invisibility compared with other existing watermarking methods. [ABSTRACT FROM AUTHOR]

Granite rocks are subjected to exogenic forces, such as tectonic stress and chemical weathering, which cause them to weather differentially, forming boulders that are common in temperate and humid regions. The presence of a large boulder on a slope can significantly impact slope stability, resulting in the formation of what is known as a solitary boulder slope. This study is dedicated to investigate the large-deformation instability characteristics of such slopes. In pursuit of this objective, our study first validates an adapted material point method through laboratory experiments. Subsequently, utilizing the inverse discrete Fourier transform theory, boulders with varying shapes were synthesized and further transformed into material point method models. Finally, the material point method was employed to analyse the impact of boulder position, shape, and incline on the large deformation characteristics of the slope. The results show that: (1) The closer a boulder is to the plastic zone of the slope, the stronger its reinforcement effect on the slope. (2) As the shape of boulders becomes more complex, the run-out distance of the slope tends to decrease, but the variability in run-out distances increases. (3) The slope plastic zone moves around the boulder, and its distribution undergoes alterations based on the boulder's shape. This study investigates the instability characteristics of large deformation in boulder slopes with varying boulder shapes, which provides a reference for the stability assessment of solitary boulder slopes. [ABSTRACT FROM AUTHOR]

The pulse-coupled neural network (PCNN) is widely used in digital image processing. Although existing studies mainly analyze the network from the time-domain perspective, there are still some limitations in revealing the characteristics of network. In this paper, from the iterative equations of PCNN, the expressions for the firing time and firing interval of neuron are given. Spectrum for the dynamic threshold subsystem and firing subsystem of PCNN is given by using the Z-transform and the discrete Fourier transform, and the effects of different parameters a E , V E and K on the frequency-domain characteristics of the two subsystems are analyzed. The edge detection phenomenon exhibited by the iterative output of the PCNN is explained by analyzing the effect of the neighbor coupling state on the firing time and interval. Finally, the correctness of the analysis is validated by simulation experiments, which provides a new idea for the further study into the characteristics of PCNN. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a fast Fourier transforms (FFTs) based preconditioner and propose a novel α -absolute value preconditioner for all-at-once systems from heat equations. Our preconditioner is symmetric positive definite and can also be computed efficiently using fast Fourier transforms or discrete sine transforms (DSTs). Furthermore, we prove that the corresponding preconditioned matrix has a unitary-plus-low-rank decomposition, and the eigenvalues of the preconditioned matrix are clustered at ±1, which leads to fast convergence rate of the preconditioned MINRES method. Finally, numerical examples on all-at-once systems from heat equations with different discretization scheme demonstrate the effectiveness of our new preconditioner. [ABSTRACT FROM AUTHOR]

• A 3D size-dependent sliding contact model is established within the couple stress elasticity. • The analytical FRFs are derived with considerations of couple stress effects and transverse effects. • The effects of two additional intrinsic length-related parameters induced by couple stress theory are investigated. • The role of couple stress effects in size-dependent frictional sliding contact is studied. This work investigates a three-dimensional size-dependent frictional contact problem of a rigid spherical punch sliding on an elastic half-space. The size-dependent material behaviors are governed by the couple stress theory, and the corresponding frequency response functions, which include two additional length-related elastic parameters, are obtained. Based on the conjugate gradient method and the discrete convolution-fast Fourier transform algorithm, a semi-analytical model is established to predict the frictional size-dependent contact behaviors and then the detailed parameter analysis is performed. The transverse effects of the couple stress, which are induced due to the considerations of the two additional elastic parameters in couple stress theory, are investigated in the contact problems for the first time. And the dependences of the size-dependent contact behavior on the coupling effects of the friction and the two additional elastic parameters are presented. [ABSTRACT FROM AUTHOR]

After a successful supernova, a proto-neutron star (PNS) cools by emitting neutrinos on ∼1–100 s time-scales. Provided that there are neutrino emission 'hotspots' or 'cold-spots' on the surface of the rotating PNS, we can expect a periodic modulation in the number of neutrinos observable by detectors. We show that Fourier transform techniques can be used to determine the PNS rotation rate from the neutrino arrival times. Provided there is no spin-down, a 1-parameter Discrete Fourier Transform (DFT) is sufficient to determine the spin period of the PNS. If the PNS is born as a magnetar with polar magnetic field strength B 0 ≳ 1015 G and is 'slowly' rotating with an initial spin period ≳100 ms, then it can spin-down to periods of the order of seconds during the cooling phase. We propose a modified DFT technique with three frequency parameters to detect spin-down. Due to lack of neutrino data from a nearby supernova except the ∼20 neutrinos detected from SN1987A, we use toy models and one physically motivated modulating function to generate neutrino arrival times. We use the false alarm rate (FAR) to quantify the significance of the Fourier power spectrum peaks. We show that PNS rotation and spin-down are detected with |$\rm FAR\,\lt\, 2~{{\ \rm per\ cent}}$| (2σ) for periodic signal content |$\rm M\gtrsim 13-15~{{\ \rm per\ cent}}$| if 5 × 103 neutrinos are detected in ∼3 s and with |$\rm FAR\,\lt\, 1{{\ \rm per\ cent}}$| for |$\rm M\,\ge 5{{\ \rm per\ cent}}$| if 5 × 104 neutrinos are detected in ∼3 s. Since we can expect ∼104−105 neutrino detections from a supernova at 10 kpc, detection of PNS rotation and spin-down is possible using the neutrinos from the next Galactic supernova. [ABSTRACT FROM AUTHOR]

One of the six‐string classical guitar's most distinctive features is its unbalanced timbre. That is to say, only four pitches of the chromatic aggregate are overtonally supported by the resonances of the guitar's three bass strings. A common pitch‐ and structure‐generating strategy, exploited by twentieth‐century composers who wrote for the guitar, is bound up with this affordance. Local and global harmonic contrasts are often built around an antagonism between open‐string sets and fretted harmonies that largely avoid duplicating notes associated with the open strings. Open‐string/fretted binaries derive their sonic antagonism not just from timbral distinction, however, but from the fact that their pitch materials often lie at diametrically opposed positions on the circle of fifths. In many instances, this colouristic conflict replaces harmonic‐functional tension as a structure‐generating agent; the latter is reduced to a surface‐level topic. Representative works by Frank Martin and Leo Brouwer are taken as exemplifications of this trend. In demonstrating the subtle interactions between instrumental affordance and abstract pitch relationships and hierarchies, an invaluable theoretical approach developed by Ian Quinn and finessed by Jason Yust entails the study of harmonic quality through the discrete Fourier transform. This mathematical formula has been processed into an easy‐to‐use but powerful web application by Jennifer Harding – the Pitch Class DFT Vector Calculator – making its explanatory power available to music analysts without mathematical expertise. Part of the raison d'être of this article is to show how this application might be used and, more important, how the information that it yields might be interpreted. [ABSTRACT FROM AUTHOR]

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established. [ABSTRACT FROM AUTHOR]

Among the invertible orthogonal transforms employed to perform the analysis and synthesis of 2D signals (especially images), the ones defined by means of John von Neumann's cardinal sinus are extremely interesting. Their definitions rely on transforms similar to those employed to process time-varying 1D signals. This article deals with the extension of John von Neumann's transforms from 1D to 2D. The approach follows the manner in which the 2D Discrete Fourier Transform was obtained and has the great advantage of preserving the orthogonality property as well as the invertibility. As an important consequence, the numerical procedures to compute the direct and inverse John von Neumann's 2D transforms can be designed to be efficient thanks to 1D corresponding algorithms. After describing the two numerical procedures, this article focuses on the analysis of their performance after running them on some real-life images. One black and white and one colored image were selected to prove the transforms' effectiveness. The results show that the 2D John von Neumann's Transforms are good competitors for other orthogonal transforms in terms of compression intrinsic capacity and image recovery. [ABSTRACT FROM AUTHOR]

To further enhance the security of image encryption, a new chaos-based image encryption algorithm (IEA) based on discrete Fourier transform and Joseph traversal is proposed to encrypt the plain image in both the frequency domain and space domain simultaneously. In the proposed IEA, the logistic map is used to generate the appropriate chaotic sequence, and the improved Joseph traversal is used to scramble the image in different starting positions and variable step sizes. Then, block diffusion is performed at the end. The main finding concerning the proposed IEA is that the combination of discrete Fourier transform and Joseph traversal can enhance the security of the image information, which has been validated by measuring the performance in resisting the common types of attacks. [ABSTRACT FROM AUTHOR]

In the realm of neural network-based noise reduction, conventional models predominantly address Gaussian and blur artifacts across entire images. However, they encounter notable challenges when directly applied to periodic noise characteristics of high-resolution infrared sequential imagery. The high resolution also complicates the construction of suitable datasets. Our study introduces an innovative strategy that transforms two-dimensional images into one-dimensional signals, eliminating the need for processing the full image. We have developed a simulated dataset that closely mirrors natural infrared line scanning images derived from the FLIR dataset. To address low-frequency periodic noise, we propose two neural-network-based denoising approaches. The first employs a neural network to deduce noise from the one-dimensional signal, while the second utilizes discrete Fourier transforms for noise prediction within the frequency domain. Our experimental results highlight the Restormer model's exemplary performance in direct noise prediction, where denoised images attain a PSNR of around 41 and an SSIM close to 0.9 on simulated data, leaving minimal residual noise in the actual denoised images. Further, we investigate the influence of Fourier coefficients, as predicted by neural networks, on the denoising process in the second approach. Employing 12 frequency coefficients, the Restormer and NAFNet models both reach a PSNR near 34 and an SSIM of approximately 0.842. [ABSTRACT FROM AUTHOR]

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. We consider two cases: the fixed bandwidth design and the adaptive bandwidth design. The general result on the asymptotic efficiency, established for linear processes, is then applied to the class of stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule–Walker system of equations in the generalised autocovariance estimator. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for the removal of dual decaying DC offsets (DDCOs) to enhance discrete Fourier transform-based phasor estimation. The method proposed in this paper uses the sum of even samples and the sum of odd samples from the input signal to remove the AC components, thereby precisely estimating the primary and secondary DDCOs. The fluctuations induced by DCCOs present in the output of traditional DFT methods are eliminated by using the estimated DCCOs. The performance of the method was evaluated in terms of both mathematical-generated signals and fault current signals from a 154 kV Korean overhead transmission system. PSCAD/EMTDC was used to generate fault current signals at various fault distances and angles. The results show that the proposed method can estimate the phasor of the fundamental frequency component accurately regardless of the primary and secondary DCCOs. The paper concludes by describing the hardware implementation in a prototype unit that considers the real-world requirements of power system protection. [ABSTRACT FROM AUTHOR]

Differences between conventional sonar and Multiple-Input Multiple-Output (MIMO) sonar systems arise in achieving high angular and range resolution. MIMO sonar uses Matched Filtering (MF) with well-correlated transmitted signals to enhance spatial resolution by obtaining virtual arrays. However, imperfect correlation characteristics yield high sidelobe values, which hinder accurate target localization in underwater imagery. To address this, a Compressed Sensing (CS) method is proposed by reconstructing echo signals to suppress correlation noise between orthogonal waveforms. A shifted dictionary matrix and a deterministic Discrete Fourier Transform (DFT) measurement matrix are used to multiply received echo signals to yield compressed measurements. A sparse recovery algorithm is applied to optimize signal reconstruction before joint transmit–receive beamforming forms a 2D sonar image in the angle-range domain. Numerical simulations and lake experimental results confirm the effectiveness of the proposed method, by obtaining a lower sidelobe sonar image under sub-Nyquist sampling rates as compared with other approaches. [ABSTRACT FROM AUTHOR]

The discrete Fourier transform (DFT) is the most commonly used signal processing method in modern digital sensor design for signal study and analysis. It is often implemented in hardware, such as a field programmable gate array (FPGA), using the fast Fourier transform (FFT) algorithm. The frequency resolution (i.e., frequency bin size) is determined by the number of time samples used in the DFT, when the digital sensor's bandwidth is fixed. One can vary the sensitivity of a radio frequency receiver by changing the number of time samples used in the DFT. As the number of samples increases, the frequency bin width decreases, and the digital receiver sensitivity increases. In some applications, it is useful to compute an ensemble of FFT lengths; e.g., 2 P − j for j = 0 ,   1 ,   2 ,   ... ,   J , where j is defined as the spectrum level with frequency resolution   2 j · Δ f . Here Δ f is the frequency resolution at j = 0 . However, calculating all of these spectra one by one using the conventional FFT method would be prohibitively time-consuming, even on a modern FPGA. This is especially true for large values of P ; e.g., P ≥ 20 . The goal of this communication is to introduce a new method that can produce multi-resolution spectrum lines corresponding to sample lengths   2 P − j for all J + 1 levels, concurrently, while one long 2 P -length FFT is being calculated. That is, the lower resolution spectra are generated naturally as by-products during the computation of the 2 P -length FFT, so there is no need to perform additional calculations in order to obtain them. [ABSTRACT FROM AUTHOR]

From the beginning of COVID-19 pandemic, numerous mutants of SARS-CoV-2 have since been evolved owing to high transmissibility and virulence. Due to the limited effectiveness of previously imposed vaccines and preventive therapies, these strains are still causing concern. This paper proposes comparative evaluation of three novel genomic signal processing-based methods employing discrete wavelet decomposition with lifting (DWT), discrete Fourier transform (DFT), and singular value decomposition (SVD) for the classification of emerging SARS-CoV-2 variants utilizing feature extraction from collected SARS-CoV-2 variants acquired from the NCBI virus database. The efficiency and accuracy of the proposed alignment-free algorithms have been tested using three Coronavirus datasets including human Coronavirus (HCoV), SARS-CoV-2 variants (CoV-Variants and Omicron). The viral nucleotide sequences which are converted into numerical representation leveraging purine-pyrimidine mapping, DNA walk & Z-curve are fed into DWT, SVD, & DFT processors, respectively. In the approach with DWT, the second-generation wavelet transform employs two best wavelet bases Daubechies (Db) and Biorthogonal (Bior) based on the validation of the HCoV dataset for the feature extraction of the CoV-Variants dataset. Various machine learning algorithms, such as Support Vector Machine, K-nearest neighbors, and ensemble, are used to classify the virus strains and evaluate the efficacy of the algorithm. Finally, hyper-parametric tuning is done utilizing the Bayesian optimization technique to select the best fit model for KNN and SVM. The proposed algorithm has successfully classified the CoV-Variants dataset with an average accuracy of 98.76% utilizing the DWT, DFT, and SVD, while the best-achieved accuracy for this dataset is 98.9% using the DWT technique employing purine–pyrimidine mapping. The best-achieved accuracy rate for predicting Omicron is 99.8% using SVD-based technique. The best-obtained accuracy for HCoV dataset is 100% resulted in all three methods. [ABSTRACT FROM AUTHOR]

The paper compares the accuracy of reconstruction using different wavelets, and the authors also use fast and discrete Fourier transforms together to calculate the forward and inverse continuous wavelet transform in the frequency domain. Due to the use of calculations in the frequency domain, it becomes possible to perform decomposition, reconstruction, image filtering, and other transformations with high performance and precision. For multiscale signal analysis, a wavelet with a rectangular amplitude-frequency response has been constructed, which allows for an increase in the accuracy of decomposition and reconstruction compared to the Mallat algorithm presented in Matlab computer mathematics. At the same time, the time of multiscale analysis is reduced several times compared to the Mallat algorithm. [ABSTRACT FROM AUTHOR]

Today, safeguarding sensitive content through encryption is crucial. This work presents a hybrid cryptosystem for images that employs both asymmetric and symmetric encryption. The asymmetric component involves applying the Diffie–Hellman protocol and the ElGamal cryptosystem to securely transmit two constants. These constants are necessary for the symmetrical aspect to generate dynamic permutations, substitution boxes, and round keys. Following an encryption process with fourteen rounds, the encrypted images are processed by an algorithm proposed to enhance entropy, a critical metric for assessing encryption quality. It increases the frequencies of the basic colors to achieve a histogram closely resembling a uniform distribution, but it increases the image size by approximately 8%. This improves the entropy values achieved by the hybrid cryptosystem, bringing them remarkably close to the ideal value of 8.0. In specific instances, the entropy values were elevated from 7.99926 to 8.0. The proposed method exhibits resilience against various attacks, including differential, linear, brute force, and algebraic attacks, as evaluated through the entropy, correlation, goodness of fit, Discrete Fourier Transform (DFT), Number of Pixels Change Rate (NPCR), Unified Average Changing Intensity (UACI), Avalanche Criteria (AC), contrast, energy, and homogeneity. Further, encrypted images are subjected to noise attacks ranging from 20% to 50% noise, including additive, multiplicative, occlusion noise, as well as the newly introduced χ 2 noise. The noise damage is quantified using the proposed Similarity Parameter (SP), and a 3 × 3 median filter is employed to enhance the visual quality. [ABSTRACT FROM AUTHOR]

Secure transmission of images over wireless communications systems can be done using RSA, the most known and efficient cryptographic algorithm, and OFDMA, the most preferred signal processing choice in wireless communications. This paper aims to investigate the performance of OFDMA system for wireless transmission of RSA-based encrypted images. In fact, the performance of OFDMA systems; based on different signal processing techniques, such as, discrete sine transforms (DST) and discrete cosine transforms (DCT), as well as the conventional discrete Fourier transforms (DFT) are tested for wireless transmission of gray-scale images with/without RSA encryption. The progress of transmitting the image is carried by firstly, encrypting the image with RSA algorithm. Then, the encrypted image is modulated with DFT-based, DCT-based, and DST-based OFDMA systems. After that, the modulated images are transmitted over a wireless multipath fading channel. The reverse operations will be carried at the receiver, in addition to the frequency domain equalization to overcome the channel effect. Exhaustive numbers of scenarios are performed for study and investigation of the performance of the different OFDMA systems in terms of PSNR and MSE, with different subcarriers mapping and modulation techniques, is done. Results indicate that the ability of different OFDMA systems for wireless secure transmission of images. However, the DCT-OFDMA system showed superiority over the DST-OFDMA and the conventional DFT-OFDMA systems. [ABSTRACT FROM AUTHOR]

The results of the analysis of the single-color (B) observations undertaken at the Stefanion Observatory for the red dwarf star AD Leo at any stage of the stellar activity (quiescence, weak flares, strong flares) indicate that (a) transient high-frequency oscillations occur during the flare event and during the quiet-star phase as well, and (b) the observed frequencies range between 0.0005 Hz (period 33 min) and 0.3 Hz (period 3 s) not rigorously bounded. However, the quiescence parts of the light curve that were analyzed belong to the preflare or after-flare state, that is, are connected with a major magnetic event (the observed flare). In this work, we find that transient oscillations appear far apart from the observed flares, during the quiet state of the star, as a result of the general magnetic activity of the star. The power spectrum of these oscillations resembles those of the solar-like oscillation spectra, that is, the sunquake spectra. Finally, a tentative estimation of the main physical parameters of the star, using asteroseismic analysis, is performed. [ABSTRACT FROM AUTHOR]

Sea ice is important for Earth's energy budget as it influences surface albedo and air–sea fluxes in polar regions. On its margins, waves heavily impact sea ice. Routine and repeat observations of waves in sea ice are currently lacking, and therefore a comprehensive understanding of how waves interact with sea ice and are attenuated by it is elusive. In this paper, we develop methods to separate the two-dimensional (2D) surface wave spectra from sea-ice height observations made by the ICESat-2 (IS2) laser altimeter, a polar-orbiting satellite. A combination of a linear inverse method, called generalized Fourier transform (GFT), to estimate the wave spectra along each beam and a Metropolis–Hastings (MH) algorithm to estimate the dominant wave's incident angle was developed. It allows us to estimate the 2D wave signal and its uncertainty from the high-density, unstructured ATL03 ICESat-2 photon retrievals. The GFT is applied to re-binned photon retrievals on 25 km segments for all six beams and outperforms a discrete Fourier transform (DFT) in accuracy while having fewer constraints on the data structure. The MH algorithm infers wave direction from beam pairs every 25 km using coherent crests of the most energetic waves. Assuming a dominant incident angle, both methods together allow a decomposition into 2D surface wave spectra with the advantage that the residual surface heights can potentially be attributed to other sea-ice properties. The combined GFT–MH method shows promise in routinely isolating waves propagating through sea ice in ICESat-2 data. We demonstrate its ability on a set of example ICESat-2 tracks, suggesting a detailed comparison against in situ data is necessary to understand the quality of retrieved spectra. [ABSTRACT FROM AUTHOR]

We present an asymptotic approach to solving the water wave scattering by undulating bottom topography in the presence of uniform currents where a flexible membrane covers the upper surface. The interest in this problem lies in the development of asymptotic solutions using the Fourier transform under the action of uniform currents. The method allows the physical processes involved in the sea-bed topography and membrane-covered surface and wave interactions to be studied. In particular, we identify the existence of Bragg resonance between gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We consider the impacts of uniform current, and we highlight the central role of the asymptotic expansion method in the evolution of the response of current. For depth Froude number in the range of 0.4–0.7, the effects of bottom topography on Bragg resonance dominate. The current shifts the frequency of the most reflected wave components, and wave action conservation results in amplified reflected wave energies for the following currents. The theory developed in the frequency domain is illustrated in the time domain using discrete Fourier transform with the Joint North Sea Wave Observation Project (JONSWAP) spectrum [Hasselmann et al., J. Phys. Oceanogr. 10, 1264 (1980)] to analyze wave propagation through the whole system. [ABSTRACT FROM AUTHOR]

We present estimated images of a hypothetical satellite system in the presence of Seidel aberrations. We exemplified the effects of defocus, spherical, coma, astigmatism, field curvature, and distortion aberrations. Our analysis methods allow the investigation of aberration effects individually and in any combined form. Thereby, we introduce a potentially rich analysis dimension for the imaging chain, image processing, and image quality analyses. For a number of degrees of aberrations, we provide image estimates, show and quantify how individual (field‐dependent) aberrations would introduce degradations on the satellite images. We got correlation coefficient values for defocus, spherical, coma, astigmatism, field curvature, and distortion aberrations as 0.69, 0.71, 0.66, 0.79, 0.76, and 0.60, respectively. Thereby, we identfied the significance order of Seidel aberrations for this system by using image simulations and image similarity measures. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform (FrFT) is a useful mathematical tool for signal and image processing. In some applications, the eigendecomposition-based discrete FrFT (DFrFT) is suitable due to its properties of orthogonality, additivity, reversibility and approximation of continuous FrFT. Although recent studies have introduced reduced arithmetic complexity algorithms for DFrFT computation, which are attractive for real-time and low-power consumption practical scenarios, reliable hardware architectures in this context are gaps in the literature. In this paper, we present two hardware architectures based on the referred algorithms to obtain N-point DFrFT ( N = 4 L , L is a positive integer). We validate and compare the performance of such architectures by employing field-programmable gate array implementations, co-designed with an embedded hard processor unit. In particular, we carry out computer experiments where synthesis, error and latency analyses are performed, and consider an application related to compact signal representation. [ABSTRACT FROM AUTHOR]

The magneto-electro-elastic (MEE) medium is a typical intelligent material with promising application prospects in sensors and transducers, whose thermal contact response is responsible for their sensitivity and stability. An effective thermal contact model between a moving sphere and a coated MEE medium with transverse isotropy is established via a semi-analytical method (SAM) to explore its thermal contact response. First, a group of frequency response functions for the magneto-electro-thermo-elastic field of a coated medium are derived, assuming that the coating is perfectly bonded to the substrate. Then, with the aid of the discrete convolution–fast Fourier transform algorithm and conjugate gradient method, the contact pressure and heat flux can be determined. Subsequently, the induced elastic, thermal, electric and magnetic fields in the coating and substrate can be obtained via influence coefficients relating the induced field and external loads. With the proposed method, parametric studies on the influence of the sliding velocity and coating property are conducted to investigate the thermal contact behavior and resulting field responses of the MEE material. The sliding velocity and thermal properties of the coating have a significant effect on the thermal contact response of the MEE material; the coupled multi-field response can be controlled by changing the coating thickness between ~0.1 a0 and a0. [ABSTRACT FROM AUTHOR]

• A semi-analytical solution to incipient sliding contact on viscoelastic layered solids is proposed. • The spring-like interface between the coating and substrate is considered. • A two-step procedure is performed to verify the validity of the solution. • The influences of variation in parameters on the contact behaviors are analyzed. This paper presents a semi-analytical solution to the incipient sliding contact between a rigid sphere and a viscoelastic layer-elastic substrate with spring-like interface. In this solution, Papkovich–Neuber potentials in frequency domain are applied to calculate the influence coefficients (ICs) based on the surface and interface conditions; the elastic constant in the elastic solution is replaced by the creep function to cover the time-dependent characteristics of the viscoelastic layer; discrete convolution-fast Fourier transform (DC-FFT) is used to obtain the displacements and stresses caused by the external load; conjugate gradient method (CGM) is employed to compute the pressure on the top surface of the viscoelastic layer. The validity of the solution is verified by a previous method and the finite element method (FEM), where the relaxation and creep functions of the viscoelastic layer are expressed as Prony series. Influences of variations in parameters on the surface pressure, normal displacement and von Mises stress are analyzed by a parametric study of the time, relaxation time, relaxation modulus, stiffness coefficients, coating thickness and friction coefficient. The results indicate that this solution can predict the contact behaviors of viscoelastic layered half-space and may serve as a numerical experiment in research of coating materials. [ABSTRACT FROM AUTHOR]

In this study, a speech enhancing neural network (NN) is proposed, which is designed for monaural auditory devices, specifically designed for use in hearing aids. Herein, a 32‐channel auditory filterbank (FB) is first implemented with an algorithm processing delay of 8 ms, which is tailored to meet the requirements of auditory devices. The proposed method primarily aims to integrate a denoising NN within the analysis phase of a uniform polyphase discrete Fourier transform (DFT) FB, aimed at enhancing speech within each band. For the denoising model, complex‐valued convolutional NNs have been applied, specifically targeting the restoration of speech phase information based on the spectral components of the DFT. A multi‐loss method is introduced, which is designed to further account for the loss of analysed speech signals within the split bands during the training process, leveraging the DFT FB strategy. To evaluate the efficacy of the proposed method, objective assessments of speech intelligibility and quality scores are conducted under various noise conditions. The results demonstrate that the proposed method can outperform the existing method across all types of noise. [ABSTRACT FROM AUTHOR]

First-principles calculations have been carried out to study the electronic structure, optical properties and high-pressure structural phase transitions of rarely studied Sc2AlB6 compounds. Ground state structure calculations reveal the compound to be a non-magnetic material with Hexa, Penta, and Hepta boron layers sandwiched between metal layers. From the electronic structure and charge density calculations, it is confirmed that the compound is metallic with a mixed nature of bonding i.e. covalent, ionic, and metallic bonding. The mechanical stability of Sc2AlB6 is confirmed by the calculated elastic constants. We have systematically carried out high-pressure studies on these compounds and have not seen any change in lattice parameters, volume, electronic structures with the pressure indicating no structure phase transition in these compounds. The theoretical hardness of the compound is predicted and compared with the experimental result. We observed this hardness increase with an increase in the pressure. In addition to that optical properties, which include absorption coefficient, dielectric function, reflectivity, refractive index are discussed. [ABSTRACT FROM AUTHOR]

We apply the classical finite Fourier transform to construct an embedded functional process to a given multivariate time series model. The basic properties of the embedded functional process are presented. It appears that the embedded functional process is quite useful for the model building and prediction. We also provide a new method for approximating an infinite functional process with a finite functional process. The methodology is different from the classical one. Indeed, the active underlying frequencies are taken into consideration rather than the significant eigenvectors. The performance of our method is illustrated through simulations. Interestingly, in doing prediction, it appears that for our proposal the computation time is much shorter compared to existing ones. [ABSTRACT FROM AUTHOR]

Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step. [ABSTRACT FROM AUTHOR]

The parallels between terahertz time-domain spectroscopy (THz-TDS) and discrete Fourier transform spectroscopy (DFTS) are unraveled by showing that the same Fresnel equation links the optical properties of a material to measurements obtained using either approach. Such similarities are revealed by evaluating this same Fresnel equation incorporating either THz-TDS or DFTS measurement data, permitting the optical material properties (i.e., refractive index and extinction coefficient) to be obtained from the sample under investigation. The aim of this study is to highlight the likenesses between THz-TDS and DFTS in characterizing optical material properties and, in doing so, stimulate collaboration between individuals conducting research in the realms of infrared and terahertz spectroscopy. [ABSTRACT FROM AUTHOR]

Crispness is a textural characteristic that influences consumer choices, requiring a comprehensive understanding for product customization. Previous studies employing neural networks focused on acquiring audio through mechanical crushing of crispy samples. This research investigates the representation of crispy sound in time intervals and frequency domains, identifying key parameters to distinguish different foods. Two machine learning architectures, multi‐layer perceptron (MLP) and residual neural network (ResNet), were used to analyze mel frequency cepstral coefficients (MFCC) and discrete Fourier transform (DFT) data, respectively. The models achieved over 95% accuracy "in‐sample" successfully classifying fried chicken, potato chips, and toast using randomly extracted audio from ASMR videos. The MLP (MFCC) model demonstrated superior robustness compared to ResNet and predicted external inputs, such as freshly toasted bread acquired by a microphone or ASMR audio of toast in milk. In contrast, the ResNet model proved to be more responsive to variations in DFT spectrum and unable to predict the similarity of external audio sources, making it useful for classifying pretrained "in‐samples". These findings are useful for classifying crispness among individual food sources. Additionally, the study explores the promising utilization of ASMR audio from Internet platforms to pretrain artificial neural network models, expanding the dataset for investigating the texture of crispy foods. [ABSTRACT FROM AUTHOR]

Brain–computer interfaces (BCIs) establish a direct communication channel between the human brain and external devices. Among various methods, electroencephalography (EEG) stands out as the most popular choice for BCI design due to its non-invasiveness, ease of use, and cost-effectiveness. This paper aims to present and compare the accuracy and robustness of an EEG system employing one or two channels. We present both hardware and algorithms for the detection of open and closed eyes. Firstly, we utilize a low-cost hardware device to capture EEG activity from one or two channels. Next, we apply the discrete Fourier transform to analyze the signals in the frequency domain, extracting features from each channel. For classification, we test various well-known techniques, including Linear Discriminant Analysis (LDA), Support Vector Machine (SVM), Decision Tree (DT), or Logistic Regression (LR). To evaluate the system, we conduct experiments, acquiring signals associated with open and closed eyes, and compare the performance between one and two channels. The results demonstrate that employing a system with two channels and using SVM, DT, or LR classifiers enhances robustness compared to a single-channel setup and allows us to achieve an accuracy percentage greater than 95% for both eye states. [ABSTRACT FROM AUTHOR]

The successful construction of S-scheme heterojunction in the composite material, resulting in strong redox capacity, effective separation and photogenic carrier transfer, prolongs the life of photon-generated carrier to improve the activity of the photocatalyst and the efficiency of photoreduction of CO 2. [Display omitted] • Using a simple hydrothermal method, sheets of ZnIn 2 S 4 were grown in situ on ZnO rods. • The S-scheme heterojunction was successfully constructed, which significantly improved the charge separation ability and redox ability of the material. • ZnO@ZnIn 2 S 4 S-scheme heterojunction photocatalysts has achieved excellent results in reducing CO 2 to CO and CH 4. Step-scheme (S-scheme) heterojunctions can efficiently promote the separation of photogenerated carriers while maintaining the strong oxidation/reduction ability of photocatalysts; thus, research attention on S-scheme heterojunctions is increasing year by year. In this study, the S-scheme ZnO@ZnIn 2 S 4 (ZnO@ZIS) heterojunction was prepared successfully. Then, electron spin resonance (ESR) characterization was applied to prove the successful construction of the S-scheme heterojunction. Photoluminescence (PL), time-resolved photoluminescence (TRPL), and photoelectrochemical experiments have demonstrated efficient interfacial charge transport in ZnO@ZIS. Finally, the mechanism of CO 2 activation and electron transport was investigated by in situ Fourier transform infrared spectroscopy (FT-IR) and discrete Fourier transform (DFT) calculation analysis. The 40-ZnO@ZIS composite showed the best activity under light, and its CO and CH 4 yields reached 39.76 and 3.92 μmol∙g−1∙h−1, respectively. This study provides a solution for optimizing the photocatalytic reduction activity of semiconductor photocatalysts by constructing S-scheme heterojunction materials to improve the CO 2 reduction capacity. [ABSTRACT FROM AUTHOR]

Radio Frequency Fingerprinting (RFF) has been recently investigated by the research community to enhance wireless security. This letter proposes an RFF approach based only on the complex ramp‐up transient portion of the transmitted frame. The well known issues in literature with the use of the transients for RFF is that they have very short duration, they are strongly non‐stationary and their analysis is sensitive to noise. Then, this letter proposes the application of an ensemble of transforms to the original complex time domain representation in combination with Convolution Neural Network (CNN) to address these issues. To avoid the computing demanding calculation of all the considered transforms, a pre‐processing step based on feature extraction and filter feature selection is implemented to select the most discriminating transforms to reduce the input data to the CNN. The approach is applied to two public RFF data sets, where the complex transient ramp‐up have been extracted. The result shows that the proposed approach is able to obtain a classification accuracy higher than the baseline based on the original signal or selected transforms across various values of Signal to Noise Ratio (SNR). [ABSTRACT FROM AUTHOR]