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Discrete complex exponentials are almost periodic signals, not always periodic; when periodic, the frequency determines the period, but not viceversa, the period being a chaotic function of the frequency, expressible in terms of Thomae's function. The absolute value of the frequency is an increasing function of the subadditive functional of average variation. For discrete signals that are either sums or series of periodic complex exponentials, the decomposition into their periodic, additive components allows for their filtering according to period. Likewise, their period-frequency spectrum makes predictable the effects on period of convolution filtering. Ramanujan-Fourier series are a particular case of the signal class of series of periodic complex exponentials, a broad class of signals on which a transform, discrete both in time and in frequency, called the DFDT Transform, is defined.

Various forward models have been developed for measuring agricultural biophysical parameters from the microwave or optical data, but their application to periodic surfaces (e.g., periodic crop rows and plowed soil rows) still suffers from the need of row-feature descriptions. Accordingly, an operational method was proposed herein to estimate the orientation and period of row features from very high spatial resolution optical data, as input to multi-scale scattering models. The periodic features were estimated from the frequency domain using the Fourier magnitude spectrum by scanning in azimuth to identify the presence/absence of periodic features and the main direction. The frequency representing the dominant periodic features was then determined from the main direction of the magnitude spectrum. Moreover, the tillage type of the bare soil surface was classified into one of three types (sinusoidal, sinusoidal bench, or bench) according to the magnitudes of dominant periodic features in the main direction. The method was evaluated using two data sets consisting of various agricultural features in regional Melbourne, Australia. The retrieval Root Mean Square Error (RMSE) in orientation and period was 95% and 70.59%, respectively. The proposed method and the retrieved periodic features are expected to partly solve the ill-posed nature of microwave data inversion over periodic surfaces.

An accurate frequency estimator of complex sinusoid in additive white noise is proposed. It is based on interpolation of Fast Fourier Transform (FFT) and Discrete-Time Fourier Transform (DTFT). Zero-padding is firstly performed before the FFT of the sinusoid sampled data, and the coarse estimate is obtained by searching the discrete frequency index of the maximum FFT spectrum line. Then the fine estimate is obtained by employing the maximum FFT spectrum line and two DTFT sample values located on the left and right side of the maximum spectrum line. The correlation coefficients between the Fourier Transform of the noises on two arbitrarily spaced spectrum lines are derived, and the MSE calculation formula is derived in additive white noise background based on the correlation coefficients. Simulations results demonstrate that the proposed algorithm has lower MSE than the competing algorithms, and its signal-to-noise ratio (SNR) threshold is lower compared with Candan algorithm, AM algorithm and Djukanovic algorithms.

An efficient parabolic equation solver is proposed to predict the radio wave propagation in forest during the rainy seasons. The method starts with considering the forest model as a mixture of plants and rain particles. Then the equivalent dielectric constant model can be used to simplify the calculation process. At last, the Split-step Fourier transform (SSFT) is used to calculate the loss of radio wave propagation in the forest. Numerical results show that this model can calculate the propagation loss of forests in rainy environment.

The results of the study are proposed, showing the theoretical substantiation of the possibility of using wavelets for the synthesis of signal structures based on them. Considering that wavelets are localized in both frequency and time space, it is proposed to synthesize phase-shift keyed signals on their basis. It is shown that the conditions that determine the legality of their application for the transmission of discrete information are valid for wavelets. Signal constructions of phase shift keying, analogous to signals of double phase shift keying, are substantiated on the basis of wavelets. A structural model has been developed for the synthesis of BPSK signal structures both based on harmonic Fourier functions and based on wavelets. The stages of synthesis are described in detail. The results of the study of structural differences in signal structures formed in the basis of wavelets and Fourier functions are presented. The values of peak factors and functional changes in the energy level on the duration of the symbol for the wavelet and sinusoid are calculated. The time intervals are substantiated within which 95% of the signal energy is localized. The function of the difference in energy growth is presented, which made it possible to determine the boundaries of the passband of the filters when compensating for the aiming noise in the spectrum. It is shown that narrowing the optimal bandwidth of the receiving path by a factor of 2.29 will lead to a loss in energy for the wavelet by only 19%, while the loss in the sinusoid will reach more than 62%. It is shown how the structural differences between wavelets and sinusoids will be reflected in the behavior of the formed difference functions. An estimate of the noise immunity for wavelet signals compared to BPSK signals is presented.

The article discusses an original approach that allows the formation of signals based on wavelets by digital synthesis. Its essence lies in the fact that they take the original wavelet and reset its time counts, which have a small amplitude. Then form a spectral representation of the truncated structure and zero the spectral coefficients with a small amplitude. These procedures are repeated until no beats in the spectrum are excluded. Implementation of this approach allows one to obtain signals with new physical properties. In addition, the possibility of synthesizing signals from wavelet fragments has been investigated. It is substantiated that the signal structures obtained on the basis of wavelets have a higher noise immunity, which is achieved by increasing the occupied frequency band. The analysis of the general properties of wavelets is carried out, including the possibility of reconstruction in the time domain from their spectra. The parameters of wavelets that can be modulated by information messages are considered. It is substantiated that the operation of scaling wavelets can be considered as an analog of the frequency for harmonic oscillations. It is shown how it is possible to synthesize manipulated signals with given frequency properties from wavelet fragments. It is shown that when the time periods of the existence of a sinusoid and a wavelet are equal, the spectrum of the latter occupies a much wider band. The results of the analysis of the energy distribution of the spectrum of the sinusoid and wavelet over the duration of their period are presented. An interesting point is considered, when the 15% of the time interval for a wavelet has an energy gain of only 0.7%, while for a harmonic pulse under the same conditions, the gain will be 10%• This fact served as the basis for digital synthesis of wavelets with new properties. The synthesis itself is defined in three stages. First, the initial wavelet is formed in the time domain and the time interval is set, within which no less than 99.3% of its energy is concentrated. Then the amplitude values are zeroed outside the allocated interval. After that, by means of Fourier transform, a frequency representation of a new design is formed, in which all spectral coefficients located outside the main lobe are also zeroed, thus ensuring the smoothness of the function. And at the last stage, the modified structure is again transferred to the time domain.

This study aims to evaluate simple recurrent neural network (simple RNN), long short-term memory (LSTM) and gate recurrent unit (GRU) for steam trap opening sound classification. This study conducts three experiments using different activation functions: ReLU, tanh and sigmoid functions at the output layer. Five feature extraction methods are employed including zero-crossing rate, spectral centroid, Mel-frequency cepstral coefficient, spectral rolloff, and short-term Fourier transform. We collected 24,512 real-world audio files of steam trap operations, dividing into three subsets for evaluating the models. The results showed that an LSTM model employing sigmoid function yielded the highest precision, recall, specificity and F1 score of 75.00%, 66.67%, 99.97% and 70.59%, respectively. In addition, the LSTM model also outperformed the GRU and the Simple RNN.

We proposed a signal decomposition algorithm for the electroencephalogram (EEG), which is separated into short oscillation, long oscillation, low frequency component, and the residual component. The decomposition problem is reduced to a sparse optimization one and the four components can be estimated by minimizing a convex objective function. A high-pass filter is applied to split the low frequency from the long oscillation. Meanwhile, two inverse short-time Fourier transforms are used to reconstruct the short oscillation and the long oscillation. After the EEG signal is decomposed, the sleep spindle is extracted from the long oscillation component. An EEG database is used to evaluate our method and the average F1 score 0.633 is obtained.

In this paper, a variable step size method for the PE (Parabolic Equation) is proposed to solve the problem of low computational efficiency in the study of radio wave propagation at a large range of complex environments. Firstly, the relationship between the error and the step size, the frequency and other factors of the SSFT (Split-Step Fourier Transform) solution in the standard atmosphere is deduced, and then the basic selection range of the step size is given for the variable step size method. Secondly, the action mechanism of different environmental factors and the requirement of changing trend for step size are expounded through simulation, and the complex environment of PE application is classified according to the requirement of error. Finally, the electric wave characteristics of typical complex environment are advanced by the method. The simulation results show that compared with the small step method, the variable step size method of the PE can reduce the horizontal sampling point by 55.9% and the calculation time by 69.6% when the accuracy is close to the small step method, and the variable step size method is more accurate than the fixed large size step method. Therefore, the variable step size on the PE can not only ensure the accuracy of calculation, but also reduce the memory and time required, which greatly improves the reliability and efficiency. Moreover, it is suitable for solving the problem of the radio wave propagation in complex environment.

Fourier-processing of finite discrete two-dimensional signals (FDTD signals) (including images) in informational control systems (IC systems) is the most important method for studying processes and analyzing information. The theoretical basis of Fourier-processing of FDTD signals is two-dimensional discrete Fourier transforms. The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. In order to solve this problem, the authors have developed new discrete Fourier transforms with variable parameters were developed (2D DFT-VP). The purpose of this work is to study the properties of two-dimensional discrete exponential functions with a variable parameter in the spatial-frequency domain. The introduction of discrete exponential functions with a variable parameter makes it possible to generalize the concept of periodicity of the DEF-VP system. Recall that the periodicity of the DEF system in the classical DFT is understood as a periodic continuation of the DEF system outside the interval of N samples. Moreover, the system of discrete basis functions in the classical DFT does not contain discontinuities. In the case of DFT-VP, for the DEF-VP system to be inseparable, the periodicity should be understood as parametric periodicity. The parametric periodicity of discrete exponential functions with a variable parameter is understood as their periodic continuation with rotation in complex space by a certain angle. Note that the introduced concept of parametric periodicity is valid for 1D and 2D real and complex functions. The theorems of linearity, shift, and correlation are proved for Fourier transforms with variable parameters.

The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. The purpose of this paper is to develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals. We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. Methods are based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. New discrete Fourier transforms with variable parameters were developed. The proposed discrete Fourier transforms with variable parameters are a generalization of the classical discrete Fourier transform. The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier - processing of two-dimensional signals (including images), padded with zeros.

The article focuses on rigorously describing and analyzing the dehydration process with regard to fruits and vegetables through the use of controlled instantaneous decompression (DIC) treatment. The mathematical modeling is obtained from the discretization of Fourier's and Fick's laws by the finite difference method. This is applied to the most common geometries in drying processes - the parallelepiped and the flat cylinder geometries. The validity of the mathematical model is determined by the Pearson correlation coefficient, which provides the linear relationship between two quantitative random variables in the form of the data obtained experimentally in the laboratory of La Rochelle University (France), and the results obtained from the mathematical model presented in this article.

This paper introduces preliminary results about a novelty formulation to calculate the first reflection over a sloping terrain for a Modified Two-ray model with atmospheric effects. For a case study of a lossy wedge and standard atmosphere conditions, the obtained results through the proposed formulation and the Discrete Mixed Fourier Transform (DMFT) numerical solution are compared, in order to validate the presented approach. The early results showed similar behavior for path loss, in this canonical scenario.

Neutral-point clamped Multilevel inverter (NPC-MLI) has transformed into a broadly accepted choice in the field of electric vehicle (EV) applications. In the traditional NPC-MLI-fed EV system, the problem regarding the higher harmonic content and the lower switching frequency still persists. In this paper, a phase opposition disposition pulse width modulation (POD-PWM) technique is proposed for a three-phase NPC-MLI, which is proficient in reducing the high-frequency weighted total harmonic distortion (WTHD) content of the inverter output voltage by employing a double Fourier integration-based harmonic mitigation algorithm. The experimental results of an FPGA-based NPC-MLI prototype hold a good resemblance with the theoretical analysis and simulated results. In addition, a comparative investigation affirms that the proposed strategy exhibits the highest efficiency and lowest WTHD content among all the previously-reported PWM schemes under study, which verifies the adaptability of the proposed work in electric vehicle applications.

In order to extract the effective fatigue detection index from the weak pupil size change, the excavator driving comparative experiment was set up. The pupil information of the experimenter was detected under the given constant light environment. Fourier transform and statistical analysis of the collected data showed that the pupil of the task group decreased significantly after operation (P 0.05) and the pupil amplitude did not change significantly. Experiments show that the amplitude of pupil size oscillation is more significant in frequency domain which is conducive to the application of pupil measurement method in reality.

The fault signal of rolling bearing is non-stationary nonlinear signal, and it is difficult to extract the feature of weak fault under strong background noise. This paper uses a new filtering method-Fractional Fourier Transform (FRFT). Compared with the traditional Fourier transform (FFT), it can make the time-frequency characteristics of unstable fault signals better displayed and suppress cross-interference. In this paper, the method of feature extraction of rolling bearings combined with Variational Mode Decomposition (VMD) and Fractional Fourier Transform (FRFT) is used. First, the original vibration signal is decomposed by VMD to obtain several intrinsic mode component functions (IMF). The component with the largest correlation coefficient is selected as the optimal component for filtering in the fractional order domain. The 1.5-dimensional envelope spectrum of the filtered signal is analyzed , The frequency value corresponding to the maximum amplitude can be obtained. This frequency value is the fault characteristic frequency of the rolling bearing. The simulation results show that the method can effectively extract the fault characteristic information of the rolling bearing.

The reconstruction of super-resolution images from low-resolution images is essentially a morbid invers problem , which can often be dealt with by adding regular terms. In this paper, using the traditional total variational method for reference, fractional total variational regularities and fractional fidelity terms are added to the model to constrain the solution space, and an adaptive fractional order function based on local variance is proposed. In addition, the Fourier transform is used to calculate in the frequency domain, which reduces the computational complexity. The experimental results show that the image reconstruction model proposed in this paper can reconstruct the texture details and edges more clearly, and at the same magnification, the values of peak signal to noise ration (PSNR) and structural similarity index measure(SSIM) are higher than those of the comparison methods.

A new spectrum representation method based on short time Fourier transform (STFT) is proposed. The formula of new spectrum representation is derived base on energy cumulant of short time Fourier transform (EC-STFT), which indicates that EC-STFT has the spectrum characteristics. Simulations on the linear frequency modulation (LFM) signal show that the EC-STFT spectrum is closer to the ideal spectrum curve than FT spectrum. During calculating EC-STFT, a time frequency domain iterative mean threshold (TFD-IMT) denoising method is presented to remove the addictive white Gaussian noise (AWGN), by which the EC-STFT spectrum has better anti-noise capacity than FT spectrum. Theoretical analyses and simulations verify the advantages of the EC-STFT over FT in conditions of low SNR.

A three-dimensional parabolic equation acoustic propagation model using a higher-order operator approximation is proposed for calculating sound propagation in complex underwater environments. The higher-order approximation method improves wide-angle capability and accuracy of the parabolic equation solution. The split-step Fourier method is used to implement the model. An idealized wedge problem is presented to verify the performance of the proposed higher-order model by contrasting with the image method.

This paper presents a study of the frequency spectrum regarding the first and the second heart sounds. Therefore, measurements of the phonocardiogram under different physiological conditions are conducted. For this purpose, two different methods to analyze the spectrum, are applied to the recorded phonocardiograms, namely the short-time Fourier transform and the continuous wavelet transform. Seven different measurements are conducted from 12 subjects, where the auscultation point, posture, breathing and physical stress are varied. Moreover, the interference due to human voices is taken into account. The determined 10 dB-bandwidth, which starts at around OHz, for the first and the second heart sound is approx. 55 Hz in the short-time Fourier case and about 100Hz for the continuous wavelet transform. The analyzed parameters do not show any significant change under varying physiological conditions, whereas the frequency spectrum of the first and second heart sounds differs strongly between the individual subjects. It is concluded that for the filter design of heart sounds monitoring in wearable devices, adaptation to varying daily activities can be neglected, whereas adaptation to the user is strongly recommended.

In this paper, a linear transform that is a further generalization of the linear canonical transform (LCT) and the fractional Fourier transform is proposed. Instead of the chirp terms adopted by the LCT, we apply exponent functions with higher order phases before and after applying the scaled Fourier transform. With the proposed higher order phase linear canonical transform (HOPLCT), one can convert the time-frequency distribution with a bent shape, which is usually the case for vocal signals, into a rectangular shape. It can much reduce the bandwidth of a signal and is very helpful for signal sampling and filter design.

The paper is devoted to analysis of filtration errors in finite-frequency identification. We consider a linear control plant subject to action of an external disturbance, which is assumed to be a bounded function. The finite-frequency identification allows us to find estimates of this plant's parameters. It uses special integral filters (Fourier filters), that find estimates of the plant frequency response at specific frequencies. There are known estimates of convergence rate of Fourier filters errors, which we use as a basis to propose a new approach to determine the duration of identification. The proposed method is based on a special linear programming problem, the solution of which gives us two parameters: an estimate of the filter value and a parameter that describes the rate of the filter error convergence. The last one is used for duration determination. We develop corresponding filtration algorithm and give certain consideration to its accuracy. An illustrative example is presented.

The great role of Fourier coefficients in the digital processing of multidimensional signals is known. The classic cubic formulas for the calculation of Fourier coefficients use the values of an approximate function at individual points. Due to the advent of computerized tomograph, new information operators have emerged that do not directly use the function value at individual points in the integration area. These are projections-integrals from an approximate function along a given system of straight lines, tomograms–images of traces of an approximate function of three variables on given planes, etc. The classic Fourier coefficients calculation using projections is based on the formulas used for direct and inverse Radon conversion. For the 2D case, such studies were performed in classical computed tomography circuits, including the direct Fourier method Another approach to calculate 2D Fourier coefficients without using a direct and inverse Radon transform was proposed in 2000 by Oleg M. Lytvyn. This paper deals with the generalization of this method in the 3D case. It is a continuation of the research begun by the authors in 2019. This paper proposes explicit formulas for the limits of integration for the re-integrals arising in the proposed method. Thus, this paper is devoted to the further development of a new scientific direction in the theory of approximate Fourier coefficients of the function of three variables using new information operators.

Aimed at the problem that the transient zero sequence current of each point in distribution networks is characterized by nonlinearity and non-stationarity when a single phase to ground (SPG) fault occurs, a fault diagnosis methodology based on Fourier decomposition method (FDM) and multi-scale transient energy relative entropy is proposed. First, the zero sequence current after a SPG fault is expanded based on the generalized Fourier expansion method, and a series of Fourier intrinsic band functions (FIBFs) which are adaptive, local, orthogonal and uncorrelated are obtained. Subsequently, the Fourier-Hilbert spectrum (FHS) based on FIBFs is utilized to define the instantaneous energy (IE) density of the fault. After that, the difference between the signals on both sides of various sections is represented by energy relative entropy. Through calculation and comparison, the fault section can be accurately determined. This methodology is suitable for different transition resistances, fault angles and noise interference. Finally, numerical results verify the effectiveness of the proposed methodology.

In this paper, a new algorithm of sinusoidal signal frequency estimation based on interpolation of Fast Fourier Transform (FFT) and Discrete-Time Fourier Transform (DTFT) is proposed. Firstly, A-point FFT is used to search the position of the maximum FFT spectrum line and get the rough frequency estimation. Then three DTFT spectrum lines located at any position within the main lobe is used to obtain the fine frequency estimation value. Next, in order to further improve the estimation performance, the proposed algorithm is combined with Candan algorithm. According to the simulation results, Root Mean Square Error (RMSE) of the proposed algorithm is significantly lower than Candan algorithm, RCTSL algorithm and A&M algorithm. In a wide range of Signal-to-Noise Ratio (SNR), the proposed algorithm is closer to the Cramer-Rao lower bound (CRLB) than the competitive algorithms. In terms of computation complexity, the proposed algorithm in this paper is lower than A&M algorithm, HAQSE algorithm and RCTSL algorithm.

We propose a novel trellis-based peak frequency tracking technique for photoplethysmographic (PPG) heart rate (HR) sensing. Dealing with HR changes on the time-frequency plane obtained by the Short Time-Discrete Fourier Transform as state transitions in a trellis diagram, the proposed technique rejects outliers induced during exercise. Experimental result involving thirteen subjects reveals that the proposed peak frequency tracking technique, which is combined with the recursive least squares algorithm-based MA cancellation technique, can achieve the root mean squares percent error of 6.924 % for a series of exercises with various intensities.

In this contribution, a novel maximum likelihood (ML) based direction of arrival (DOA) estimator for concurrent speakers in a noisy reverberant environment is presented. The DOA estimation task is formulated in the short-time Fourier transform (STFT) in two stages. In the first stage, a single local DOA per time-frequency (TF) bin is selected, using the W-disjoint orthogonality property of the speech signal in the STFT domain. The local DOA is obtained as the maximum of the narrow-band likelihood localization spectrum at each TF bin. In addition, for each local DOA, a confidence measure is calculated, determining the confidence in the local estimate. In the second stage, the wide-band localization spectrum is calculated using a weighted histogram of the local DOA estimates with the confidence measures as weights. Finally, the wide-band DOA estimation is obtained by selecting the peaks in the wide-band localization spectrum. The results of our experimental study demonstrate the benefit of the proposed algorithm in a reverberant environment as compared with the classical steered response power phase transform (SRP-PHAT) algorithm.

This paper presents a new method to overbound Kalman filter (KF) based estimate error distributions in the presence of uncertain, time-correlated noise. Each noise component is a zero-mean Gaussian random process whose autocorrelation sequence (ACS) is stationary over the filtering duration. We show that the KF covariance matrix overbounds the estimate error distribution when the noise models overbound the Fourier transform of a windowed version of the ACS. The method is evaluated using covariance analysis for an example application in GPS-based relative position estimation.

Most research on drone detection and classification focus on using features from micro-Doppler signatures with blade flashes. However, these methods are limited in range and require radars with high pulse repetition frequency (PRF)–at least twice the maximum tip velocity. A different method to detect and classify drones at longer ranges using a low PRF radar is desired. In the literature, the cepstrum method was shown to be able to estimate the rotation rate when the PRF is insufficient. An alternative way of analyzing micro-Doppler is by using a long windowed Short-time Fourier transform (STFT) to generate HElicopter Rotation Modulation (HERM) lines. HERM lines exhibit similar behavior to a cepstrogram, with spectral lines separated in frequency by a value related to the rotation rate. In this paper, the separation frequency of HERM lines was estimated using a log harmonic summation algorithm. The proposed algorithm was tested on a simple HERM line model and also on real data obtained from two blade single rotor micro-helicopter drone. The algorithm was shown to be more resilient than cepstrum under Gaussian noise.

Methods of determining the envelope (instantaneous amplitude) of a real signal for continuous and discrete cases are investigated. The analysis of advantages and disadvantages of existing methods for extraction the envelope of a real signal for continuous and discrete cases are given. The efficiency of the choice of determining the conjugate signal by Hilbert transform method is substantiated. The problem of determining Hilbert envelope for real, discrete and finite signals by the methods of discrete Fourier transform (DFT) and discrete-time Fourier transform (DTFT) is considered. The reason for the large relative errors in the measurement of Hilbert envelopes by DFT method is revealed. A new effective method and an effective method for measuring the envelope of a real, discrete, finite signal is developed, which reduces the relative error of its measurement by an order of magnitude and reduces by an order of magnitude the computational cost and the necessary memory for implementing the method as well. The theoretical results obtained in the work are confirmed by the results of numerical modeling.

Methods for solving the amplitude-phase problem for one-dimensional signals are proposed. The amplitude-phase problem is investigated in the case of continuous and discrete signals. In both cases, the amplitude-phase problem is modeled by nonlinear singular integral equations. The study of continuous signals leads to nonlinear singular integral equations defined on the numerical axis; discrete ones to nonlinear singular integral equations defined on a unit circle in the plane of a complex variable. The obtained singular integral equations relate to the exceptional case — the symbols of the Frechet derivatives of singular operators degenerate in the entire domain of their definition.The continuous operator method for solution of nonlinear equations is used for solution these nonlinear singular integral equations. The numerical schemes for solving corresponding singular integral equations are constructed. Solutions of model examples have shown effectiveness of the proposed method and numerical algorithms.

This work presents beta subband (12-30 Hz) as a biomarker to distinguish between interictal and ictal states using Haralick features. Previous works has showed whole frequency spectrum for this analysis. Significance of this work is it has used only beta subband of electroencephalogram (EEG) for classification using image descriptors. One dimensional EEG data has been converted into image using Short-time Fourier Transform (STFT). Beta subband is cut from the time frequency (t-f) plane and Haralick features is fed in the decision tree classifier. The results have been evaluated using K-fold cross validation and classification accuracy of 92.5% has been calculated. Receiver operating characteristic (ROC) analysis has also been performed which shows maximum area under curve (AUC) of 0.94 to distinguish between interictal and ictal.

The fractional powers of the Fourier transform operator is a generalization of the Fourier transform. Although there are many works on the derivation of sampling about fractional Fourier transform, studies on the aliasing error are rare. This paper presents a new aliasing error estimation method of fractional Fourier transform connecting with Poisson sum formula based on windowed samples.

In this paper we deal with a special class of Lyapunov equations which we call circulant Lyapunov equations. These are Lyapunov equations arising from system matrices that are circulant. Here we bring out new theoretical insights into such Lyapunov operators. We show that, under a suitable projection map, two-variable polynomials are related to circulant Lyapunov operators. We also give necessary and sufficient conditions for the solvability of a circulant Lyapunov equation using two dimensional Fourier transform operation (2D-DFT) performed on a specially constructed matrix. Using these links among circulant Lyapunov operators, two-variable polynomials and 2D-DFT, an algorithm to solve circulant Lyapunov equations using 2D-DFT is developed in this paper.

A coordinate-free formulation of four tensor Green’s functions for the general homogeneous uniaxial anisotropic lossless and/or lossy media is developed with the Fourier transformations. In the derivation of these formulas, the product to sum approach is introduced to evaluate the Fourier integrations. For the lossy media, the double-value problem, which is existed in calculating formula of Green’s functions, is effectively avoided by the radiation condition. With a way of finding limit, the pseudo-singularities problem of four Green’s functions, which has to be treated, is discussed deeply and resolved briefly. The robust and efficient computational procedure of Green’s functions is achieved with C++ operator overloading. Modeling results of the triaxial induction well-logging demonstrate efficiency of the computation of the coordinate-free formulas.

In time-frequency representation, S-transform is unique because of its local spectral phase properties. Discrete orthogonal S-transform (DOST) is the efficient representation of redundant S-transform. The Fourier transform of DOST bases is restricted to different non-overlapping subbands, ensuring orthogonality. DOST can be simplified as the multiplication of the DFT matrix and a block diagonal matrix containing IDFT matrices along with phase term. But, the use of DFT leads to a relatively low rate of decay which results in poor localization. Therefore, a noble mathematical formulation for DCT based DOST is introduced as the symmetrical nature of DCT reduces the abruptness during the transition between subbands. The basis function and its Fourier transform of different subbands of the proposed DCT based DOST formulation are simulated and plotted using MATLAB. Moreover, the use of DCT reduces the total number of computation in the case of DCT based DOST compared to original DOST.

We considered a problem of reconstruction function in a strip, if we know the integrals of sought function in the family of straight-line segments with a given weight function of a special kind. We obtained an analytical representation of solution in the class of smooth finite functions, the uniqueness theorems for solution of the problem is proved. The stability estimate of solution in Sobolev spaces is presented, which implies its weakly ill-posedness. These theoretical results are used in order to recover the internal structure objects from their integral data. Numerical results and their graphical representation are given.

In this paper, we proposed a novel reversible watermarking scheme based on Discrete Fourier Transform (DFT) and on Even-Odd Modulation technique. The proposed scheme supports the following features: blind extraction of the watermark, multi-layer mode of the watermark embedding, high quality of the watermarked image, complete recovering of the original image. We proposed to embed the watermark into DFT magnitudes by changing the parity of the magnitude digits. During experiment we used the watermarks of sizes 512×512 and 32×32. The watermark of size 512×512 has been embedded four times into the original gray scale image of size 512×512 with the following indicators: the values of Peak Signal Noise Ratio (PSNR) of the watermarked images were about 65 dB, the Normalized Coefficient of Correlation (NCC) between the embedded and extracted watermarks was equal to one and after performing the recovering procedure, the values of PSNR between the original and recovered images tend to infinity, which indicates 100% recovering of the original images. The watermark of size 32×32 has been embedded into the same image five times with the following indicators: PSNR of the watermarked image was equal to about 91 dB, NCC=1 and PSNR of recovered images tends to infinity. Experimental results prove that the presented scheme achieves high capacity, visual quality, complete reversibility, and better robustness in comparison with the schemes proposed in the literature.

This paper presents the integrated parameterized cubic phase function (IPCPF) for noisy multicomponent cubic phase signals (CPSs) analysis. The IPCPF is based on the parameterized cubic phase function (PCPF) and uses the modulus operator to guarantee a low computational cost [O(N2log 2 N)]. The cross terms, computational cost and antinoise performance are theoretically analyzed for the IPCPF and maximum likelihood (ML) method. Theoretical analyses and numerical simulations demonstrate that, compared to the ML method, the IPCPF can obtain a higher anti-noise performance with a lower computational cost.

A unified framework for bridging the power allocation (PA) strategies of single carrier (SC) and orthogonal frequency division multiplexing (OFDM) systems is proposed through the hybrid carrier (HC) scheme based on weighted-type fractional Fourier transform (WFRFT). The performance analyses of multiple indicators are easily achieved in the proposed framework, and the expected performances could be obtained according to the flexible WFRFT order. The closed BER expressions of arbitrary PA strategies in the HC scheme are derived and verified by the simulations, where the equal gain PA and minimum mean square error (mmse) PA are given as the examples. Compared with the OFDM and SC systems, the tradeoff between BER, achievable rate, and peak-to-average power ratio is achieved in the HC system. In addition, an approximate minimum bit error rate method through joining the PA and subcarrier allocation is investigated in the HC scheme as a generalization of the proposed framework. Considering the novel structure, the dynamic threshold could be established in the HC scheme, and thus the tradeoff between BER and complexity is obtained in comparison with the OFDM and SC systems. In order to illustrate the anti-interference characteristics of the proposed HC framework with PA, the BER performance with insufficient cyclic prefix is investigated. The better BER performance is captured in the HC system with the proposed mmse PA strategy.

This paper proposes a fast Taylor Weighted Least Squares (fast-TWLS) algorithm for accurate real-time synchrophasor estimation. It requires the calculation at runtime of only two Discrete Time Fourier Transforms (DTFTs) at nominal frequency and further few elementary arithmetic operations. The accuracies - as expressed by the Total Vector Error (TVE) - of the proposed fast-Twls algorithm and the real-valued TWLS algorithm recently proposed in the literature are compared to each other under several static and dynamic testing conditions suggested in the IEEE Standard C37.118.1-2011 for synchrophasor measurements for power systems.

Considering nonlinear and non-Gaussian environments that are frequently encountered in real-world scenarios, we thus propose the approximated kernel least mean p-power (AKLM-P) algorithm to address these tough factors for nonlinear distributed systems. In order to efficiently devise this algorithm, we use the property of shift-invariant kernel function that can be approximated by random Fourier features, and consider the impulsive noise characterized by the α-stable distribution. The stability in the mean of proposed distributed algorithm is then studied. Finally, the superiorities of proposed approach compared to diffusion kernelleast-mean-square (KLMS) algorithm are confirmed via the experimental results.

In this paper a low computation effort Taylor-based Weighted Least Squares (TWLS) synchrophasor estimator that takes into account the contribution of the 2nd harmonic is proposed. That algorithm, called Simplified TWLS-H(2) algorithm, requires the evaluation of only four Discrete-Time Fourier Transforms samples and the related weighting coefficients. The achieved estimation accuracy is compared with that provided by different versions of the TWLS algorithm published in the literature through computer simulations performed under different steady-state and dynamic conditions suggested in the IEEE Standard C37.118.1-2011 for synchrophasor measurements for power systems for the M-class of performance. The comparison is based on the Total Vector Error (TVE), the Frequency Error (FE), and the Rate-of-change-of-Frequency Error (RFE) parameters.

In this paper we study the Fourier transform and the possibility to determine the binary expansion of the values of the Fourier transform in the Zermelo–Fraenkel set theory with the axiom of choice included (ZFC). We construct a computable absolutely integrable bandlimited function with continuous Fourier transform such that ZFC (if arithmetically sound) cannot determine a single binary digit of the binary expansion of the Fourier transform at zero. This result implies that ZFC cannot determine for every precision goal a rational number that approximates the Fourier transform at zero. Further, we discuss connections to Turing computability.

The technique for increasing the robustness of the photometric method of ellipsometry according to Malus law is considered. The parameters of the polarization ellipse by the ordinary least squares method and the Fourier method are calculated. The data processing procedure using the bootstrap method is described. The spread of the measured parameters of the polarization ellipses is calculated. The stability and reliability of the obtained results is appreciated. The increase in the robustness of the photometric method is confirmed.

Firstly, based on basic properties of the kernel function of the quaternion Fourier transform we derive in detail relationships among three definitions of the quaternion Fourier transforms. Secondly, based on the quaternion Fourier transform of the quaternion Gaussian function we derive an inversion formula to recovering a quaternion function from the quaternion Fourier transform.