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Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + \epsilon, $$ where $\epsilon = \epsilon(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $\mu$ is a Borel measure on $\mathbb{P}$ satisfying $\mu(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$ and $r > 0$, then there exists $\epsilon = \epsilon(s) > 0$ such that $$ \|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem., Comment: 26 pages, 2 figures

Актуальность исследования обусловлена необходимостью разработки перспективного метода прогноза удароопасности массива по параметрам естественного электромагнитного излучения, возникающего при трещинообразовании в горных породах. Существующие методики интерпретации данных геофизического мониторинга не учитывают связи параметров электромагнитного излучения с основными геомеханическими критериями, определяющими вероятность проявления горного удара: концентрацией напряжений в массиве и расположением их максимума. Цель: установление рациональных параметров для дальнейшего экспериментального обоснования критериев локального прогноза удароопасности методом регистрации электромагнитного излучения на базе экспериментальной аппаратуры «Ангел-М», разработанной в МНЦ ВНИМИ. Объекты: массивы горных пород, склонных к динамическим проявлениям горного давления на горнодобывающих предприятиях Кузбасса (шахта «Таштагольская», глубина 700 м; шахта «Зиминка», глубина 300 м; шахта № 12, глубина 340 м). Методы: анализ физических механизмов электризации пород в поле механических напряжений на основе фундаментальных закономерностей физики твердого тела и электродинамики, экспериментально-теоретические основы метода регистрации электромагнитного излучения в лабораторных и натурных условиях. Результаты. Рассмотрены физические предпосылки возникновения электромагнитного излучения при разрушении твердых тел и горных пород, основные теоретические и экспериментальные результаты, полученные в данной области ведущими научными коллективами. Изложена методика расчета амплитудно-частотной характеристики электромагнитного излучения при трещинообразовании в горных породах, включающая решение волнового уравнения, прямое и обратное преобразование Фурье, эффекты суперпозиции систем трещин и затухания электромагнитного излучения в проводящей среде. Обоснован перспективный метод прогноза удароопасности массива методом регистрации электромагнитного излучения на основе комплексных критериев, косвенно взаимосвязанных с фундаментальными геомеханическими параметрами массива (концентраций напряжений в точке их максимума и расстоянием от контура до этой точки): средняя амплитуда импульсов за интервал регистрации и крутизна графика скорости нарастания импульсов. The relevance of the study is caused by the need to develop a promising method for predicting the impact hazard of the array according to the parameters of natural electromagnetic radiation that occurs during cracking in rocks. The existing methods of interpretation of geophysical monitoring data do not take into account the relationship of electromagnetic radiation parameters with the main geomechanical criteria determining the probability of a mountain impact: the concentration of stresses in the array and the location of their maximum. The main aim is to establish rational parameters for further experimental substantiation of the criteria for local prediction of impact hazard by the electromagnetic radiation registration method based on the experimental equipment «Angel-M» developed at the MNC. Objects: massifs of rocks prone to dynamic manifestations of mountain pressure at Kuzbass mining enterprises (Tashtagolskaya mine, depth 700 m; Ziminka mine, depth 300 m; mine no. 12, depth 340 m). Methods: analysis of the physical mechanisms of electrification of rocks in the field of mechanical stresses based on the fundamental laws of solid state physics and electrodynamics, experimental and theoretical foundations of the method of recording electromagnetic radiation in laboratory and field conditions. Results. The paper considers the physical prerequisites for electromagnetic radiation occurrence during the destruction of solids and rocks, the main theoretical and experimental results obtained in this field by leading scientific teams. A method for calculating the amplitude frequency response of electromagnetic radiation during fracturing in rocks is described, including the solution of the wave equation, the forward and inverse Fourier transform, the effects of superposition of crack systems and electromagnetic radiation attenuation in a conductive medium. A promising method for predicting the impact hazard of an array by the electromagnetic radiation registration method is substantiated on the basis of complex criteria indirectly interrelated with the fundamental geomechanical parameters of the array (stress concentrations at their maximum point and the distance from the contour to this point): the average amplitude of pulses during the registration interval and the steepness of the pulse rate graph.

Waste incineration plants are complex dynamical systems that rely on expert human operators to maintain steady combustion, by observing real-time in-chamber video feeds. Real-time plant forecasting provides vital operational support in decision making, and applying machine learning to automatically learn dynamics forecast models from video feeds is an attractive means to realise this. However, learning complex dynamics in systems that requires cost-efficiency remains an open research problem. Specifically, modelling plant dynamics in real-time is challenging due to uncertainties caused by inhomogeneous waste inputs, requiring complex learning that impedes real-time modelling. To address this, this paper presents a real-time data-driven framework for generating video forecasts, by incorporating task-relevant domain-knowledge, during learning. Specifically, this method combines dynamics modelling and forecasting using dynamic mode decomposition, with Fourier transformations informed by expert operator heuristic knowledge for encoding task-relevant frequency information inside the learning process. Experiments in this paper demonstrate that the proposed framework captures intuitive physical aspects of the underlying physiochemical process, with a greatly reduced computational runtime in comparison to standard approaches, allowing for application in real-time domains. Forecasted video predictions are accurate over short time horizons, and capture important system characteristics over longer time periods.

Several signal processing tools are integrated into machine learning models for performance and computational cost improvements. Fourier transform (FT) and its variants, which are powerful tools for spectral analysis, are employed in the prediction of univariate time series by converting them to sequences in the spectral domain to be processed further by recurrent neural networks (RNNs). This approach increases the prediction performance and reduces training time compared to conventional methods. In this letter, we introduce fractional Fourier transform (FrFT) to time series prediction by RNNs. As a parametric transformation, FrFT allows us to seek and select better-performing transformation domains by providing access to a continuum of domains between time and frequency. This flexibility yields significant improvements in the prediction power of the underlying models without sacrificing computational efficiency. We evaluated our FrFT-based time series prediction approach on synthetic and real-world datasets. Our results show that FrFT gives rise to performance improvements over ordinary FT.

We propose a novel method to determine the average water depth from shallow, weakly nonlinear water waves that are approximated by the Korteweg-de Vries equation. Our identification method only requires free-surface measurements from two wave gauges aligned in the direction of wave propagation. The method we propose is based on comparing solitonic components in wave packets, which are computed using the nonlinear Fourier transform (NFT) (typical time-series data often contains at least some solitonic components, even when these components are not directly visible). When the correct water depth is used for the normalisation of the wave, the solitonic components found by the NFT remain constant as the wave packet propagates, whereas any other water depth will result in solitonic components that do not remain constant. The basic idea is thus to iteratively determine the water depth that leads to a best fit between the solitonic components of time series measurements at two different gauge positions. We present a proof-of-concept on experimental bore data generated in a wave flume, where the identified water depth is within 5% of the measured value.

A general and flexible technique is proposed for sampling the complex envelope of a bandpass signal by using a nonuniformly interleaved two-channel analog-to-digital converter (TIADC). The signal is sampled directly without any demodulation operation, but the two sampling channels are not uniformly interleaved, since some TIADC timing offsets are rendered forbidden. The proposed complex envelope sampling scheme requires the implementation of two digital FIR filters and a discrete-time modulator. Quadrature sampling, a special case that assumes certain parameter relations, is investigated as an approximation technique. Computer simulations are presented and illustrate a -57dB mean-square error (MSE) can be achieved via quadrature approximation for sufficiently large filter orders.

Maximal and partial quantum yields of photosystem II (Fv/Fm and Fv–/Fp) in Phaseolus vulgaris leaves were determined in response to the turning on saturating and low (undersaturating) actinic light, accordingly. Measuring lights (ML) of blue, green, and red colors were applied simultaneously using a novel method of the polychromatic fast Fourier transforming pulse amplitude modulation (FFT PAM) chlorophyll fluorometry. Colors of ML were cross-combined with the colors of low and saturating actinic light. Fv/Fm values measured with the green ML were found to be close to that of red light, whereas application of blue ML leads to lower Fv/Fm. In addition, 5-nm resolved excitation spectra were measured to evaluate the dependence between red/far red fluorescence ratios (peak height based – F685/F730, and deconvolved peak square based – F685sq/F730sq) and wavelength of the excitation light. It demonstrates that chlorophyll fluorescence ratio red/far red may be dependent on the spectral absorbance of chlorophyll. The data obtained confirm the results of the previous studies which explain high photosynthetic activity of the green light in terms of redistribution of absorbed quanta throughout thickness of the leaf, thus suggesting possible advisability of green light application in greenhouse lighting, especially for the plants having thick leaves.

This article proposes the augmentation of the adjacency model of networks for graph signal processing. It is assumed that no information about the network is available, apart from the initial adjacency matrix. In the proposed model, additional edges are created according to a Markov relation imposed between nodes. This information is incorporated into the extended-adjacency matrix as a function of the diffusion distance between nodes. The diffusion distance measures similarities between nodes at a certain diffusion scale or time, and is a metric adopted from diffusion maps. Similarly, the proposed extended-adjacency matrix depends on the diffusion scale, which enables the definition of a scale-dependent graph Fourier transform. We conduct theoretical analyses of both the extended adjacency and the corresponding graph Fourier transform and show that different diffusion scales lead to different graph-frequency perspectives. At different scales, the transform discriminates shifted ranges of signal variations across the graph, revealing more information on the graph signal when compared to traditional approaches. The scale-dependent graph Fourier transform is applied for anomaly detection and is shown to outperform the conventional graph Fourier transform. © 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

The spectral analysis of signals is currently either dominated by the speed–accuracy trade-off or ignores a signal’s often non-stationary character. Here we introduce an open-source algorithm to calculate the fast continuous wavelet transform (fCWT). The parallel environment of fCWT separates scale-independent and scale-dependent operations, while utilizing optimized fast Fourier transforms that exploit downsampled wavelets. fCWT is benchmarked for speed against eight competitive algorithms, tested on noise resistance and validated on synthetic electroencephalography and in vivo extracellular local field potential data. fCWT is shown to have the accuracy of CWT, to have 100 times higher spectral resolution than algorithms equal in speed, to be 122 times and 34 times faster than the reference and fastest state-of-the-art implementations and we demonstrate its real-time performance, as confirmed by the real-time analysis ratio. fCWT provides an improved balance between speed and accuracy, which enables real-time, wide-band, high-quality, time–frequency analysis of non-stationary noisy signals.

A novel approach to implementing Galois field Fourier transform (GFT) is proposed that completely eliminates the need for any finite field multipliers by transforming the symbols from a vector representation to a power representation. The proposed method is suitable for implementing GFTs of prime and nonprime lengths on modern FPGAs that have a large amount of on-chip distributed embedded memory. For GFT of length 255 that is widely used in many applications, the proposed memory-based implementation exhibits 25% improvement in latency, 27% improvement in throughput, and 56% reduction in power consumption compared to a finite field multiplier-based implementation.

Over the past years, nonlinear frequency-domain methods have become a state-of-the-art technique for the numerical simulation of unsteady flow fields within multistage turbomachinery as they are capable of fully exploiting the given spatial and temporal periodicities, as well as modelling flow nonlinearities in a computationally efficient manner. Despite this success, it still remains a significant challenge to capture nonlinear interaction effects within the context of configurations with multiple fundamental frequencies. If all frequencies are integer multiples of a common fundamental frequency, the interval spanned by the sampling points typically resolves the period of the common base frequency. For configurations in which the common frequency is very low in relation to the frequencies of primary interest, many sampling points are required to resolve the highest harmonic of the common fundamental frequency and the method becomes inefficient. In addition when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations or when simulating the nonlinear interaction in the context of forced response or flutter, then the standard discrete Fourier transform is no longer suitable and the basic harmonic balance method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or non-equidistant time sampling. However, the definition of suitable sampling points that lead to well-conditioned Fourier transform matrices and small aliasing errors is an intricate issue and far from straightforward. To overcome the issues regarding multi-frequency problems described above, a new harmonic balance approach based on multidimensional Fourier transforms in time is presented. The basic idea of the approach is that, instead of defining common sampling points in a common time period, separate time domains, one for each base frequency, are spanned and the sampling points are computed equidistantly within each base frequency’s period. Since the sampling domain is now extended to a multidimensional time-domain, all time instant combinations covering the whole multidimensional domain are computed as the Cartesian product of the sampling points on the axes. In a similar fashion the frequency-domain is extended to a multidimensional frequency-domain by the Cartesian product of the harmonics of each base frequency, so that every point defined by the Cartesian product is an integer linear combination of the occurring frequencies. In this way the proposed method is capable of fully integrating the nonlinear coupling effects between higher harmonics of different fundamental frequencies by using multidimensional discrete Fourier transforms within the harmonic balance solution procedure. The aim of this paper is to introduce the multidimensional harmonic balance method in detail and demonstrate the capability of the approach to simultaneously capture unsteady disturbances with arbitrary excitation frequencies. Therefore the well established aeroelasticity testcase standard configuration 10 in the presence of an artificial inflow disturbance, that mimics an upstream blade wake, is investigated. The crucial aspect of the proposed testcase is that a small ratio of the frequency of the inflow disturbance and the blades vibration frequency is chosen. To demonstrate the advantages of the newly proposed multidimensional harmonic balance approach, the results are compared to unsteady simulations in the time-domain and to state-of-the-art frequency-domain methods based on one-dimensional discrete Fourier transforms.

This paper proposes a generative hierarchical probabilistic model for acoustic signals where both the frequency decomposition and log-power spectrum appear as latent variables. In order to facilitate efficient inference, we represent the model in a factor graph that includes a probabilistic Fourier transform and a Gaussian scale model as modules. We derive novel ways of performing variational message passing-based inference in the Gaussian scale model. As a result, in this model a probabilistic representation of the log-power spectrum of an acoustic signal can be effectively inferred online. The proposed model may find applications as a front end wherever probabilistic log-power spectral features of a signal are needed. We validate the model and message passing-based inference methods by tracking the log-power spectrum of a speech signal.

Convolutional neural network (CNN) is an extremely powerful machine-learning tool, especially when dealing with computer vision problems. Here, the authors present a CNN-based modulation recognition model. In order to fully elaborate the powerful image feature extraction ability of CNN, the authors have created an image dataset of different complex signal spectrograms using short-time Fourier transform (STFT). In this case, the complex modulation recognition problem is converted to an image recognition problem. To study the accuracy of automatic recognition of signal spectrograms, the authors have applied two approaches recently developed for image classification. The first approach is to optimise activation functions. Experiments show that best performance can be achieved when using sigmoid as activation function. The second approach is using optimisation functions. At last, the authors compared the recognition accuracy under different signal-to-noise ratios (SNRs). The result shows that authors’ model achieves higher recognition accuracy under low SNR and stronger generalisation ability than other recognition methods.

In this study, a coherent detection method is proposed to solve the problems during the coherent integration (CI) process, including the range cell migration (RCM) and Doppler spread (DS) effects which are caused by the complex motions of the maneuvering target. Particularly, this method applies the third-order keystone transform firstly, which could remove the third-order RCM induced by the jerk motion. Then target's defined velocity can be estimated by performing the frequency-shifting cross-correlation function and scaled Fourier transform. Thereafter, utilising the estimated defined velocity to construct a velocity compensation function, which could correct the first-order RCM (FRCM). Subsequently, the matched filtering process is performed to eliminate the second-order RCM and DS effects. Finally, applying the Fourier transform with respect to the slow time to realise the CI.

In the radar systems, such as frequency source, power amplifier, mixer, filter and A/D converter, ageing devices, temperature and environmental changes etc., it is inevitable to introduce wideband amplitude/phase error. These can cause seriously channel mismatch that will affect the performance of digital beam-forming, synthetic aperture radar image, space time adaptive processing (STAP), automatic target recognition (ATR), target detection, clutter/interference suppression, pulse compression results and cancellation ratio etc. So this study mainly discusses that the channel amplitude/phase inconsistency of wideband digital array radar affects the pulse compression, beam forming and so forth. It also researches and analyses channel equalisation method, and the simulation and experimental results based on the experimental data show that the method based on Fourier transform is effectively and convenient. The Fourier transform method is easier for hardware implementation that can be applied to practical engineering.

This study proposes a multi-ary modulation chirp spread spectrum algorithm specific for the unmanned aerial vehicles field. It further introduces the corresponding fractional Fourier transform de-spreading algorithm and the hardware implementation of the software radio-based communication system to validate the algorithm's better anti-interference performance in the wide-band non-stationary interference environment.

A novel coherent integration method for high dynamics direct sequences spread spectrum (DSSS) signal is presented. The presented method is based on adjacent cross-correlation function (ACCF) and Lv's distribution (LVD). ACCF is firstly used to compensate the code phase drift and reduce the Doppler frequency migration order. After that, LVD is used to accomplish the integration and obtain the parameter estimates. The presented method can estimate motion parameters without any searching procedure. Compared with the existing methods, the presented method can achieve a good balance between the computational cost and the detection and estimation performance. Numerical simulations are provided to verify the effectiveness of the proposed method.

Digital array radars with conformal arrays are attractive for platforms with limited installation space such as aircraft and unmanned aerial vehicles. Low-complexity digital beamforming (DBF) scheme is of great importance for conformal arrays with arbitrary geometry for real-time applications. This study presents two kinds of two-dimensional (2D) Fourier bases for the orthogonal decomposition of arbitrary array manifold. The sampling matrix (2D Fourier spectra) can be obtained from the 2D DFT of the array calibration data, facilitating fast DBF scheme for conformal arrays. Moreover, the second 2D Fourier basis can be used to rotate a function on [inline-formula]. The modelling accuracy and complexity of different 2D Fourier representations are compared by numerical results and the rotation of functions on [inline-formula] is demonstrated.

The array thinning technique can greatly reduce the number of the array elements while keeping the performance of the array almost the same. However, the existing algorithms have slow convergence rates and are easy to fall into local optimum. To improve the optimisation performance, a hybrid method based on improved genetic algorithm (GA) and iterative Fourier transform (IFT) technique for linear thinned array is proposed in this study. The population is divided into improved GA group and IFT group according to the convergence of the population and different operations can be done paralleled to generate offspring in each iteration. In the improved GA processing, adaptive crossover rate and mutation rate are used. The mechanism that keeps the fill factor stable is removed for larger search range. The IFT processing is executed paralleled for fast convergence velocity. The proposed hybrid method can obtain the fast convergence velocity and avoid being trapped into the local optimum by the combination of the two approaches. Several examples are simulated to validate the performance of the proposed method.

This study proposes a detection algorithm for a three-channel SAR system with weak ground moving target (GMT)-detection and focusing imaging under strong noise background. First, the authors use DPCA cancellation technology to eliminate ground clutter signals. By accumulating the DPCA signal amplitude in the azimuth direction and setting a suitable threshold, they realise GMT detection under strong noise background. After extracting the target range cell DPCA signal, they use the interference processing method to estimate the real-azimuth position of GMT. Then, they perform fractional Fourier transform processing on the extracted DPCA signal. Through the two-dimensional search of the peak point on the [inline-formula] plane of the parameter, the Doppler parameters and velocity estimation of the GMT are obtained. Finally, the azimuth-matching filter is modified to achieve focusing imaging. Simulation results show that the detection probability of GMT is >95% when [inline-formula]. When [inline-formula], the authors can achieve fairly accurate parameter estimation of GMT and ideal focusing imaging.

Based on the three-antenna interferometric radar system, a novel micro-Doppler feature extraction method for vibration targets is proposed in this study. First, the theory of interferometric processing of echo signal for vibration targets is introduced and explained. Second, short-time Fourier transform is conducted with the time–frequency diagram of echo signal obtained. Then, the signals received by three-antenna model are decomposed based on the segmental intrinsic chirp component decomposition. Finally, the 2D trajectory of vibration target is reconstructed and its micro-Doppler features are effectively extracted via the interferometric phase. Besides, the micro-Doppler parameters can also be estimated with the method. Two simulation experiments are conducted to validate the proposed method.

Radar moving target detection (MTD) technology is a key technology in the field of radar signal processing. The MTD method of radar has a better performance when applied to uniform motion detection although it has limited performance in other aspects, and it is also difficult to distinguish the type of the moving target. This article presents a new method for detecting and classifying moving targets based on convolutional neural network, which uses the convolutional neural network to learn the motion characteristics of the moving target echo so that realises the detection and classification of moving targets. At first, the authors model the moving target echo. Then the short-time Fourier transform (STFT) is used to perform time–frequency analysis. The obtained time–frequency graph is used as the input of the convolutional neural network which can automatically learns moving target features through training. Finally, the authors use the trained convolutional neural network model to detect and classify moving targets. The simulation verifies that the method has a great improvement in the detection accuracy rate compared with the traditional MTD.

Microplastics (MPs) are widely recognised as a contaminant of emerging concern in the marine environment. This work provides original data of the presence of MPs in coastal sediments along the Spanish Mediterranean continental shelf. Ten surface sediment samples were collected in order to document baseline microplastic distribution from Algeciras to Barcelona. Microplastics were extracted from bulk sediments by density separation. The number of microplastics per kilogramme of dry weight ranged from 45.9 ± 23.9 MPs/kg d.w. observed at Palma de Mallorca to 280.3 ± 164.9 MPs/kg d.w. noted at Málaga, with an average value of 113.2 ± 88.9 MPs/kg d.w. The lower limit is defined by the pore filter size used (1.2 μm). For all analysed locations, the dominant microplastic type was fibres (82.9%), followed by fragments, and the main colours were transparent and blue. Microplastic size distribution was presented; in the case of fragments, 85% was lower than 0.5 mm, and in the case of fibres, the three studied intervals (0.5–1, 1–2, 2–5 mm) had similar distribution (35, 34 and 31%, respectively). Attending to all available data, no statistically significant relationship (Spearman’s correlation) was found between microplastic average size and distance to the coast, the depth, density population and sediment grain size. Neither relationship was observed between these variables and microplastic concentration using the non-parametric Kruskal-Wallis H test. This study has confirmed the widespread distribution of MPs in surface sediments from the Spanish Mediterranean continental shelf, and these data are useful to define baselines for MPs in the Western Mediterranean region., Sí

A novel mission concept applying satellite formation flight to passive microwave interferometry was recently proposed to significantly improve the interferometer’s spatial resolution. This concept was shown to sample the visibility in a hexagonal tile of polar grids, and to recover the brightness map, this visibility must be inverted via a discrete polar inverse Fourier transform. For a fast and accurate solution, this letter develops a modified hexagonal variant of the pseudo-polar fast Fourier transform (PPFFT) and its inverse and explores its performance when applied to the proposed formation-flight radiometer. Compared to the conventional rectangular PPFFT, we find approximately a fivefold improvement in the recovered radiometric accuracy, where the rms radiometric error is in the order of $10^{-2}$ K. The impact of visibility interpolation method is also explored, showing that an FFT-based interpolation technique leads to the most accurate final image recovery.

The disturbance of the AC system has become one of the main causes of commutation failure in the high voltage direct current (HVDC) transmission system. Here, the time–frequency transformation method is used to analyse the AC-bus voltage disturbance from the perspective of the harmonic content of the AC power network. The time–frequency transformation for AC-bus voltage disturbance is difficult due to its characteristics of time variability and time fixity. With comparative analysis of various transformation methods, the method of combining the wavelet packet transform with the short-time Fourier transform can meet the requirement of time–frequency analysis of the AC voltage disturbance very well, and this method was used to analyse the fault recording data of Fengjing Convert Station when commutation failure occurred in recent years. The result shows that the AC-bus voltage mainly contained third and fifth harmonics, and there was a certain amount of even harmonics, and the low-order harmonics would be more in the case of a remote fault. In addition, there were more harmonics in the voltage when faults occurred in the 500 kV system, while there were more low-order harmonics when faults occurred in the other voltage class systems.

The converter is the main component of the converter station and the main noise source of the PLC interference. The accurate modelling of the high-frequency characteristics of the converter is the foundation for the design of the ± 1100 kV UHVDC PLC filter. Based on deep analysis of the high-frequency characteristics of the converter valve, this paper first proposes three high-frequency models of the converter valve. Compared with a real converter valve used in similar HVDC project, model two is chosen to analyse the high-frequency characteristics. Second, the model of Zhundong–Wannan UHVDC transmission project is established in the PSCAD simulation platform to determine the actual high-frequency characteristics of the converter valve. The voltage before the smoothing reactor is calculated, and the high-frequency noise level is obtained by Fourier transform. Considering the practical application of the high-frequency model of the converter valve, this paper finally proposes a formula to express the measured noise level. The noise source expressed by the formula is more practical for research on high-frequency characteristics of the converter station.

This paper presents a novel method, ITM- DSM, for the soil-foundation interaction problems analysis. It is a semi-numerical method based on the coupling of the Integral Transform Method (ITM) and the Dynamic Stiffness Method (DSM). The stiffness matrix of the soil-foundation system is obtained using the substructure technique. The ITM is used to obtain the solution of the wave propagation in the soil, while the DSM is used to calculate the dynamic stiffness matrix of the foundation. Both methods are operating in the frequency domain what makes them suitable for coupling. The number of numerical operations in the frequency domain is reduced by the application of the modal superposition technique. The analysis of vertical vibrations of flexible foundations resting on the viscoelastic half-space is presented. The formulation of the method could be generalized for horizontal and rocking vibrations. It could be also reduced to the problem of flexible strip foundations of infinite length.

Hyperspectral image registration is a relevant task for real-time applications like environmental disasters management or search and rescue scenarios. Traditional algorithms were not really devoted to real-time performance, even when ported to GPUs or other parallel devices. Thus, the HYFMGPU algorithm arose as a solution to such a lack. Nevertheless, as sensors are expected to evolve and thus generate images with finer resolutions and wider wavelength ranges, a multi-GPU implementation of this algorithm seems to be necessary in a near future. This work presents a multi-device MPI $$+$$ CUDA implementation of the HYFMGPU algorithm that distributes all its stages among several GPUs. This version has been validated testing it for 5 different real hyperspectral images, with sizes from about 80 MB to nearly 2 GB, achieving speedups for the whole execution of the algorithm from 1.18 $$\times $$ to 1.59 $$\times $$ in 2 GPUs and from 1.26 $$\times $$ to 2.58 $$\times $$ in 4 GPUs. The parallelization efficiencies obtained are stable around 86 $$\%$$ and 78 $$\%$$ for 2 and 4 GPUs, respectively, which proves the scalability of this multi-device version.

The modeling of source distributions of finite spatial extent in ultrasound and medical imaging applications is a problem of longstanding interest. In time domain methods, such as the finite difference time domain or pseudospectral approaches, one requirement is the representation of such distributions over a grid, normally Cartesian. Various artefacts, including staircasing errors, can arise. In this short contribution, the problem of the representation of a distribution over a grid is framed as an optimisation problem in the Fourier domain over a preselected set of grid points, thus maintaining control over computational cost, and allowing the fine tuning of the optimisation to the wavenumber range of interest for a particular numerical method. Numerical results are presented in the important special case of the spherical cap or bowl source.

The paper presents a new theory of combined moment invariants to Gaussian blur and spatial affine transformation. The blur kernel may be arbitrary oriented, scaled and elongated. No prior information about the kernel parameters and about the underlaying affine transform is required. The main idea, expressed by the Substitution Theorem, is to substitute pure blur invariants into traditional affine moment invariants. Potential applications of the new descriptors are in blur-invariant image recognition and in robust template matching.

This study presents the initialisation of a hybrid alternating current/direct current (AC/DC) power system to determine the operation point around which the system is linearised as a starting point for an electromagnetic or harmonic stability analysis. To this end, each AC or DC power system component is modelled in the Fourier transform domain and further adjusted to a simplified representation compatible with a power flow formulation. The study also presents how voltage source converter (VSC) high-voltage DC-based systems can be analysed when the different VSC controls are applied. The result of the power flow solution is then used for the power converters’ initialisation.

Distributed computing systems often consist of hundreds of nodes (machines), executing tasks with different resource requirements. Efficient resource provisioning and task scheduling in such systems are non-trivial and require close monitoring and accurate forecasting of the state of the system, specifically resource utilisation at its constituent machines. Two challenges present themselves towards these objectives.First, collecting monitoring data entails substantial communication overhead. This overhead can be prohibitively high, especially in networks where bandwidth is limited. Second, forecasting models to predict resource utilisation should be accurate and also need to exhibit high inference speed. Mission critical scheduling and resource allocation algorithms use these predictions and rely on their immediate availability.To address the first challenge, we present a communication-efficient data collection mechanism. Resource utilisation data is collected at the individual machines in the system and transmitted to a central controller in batches. Each batch is processed by an adaptive data-reduction algorithm based on Fourier transforms and truncation in the frequency domain. We show that the proposed mechanism leads to a significant reduction in communication overhead while incurring only minimal error and adhering to accuracy guarantees. To address the second challenge, we propose a deep learning architecture using complex Gated Recurrent Units to forecast resource utilisation. This architecture is directly integrated with the above data collection mechanism to improve inference speed of the presented forecasting model. Using two real-world datasets, we demonstrate the effectiveness of our approach, both in terms of forecasting accuracy and inference speed.Our approach resolves several challenges encountered in resource provisioning frameworks and can also be generically applied to other forecasting problems.

This paper presents the broken rotor bar fault diagnostics by time-frequency analysis of motor current under extended startup transient time achieved by reducing the applied voltage. The fault diagnostics under a steady stage regime has been a topic of interest since the past few decades. The main aim has been focused on the detection of fault-based frequencies which are the function of slip. Those frequencies become very less legible under low load conditions and totally disappear under no-load conditions. Moreover, the stator and rotor slot skews have a potential attenuation impact on them. To avoid these problems, the time-frequency analysis of motor startup current is investigated in this paper using a wavelet approach. To improve the legibility of the spectrum, the transient time is extended by reducing the supply voltage of the machine under no external load. By reducing the supply voltage, the inertia of the rotor acts as a load to increase the transient time which is essential for better resolution. The results are based on the practical measurements taken from the laboratory setup under healthy and faulty conditions.

Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its nodes. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures., Submitted to IEEE Transactions on Signal and Information Processing over Networks

The objective of this study is to investigate the stability of the active layer of polymer solar cells from the effect of UV–visible light irradiation using different conditions with respect to time. The active layers were composed of poly(3-hexylthiophene) (P3HT) and [6,6]-phenyl C61 butyric acid methyl ester (PCBM), deposited on conductive glass substrates through spin coating. These samples are placed in a UV–visible light exposure chamber using different conditions (heat and water) over the specific periods of time. The samples are analyzed by UV–visible absorption spectroscopy, X-ray photoelectron spectroscopy and Fourier transforms infrared spectroscopy (FTIR) measurements. The results indicate that after continuous exposure to UV irradiation for 72 and 120 h, the sample shows a significant decrease in absorption of the main peak. The sample shows around 25% loss in absorption (main peak) after 72 h of irradiation. The FTIR results illustrate a progressive decrease in intensities of all typical absorption peaks owing to P3HT ring scission, side chain oxidation as well as degradation of the side groups of PCBM. Acknowledgements The authors would like to acknowledge the support provided by King Abdulaziz City for Science and Technology (KACST) through the Science and Technology Unit at King Fahd University of Petroleum and Minerals (KFUPM) for funding this work through Project # ENE2379-04 as part of the National Science, Technology and Innovation Plan. KFUPM is also acknowledged for supporting this research. The authors would like to acknowledge the Center of Research Excellence for Renewable Energy at KFUPM. Scopus

In this paper, an optical correlator which adopts planar integrated 2F structure and optimal tradeoff synthetic discriminant function filter is proposed to improve the system integration and the recognition ability. The specialized digital microlens is designed for the correlator to replace the physical lens. By the theoretical analysis, the structural design conditions and parameters of the correlator are given, and the system volume is 63.1 cm3 which is only half of the planar integrated 4F optical correlator. Moreover, simulations and experimental research works reveal that the correlator can achieve stable recognition to the input object of distortion with scaling (52%, 150%), rotation (-50°, +42°), and existed scaling (72%, 130%) and rotation (-36°, +38°) simultaneously. Theoretical analysis and experimental results indicate that the planar integrated optical correlator with compact structure and high integration has good performance of distortion invariant recognition.

An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.

We propose and experimentally demonstrate a photonic technique for Fourier transformation of broadband optical spectrum and compression of time-bandwidth product (TBP) simultaneously. Conventionally, an optical spectrum could be Fourier transformed based on the so-called time-spectrum convolution technique with a linearly dispersive delay line. In this paper, a nonlinearly dispersive delay line is implemented by the convolution of an input optical spectrum in the spectral domain with a nonlinearly chirped temporal sinusoidal waveform in the time domain. By specially designing the chirp rate variation of the temporal waveform, an anamorphic Fourier transformation functionality is realized to compress the TBP of output waveform. The key feature of this paper is that it is the first time to our knowledge to carry out the Fourier transformation of an optical spectrum with TBP compression. In addition, since the dispersive delay line could be programmable with high resolution, this technique is adaptable for different optical spectra and realizable for a reconfigurable TBP compression ratio. Experimental results show that the TBP compression ratio is programmable from 1.6 to 3 by engineering the nonlinear dispersive delay line. After reconstructing the Fourier transform of optical spectrum from the output waveforms, the error rate of the recovered waveform is calculated quantitatively. Also, we discussed and proved the possibility to recover the input spectrum only with intensity information. This proposed method is promising to break the big data limitation of the conventional real-time optical spectrum Fourier transformation technique based on the TBP engineering for data compression.

This paper presents an analytical solution for static analysis of thick rectangular beams with different boundary conditions. Carrera’s Unified Formulation (CUF) is used in order to consider shear deformation theories of arbitrary order. The novelty of the present work is that a boundary discontinuous Fourier approach is used to consider clamped boundary conditions in the analytical solution, unlike Navier-type solutions which are restricted to simply supported beams. Governing equations are obtained by employing the principle of virtual work. The numerical accuracy of results is ascertained by studying the convergence of the solution and comparing the results to those of a 3D finite element solution. Beams subjected to bending due to a uniform pressure load and subjected to torsion due to opposite linear forces are considered. Overall, accurate results close to those of 3D finite element solutions are obtained, which can be used to validate finite element results or other approximate methods.

The spectroscopic measurement of the Cosmic Microwave Background at mm and sub-mm wavelengths received significant attention recently, aimed at measuring tiny spectral distortions of the Cosmic Microwave Background (CMB) relevant for cosmology. Several experiments, including OLIMPO (Masi et al. 2003), PRISM (Andre et al., 2014), MILLIMETRON (Smirnov and Baryshev, 2012), PIXIE (Kogut and Fixsen, 2011) are based on a Martin-Puplett Fourier-transform spectrometer. Its differential capabilities are the key to success in these difficult measurements. The polarizing beam splitter is the optical core of a Martin-Puplett interferometer. In this paper we analyze, analytically and experimentally, the systematic effects induced by a beam splitter orientation different from the canonical 45 ° . These effects are potenitally important for the delicate measurements of CMB spectral distortions. We find an analytical formula describing the effect, and verify experimentally, in the range 150–600 GHz, that our formula correctly describes the results (with a C.L. of 88 % ). We also demonstrate that the rotation of the beam splitter does not induce distortions in the measured spectra.

In this article, an efficient analytical method for elastically restrained Euler columns embedded in an elastic medium has been proposed to calculate buckling loads. The lateral deflection function under compression is represented by a Fourier sine series. Stokes’ transformation is employed to develop the legitimized stability equations. Explicit analytical expressions are derived, which can be used for any type of boundary conditions. The efficiency of present formulation is demonstrated by comparing the results to those obtained by imposing three well-known boundary conditions available in the literature. A very good agreement has been obtained. The present formulation permits to have more efficient coefficient matrix for calculating the buckling loads of Euler columns with any desired boundary conditions.

In the last decades, more and more approaches of economic issues have used mathematical tools, and among the most recent ones, spectral and wavelet methods are to be distinguished. If in the case of spectral analysis the approaches and results are sufficiently clear, while the use of wavelet decomposition, especially in the analysis of time series, is not fully valorized. The purpose of this paper is to emphasize how these methods are useful for time series analysis. After theoretical considerations on Fourier transforms versus wavelet transforms, we have presented the methodology we have used and the results obtained by using wavelets in the analysis of wage-price relation, based on some empirical data. The data we have used is concerning the Romanian economy - the inflation and the average nominal wage denominated in US dollars, between January 1991 and May 2016, highlighting that the relation between nominal salary and prices can be revealed more accurately by use of wavelets

The main objective of the Offshore Code Comparison Collaboration Continuation, with Correlation (OC5) project is validation of aero-hydro-servo-elastic simulation tools for offshore wind turbines (OWTs) through comparison of simulated results to the response data of physical systems. Phase III of the OC5 project validates OWT models against the measurements recorded on a Senvion 5M wind turbine supported by the OWEC Quattropod from the alpha ventus offshore wind farm. The following operating conditions of the wind turbine were chosen for the validation: (1) idling below the cut-in wind speed, (2) rotor-nacelle assembly (RNA) rotation maneuver below the cut-in wind speed, (3) power production below and above the rated wind speed, and (4) shutdown. A number of validation load cases were defined based on these operating conditions. The following measurements were used for validation: (1) strains and accelerations recorded on the support structure and (2) pitch, yaw, and azimuth angles, generator speed, and electrical power recorded from the RNA. Strains were not directly available from the majority of the OWT simulation tools; therefore, strains were calculated based on out-of-plane bending moments, axial forces, and cross-sectional properties of the structural members. The simulation results and measurements were compared in terms of time series, discrete Fourier transforms, power spectral densities, and probability density functions of strains and accelerometers. A good match was achieved between the measurements and models setup by OC5 Phase III participants. OWEC Tower is acknowledged for releasing the jacket substructure, transition piece, and foundation design data for Phase III. Senvion is acknowledged for sharing the definition of the Senvion 5M wind turbine and its tower. The RAVE consortium is acknowledged for releasing the measurement data from the alpha ventus wind farm. This work was authored (in part) by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. http://energy.gov/downloads/doe-public-accessplan Peer Reviewed Wojciech Popko / Fraunhofer IWES, Fraunhofer Institute for Wind Energy Systems / , Division Wind Turbine and System Technology, Am Luneort 100, Bremerhaven 27572 / , Germany / Amy Robertson / National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401 / Jason Jonkman / National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401 / Fabian Wendt / National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401 / Philipp Thomas / Fraunhofer IWES, Fraunhofer Institute for Wind Energy Systems / , Division Wind Turbine and System Technology, Am Luneort 100, Bremerhaven 27572 / , Germany / Kolja Müller / University of Stuttgart / , Allmandring 5b, Stuttgart Wind Energy, Stuttgart 70569 / , Germany / Matthias Kretschmer / University of Stuttgart / , Allmandring 5b, Stuttgart Wind Energy, Stuttgart 70569 / , Germany / Torbjørn Ruud Hagen / OWEC Tower AS, Sommerrogata 17, Oslo 0255 / , Norway / Christos Galinos / Technical University of Denmark / , Department of Wind Energy, Frederiksborgvej 399, Roskilde 4000 / , Denmark / Jean-Baptiste Le Dreff / Electricitéde France, Recherche et Développement, 13 Boulevard Gaspard Monge, Palaiseau 91120 / , France / Philippe Gilbert / IFP Energies Nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex / , France / Bertrand Auriac / Principia La Ciotat, 215 Voie Ariane, La Ciotat 13600 / , France / Sho Oh / Nippon Kaiji Kyokai (ClassNK), 4-7 Kioicho, Chiyoda-Ku, Tokyo 102-8567 / , Japan / Jacob Qvist / 4Subsea, Hagaløkkveien 26, 1383 Asker, Hvalstad / , Norway / Stian Høegh Sørum / Norwegian University of Science and Technology / , Department of Marine Technology, Trondheim 7491 / , Norway / Loup Suja-Thauvin / Simis AS, Leonardveien 3, Malm 7790 / , Norway / Hyunkyoung Shin / University of Ulsan / , School of Naval Architecture and Ocean Engineering, Ulsan / , South Korea / Climent Molins / Polytechnic University of Catalonia / , Campus Nord, Carrer de Jordi Girona, 1, 3, Barcelona 08034 / , Spain / Pau Trubat / Polytechnic University of Catalonia / , Campus Nord, Carrer de Jordi Girona, 1, 3, Barcelona 08034 / , Spain / Paul Bonnet / Siemens Industry Software, Luis, Carrer Lluís Muntadas, No. 5, Cornellà de Llobregat, Barcelona 08940 / , Spain / Roger Bergua / Envision Energy Limited, 8/F, Building B, SOHO Zhongshan Plaza, 1065 West Zhongshan Road, Shanghai 200051 / , China / Kai Wang / Envision Energy Limited, 8/F, Building B, SOHO Zhongshan Plaza, 1065 West Zhongshan Road, Shanghai 200051 / , China / Pengcheng Fu / China General Certification Center, Room 1108, Yiheng Building No. 28, North 3rd Ring Road East, Chaoyang District, Beijing 100013 / , China / Jifeng Cai / China General Certification Center, Room 1108, Yiheng Building No. 28, North 3rd Ring Road East, Chaoyang District, Beijing 100013 / , China / Zhisong Cai / China General Certification Center, Room 1108, Yiheng Building No. 28, North 3rd Ring Road East, Chaoyang District, Beijing 100013 / , China / Armando Alexandre / DNV GL, One Linear Park, Avon Street, Temple Quay, Bristol BS2 0PS / , UK / Robert Harries / DNV GL, Edificio Trovador, Plaza de Antonio Beltrán Martínez, Zaragoza 50002 / , Spain

International audience; Characterizing very small particles, from a few dozen micrometers to the nanometric scale, is a very challenging application in a wide range of domains. In this work, we demonstrate, through the recovery of silica and polystyrene bead properties (i.e. their size and refractive index) that Coherence Scanning Interferometry (CSI), in addition of being contactless, non-destructive, label-free and very well spatially resolved, is a very interesting and promising tool for such complex characterization. The CSI system is used as an imaging Fourier transform spectrometer meaning that the characterizations are achieved by analyzing the interference signal in the spectral domain. Some simulations of the proposed technique are presented and show that the accuracy of such characterization, in particular the measurement of the refractive index, are closely related to the signal to noise ratio. This observation is thereafter confirmed by the experimental results of beads buried within the depth of a transparent sample. Finally, the method is theoretically tested in the case of a scattering medium in which the quality of the signal is highly degraded. In this context, a geometrical approach enabling the simulation of an interference signal from a scattering layer is first proposed and then validated by means of comparison with experimental data.

The fault diagnosis of rotating machinery is generally performed using methods that employ vibration and sound. These methods are simple and accurate. However, all of these methods measure vibration data on the basis of the sampling theorem. Thus, they require a high measurement frequency, resulting in a large data volume and expensive measurement equipment. In recent years, a method that uses compressed sensing has been proposed to solve this problem, but it requires dedicated hardware to realize random sampling. To overcome this drawback, we developed a random start uniform sampling method (RSUSM) and combined it with compressed sensing (CS). RSUSM is a method of measuring data at a fixed frequency with a random start time. Numerical experiments demonstrate how the specific constant changes for each RSUSM parameter. This allows us to know the limit of how many measurement points are required for the number of non-zero components. We also applied CS by RSUSM to the sound pressure measurement results of the failed propeller, and found that the signal could be recovered less than 25% error even in a noisy real environment within the aforementioned limit. In this case, we found that the measurement frequency could be compressed to 1/80th of the frequency required by the sampling theorem, and the measurement data size to 1%. This approach is expected to diagnose faults in more rotating machines by significantly reducing the costs associated with data collection and storage.

The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order $n$ and $m$ are derived and expressed as a bounded, weighted sum of $n+m$ Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.

International audience; This paper discusses on off-axis digital holography applied to quantitative imaging of flows. The close link between phase measurement and measurands of interest is discussed. Numerical processing of digital holograms and phase retrieval is summarized. The paper presents different architectures that can be found in literature according to different operating modes: speckle-free mode, speckle mode, reduced-sensitivity, and combined approaches.

El propòsit d'aquesta tesi és el d'estudiar les propietats d'integrabilitat i convergència de sèries i transformades de Fourier. Els resultats principals són els següents: 1. Incestiguem les propietats d'integrabilitat de sèries trigonomètriques amb coeficients que satisfan una condició de monotonia general i demostrem resultats del tipus Hardy-Littlewood, és a dir, equivalències entre normes de les sumes de sèries trigonomètriques i normes amb pesos dels seus coeficients de Fourier. Demostrem aquestes equivalències en espais de Lorentz i espais de Lebesgue amb pesos. 2. Estudiem propietats de suavitat de funcions que poden ser representades per mitjà de sèries trigonomètriques amb coefficients que satisfan una condició de monotonia general. Es demostra una equivalència del Lp-mòdul de suavitat d'aquestes funcions i les sumes amb pesos dels seus coeficients de Fourier. 3. Obtenim versions multi-dimensionals de teoremes de tipus Boas en relació a les propietats d'integrabilitat de transformades de Fourier de funcions que són monòtones en totes les variable. 4. Finalment, estudiem criteris per a la convergència uniforme de sèries trigonomètriques amb coefficients que satisfan una condició de monotonia general. En particular, generalitzem el conegut criteri de Chaundy-Jolliffe per a la convergència uniforme de sèries sinusoidals i obtenim el resultat corresponent per sèries cosinusoidals. A més, provem condicions necessàries i suficients per tal que les sumes partials de Fourier d'aquestes sèries tinguin un cert ordre de convergència., This thesis is devoted to the study of integrability and convergence properties of Fourier series and transforms. The main results are the following. 1. We investigate the integrability properties of trigonometric series with general monotone coefficients and prove the Hardy-Littlewood-type results, i.e., equivalences of the norms of sums of trigonometric series and weighted norms of their Fourier coefficients. We prove such equivalences for the Lorentz and weighted Lebesgue spaces. Here we deal with the trigonometric series with general monotone coefficients. 2. We study the smoothness properties of functions that can be represented by trigonometric series with general monotone coefficients. The equivalence of the Lp-modulus of smoothness of such functions and weighted sums of their Fourier coefficients is proved. 3. We obtain the multidimensional versions of Boas-type theorem on integrability properties of the Fourier transforms of monotone in each variable functions. 4. Finally, we study criteria for the uniform convergence of trigonometric series with general monotone coefficients. In particular, we generalize the well-known Chaundy-Jolliffee criterion for the uniform convergence of sine series and obtain the corresponding result for cosine series. Moreover, we prove necessary and sufficient conditions for partial Fourier sums of such series to have certain convergence rate.

Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.