Fast Fourier Algorithm Research Articles

Filter-processing-assisted Fourier transform Raman spectroscopy for forensic discrimination lubricant machinery oils

The primary objective of this study was to investigate the potential value of filtering algorithms in enhancing spectral model performance. 290 mechanical lubricants oils samples from various applications underwent measurement using Fourier transform Raman spectroscopy (FT-Raman). Subsequently, three discrimination models were developed employing C5.0, Bayes discriminant analysis, and support vector machine algorithms. The emphasis was on evaluating the efficacy of four filter algorithms—Fast Fourier Transform (FFT), Discrete Wavelet Transform (DWT), Finite Impulse Response (FIR), and Hilbert Transform (HT)—for enhancing model performance. Among these, the FFT filter-SVM model, DWT filter-SVM model, and FIR (low-pass or band-stop filter)-SVM model emerged as optimal choices for identification. These models demonstrated outstanding performance, achieving 100 % accuracy across all 290 samples. Modelling using FT-Raman and chemometrics offered a simpler, more robust, and valuable approach for identifying mechanical lubrication oils across diverse applications, exploiting the potential of filtering algorithms in forensic spectroscopy.

Turbine bearing fault diagnosis based on embedded system

Abstract Failure of the main components of the transmission mechanism of large machinery and equipment usually uses sensors to collect real-time vibration or audio information, followed by data processing on the computer to analyze whether the equipment is operating normally. In this paper, a multi-core embedded fault diagnosis platform based on ARM and DSP is designed. It is small in size. It can independently set the signal types, ranges, and filter parameters of multiple acquisition channels, and synchronously set high-precision acquisition and storage of voltage signals output from various sensors. The DSP adopts the fast Fourier algorithm to analyze the multi-channel sampling data in real time and extracts the fault characteristics through the analysis of envelope demodulation. The operation results show that the system can meet the routine online fault diagnosis of mechanical equipment transmission mechanism, and the fault analysis and diagnosis can reach 95% accuracy within one minute, with high real-time accuracy.

New Algorithm for Real-Valued Fourier Transform

This paper presents a direct algorithm for fast real discrete Fourier transform (RDFT) computing, using the discrete Fourier transform (DFT) conjugate symmetric property to reduce redundancies. In RDFT, all the input and output signals were real, which differed from complex DFT. Therefore, the structure of the proposed algorithm showed only real-data operations. The developed algorithm showed the desired properties, such as in-place computation, regularity, simplicity, and arithmetic operations reduction. The RFFT performance was compared with other related transforms, such as the fast Hartley transform (FHT) for the computation in the radix-2 algorithm. It was found that FHT showed the best performance in terms of arithmetic complexity.

On the reconstruction of functions from values at subsampled quadrature points

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L 2 L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d d -dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process

The paper builds a Variance-Gamma (VG) model with five parameters: location (μ), symmetry (δ), volatility (σ), shape (α), and scale (θ); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a Γ(α,θ) Ornstein–Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order ν=0, intensity α, and steepness parameters δσ2−δ2σ4+2θσ2 and δσ2+δ2σ4+2θσ2; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean (μ+αθδ) and variance α(θ2δ2+σ2θ). The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton–Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black–Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options.

Ultra-short pulse focusing algorithm for optical parametric chirp pulse amplification numerical simulation platform

The development of optical parametric chirp pulse amplification (OPCPA) numerical simulation platform involves physical models such as broadening and compression of optical pulse, parametric amplification and focusing output. In the simulation platform, the Fresnel far-field diffraction equation is usually used to simulate the characteristics of ultrashort pulse focusing. Firstly, we need to calculate the optical field distribution of different wavelength components in the ultrashort pulse, and then use the inverse Fourier transform to obtain the temporal and spatial distribution characteristics of the pulse. However, for different wavelength components, the sizes of focused field grids obtained by the far-field algorithm are not equal, and subsequent resampling is required, which will increase the amount of calculation. In addition, due to the limitation of the calculation range of the light field in the pulse broadening and compression, there is also a problem of poor resolution of the focused field. In this work, the mathematical expression that can adjust the range of the output light field and use the fast fourier algorithm is derived. The main mechanism of this algorithm is as follows. Based on the Fresnel far-field diffraction equation, the output field is sampled independently in the discrete calculation process to meet the requirements for adjustable range of the output field. After identity transformation, the output field results can be calculated by the fast Fourier algorithm. Furthermore, the sampling conditions that need to be satisfied when using the algorithm are further analyzed and discussed. It solves the problem of how to improve the resolution of light field and keep the computational grid size of each wavelength component consistent when the traditional Fresnel far field diffraction is used to simulate the focusing process, which provides the convenience for the subsequent direct time-frequency inverse transformation. The numerical simulation results reveal that the dark ring region of the ultrashort pulse focusing field shows strong spatiotemporal coupling characteristics. This algorithm has been successfully applied to the development of OPCPA numerical simulation platform, and is expected to play an important role in optimizing the design of ultrashort laser pulse device.

Algorithm for Calculating Primary Spectral Density Estimates Using FFT and Analysis of its Accuracy

Introduction. Fast algorithms for solving problems of spectral and correlation analysis of random processes began to appear mainly after 1965, when the algorithm of fast Fourier transform (FFT) entered computational practice. With its appearance, a number of computational algorithms for the accelerated solution of some problems of digital signal processing were developed, speed-efficient algorithms for calculating such estimates of probabilistic characteristics of control objects as estimates of convolutions, correlation functions, spectral densities of stationary and some types of non-stationary random processes were built. The purpose of the article is to study a speed-efficient algorithm for calculating the primary estimate of the spectral density of stationary ergodic random processes with zero mean. Most often, the direct Fourier transform method using the FFT algorithm, is used to calculate it. The article continues the research and substantiation of this method in the direction of obtaining better estimates of rounding errors. Results. The research and substantiation of the method in the direction of obtaining more qualitative estimates of rounding errors, taking into account the errors of the input information specification, has been continued. The main characteristics of the given algorithm for calculating the primary estimate of the spectral density are accuracy and computational complexity. The main attention is paid to obtaining error estimates accompanying the process of calculating the primary estimate of the spectral density. The estimates of the rounding error and ineradicable error of the given algorithm for calculating the primary estimate of the spectral density, which appear during the implementation of the algorithm for the classical rounding rule for calculation in floating-point mode with τ digits in the mantissa of the number, taking into account the input error, are obtained. Conclusions. The obtained results make it possible to diagnose the quality of the solution to the problem of calculating the primary estimate of the spectral density of stationary ergodic random processes with a zero mean value by the described method and to choose the parameters of the algorithm that will ensure the required accuracy of the approximate solution of the problem. Keywords: primary estimation of spectral density, fast Fourier transform, discrete Fourier transform, rounding error, input error.

Algorithms for Direct and Inverse Parametric Fast Fourier Transform

Русский Main page New issue Archive of articles Editorial board For the authors Publishing house ABSTRACTS OF ARTICLES OF THE JOURNAL "INFORMATION TECHNOLOGIES". No. 1. Vol. 28. 2022 DOI: 10.17587/it.28.9-19 O. V. Ponomareva, Dr. Sc., Tech., Professor, A. V. Ponomarev, PhD, Econ., Associate Professor, Kalashnikov Izhevsk State Technical University, Izhevsk, 426069, Russian Federation, N. V. Smirnova, PhD, Tech., Associate Professor, Sevastopol State University, Sevastopol, 299053, Russian Federation Algorithms for Direct and Inverse Parametric Fast Fourier Transform Classical Fourier processing of finite information discrete signals (FID signals) is the most important method of digital analysis, modeling, optimization, improvement of control and decision making. The theoretical basis of classical Fourier processing of FID signals is the discrete Fourier transform (DFT). The practical basis of classical Fourier processing of FID signals is the Fast Fourier Transform (FFT). The practice of using classical Fourier processing of FID signals, having confirmed its effectiveness, revealed a number of negative effects inherent in this type of digital signal processing (DSP). The aliasing effect, scalloping effect, picket fence effect, significantly affect the effectiveness of analysis, modeling, optimization, improvement of management and decision making. To increase the efficiency of Fourier processing of FID signals, the authors of the paper have developed a generalization of DFT in the form of a parametric DFT (DFT-P). Since the direct application of parametric Fourier processing of FID signals (as well as the use of classical Fourier processing of FID signals) requires complex multiplications, fast procedures are required for the practical implementation of this type of FID signals. Purpose of the research is to develop algorithms for the fast parametric discrete Fourier transform (FFT-P). The work developed fast procedures for the implementation of DFT-P by time decimation. Parametric FFT-P with substitution (in place) and without substitution (no place) are proposed. The estimation of the efficiency of the FFT-P algorithms is given. The practical significance of the work is in the fact that developing algorithms for the parametric fast Fourier transform can reduce the computational costs of performing parametric discrete transformations by three or more orders of magnitude.

Multiresolution Preconditioners for Solving Realistic Multi-Scale Complex Problems

In this work, the hierarchic multiresolution (MR) preconditioner is combined with the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) and efficiently parallelized in multicore computers for computing electromagnetic scattering and radiation from complex problems exhibiting deep multi-scale features. The problem is formulated using the thin-dielectric-sheet (TDS) approximation for thin dielectric materials and the electric and combined field integral equations (EFIE/CFIE) for conducting objects. The parallel MLFMA-FFT is tailored to accommodate the MR hierarchical functions, which provide vast improvement of the matrix system conditioning by accurately handling multi-scale mesh features in different levels of detail. The higher (coarser) level hierarchical functions are treated by an algebraic incomplete LU decomposition preconditioner, which has been efficiently embedded into the parallel framework to further accelerate the solution. Numerical examples are presented to demonstrate the precision and efficiency of the proposed approach for the solution of realistic multi-scale scattering and radiation problems.

Fast Construction of Dual-Field Superfocusing Based on Forward-Propagation Design

A three-port plasmonic array is proposed to achieve a dual-field superfocusing using the forward-propagation method based on spatial Fourier transform. The algorithm of fast Fourier transform is used to facilitate the design process. Both the electric and magnetic fields can be superfocused on the focal plane of z = 0.1λ (λ is the free-space wavelength) based on an effect of radiationless electromagnetic interference. The full widths of the half-maximum of both two fields are 0.060λ and 0.072λ, respectively. Besides, the metrics of the dual-field superfocusing are 1.67 and 1.39. The proposed theory can simplify the design process of superfocusing, and the plasmonic array with brief three ports can sufficiently facilitate the construction of the desired source. This letter can be beneficial for the development of beam steering, sophisticated holography, super-resolution imaging, and other applications of near- or far-field devices.

A new multilevel compensation system for neutral active grounding in distribution grid

In order to achieve effect of full compensation and zero residual-current, a multilevel compensation system for neutral point active grounding is proposed. The system is composed of three-level compensators. When power grid is grounded, the first level compensates its capacitive current, the second level compensates its active current, and the third level compensates not only its harmonic current, but also the harmonics generated by the former two level compensators, so effect of full compensation is very obvious. When distribution grid is in normal operation, the system uses the asymmetric voltage method to realize accurate detection of capacitive current and active current. In case of single-phase grounded fault, the harmonic current is extracted by fast Fourier algorithm. Then three-level compensators are quickly put into operation for corresponding compensation. A low-voltage simulated power grid is built in the laboratory, and a compensation test is carried out. The test results show that the compensation of capacitive current and harmonic current in grounding fault current is accurate. After further improvement, the system can reach the practical level.

A mine digital integrated protector with harmonic detection and analysis function

Aiming at the protection problems and harmonic detection problems of motor circuit, lighting signal circuit in coal mine, a mine digital comprehensive protector based on STM32 is designed. The protector can realize the protection and harmonic detection of motor circuit, lighting circuit and signal circuit in coal mine, and use windowed interpolation fast Fourier algorithm to calculate and analyze harmonic. The protector has the functions of under voltage, over-voltage, overload, short circuit, three-phase unbalance, open phase, leakage locking, insulation danger indication, over temperature, under temperature, wind power and gas locking, harmonic analysis, local storage, remote control and other functions. The experimental results show that the system design scheme is feasible and effective.

Oversampled filter bank channelizer for cryogenic detectors.

We present the field-programmable gate array (FPGA) implementation of an oversampled generalized discrete Fourier transform (GDFT) filter bank (FB), optimized for its usage in the readout of cryogenic detectors, such as microwave kinetic inductance detectors. A polyphase structure is derived underlining the efficiency of complex-valued subband processing, raised by a GDFT modulation. For the latter, a fast Fourier algorithm realization may be used, yielding a highly efficient polyphase implementation for arbitrary integer decimation ratios. The FB is tested through a 16-channel GDFT channelizer. The implementation is based on the Xilinx Zynq family of FPGAs and aims to show the data reuse and flexibility offered by the GDFT channelizer structure. General design criteria are summarized for the 16-channel polyphase FB channelizer. The performance in hardware resource usage is also presented, taking into account one of the main objectives of the current physics experiments, aiming to have an extremely large readout format.

Algorithm for B-scan Image Reconstruction in Optical Coherence Tomography

Optical coherence tomography (OCT) is an evolving medical imaging technology that offers in vivo cross-sectional, sub-surface images in real-time. OCT has become popular in the medical as well as non-medical fields. The technique extensively uses for food industry, dentistry, dermatology, and ophthalmology. The technique is non-invasive and works on the Michelson interferometry principle, i.e., dependent on back reflections of the signal and its interference. The objective is to develop an algorithm for signal processing to construct an OCT image and then to enhance the quality of the image using image processing techniques like filtering. The image construction was primarily based on the Fourier transform (FT) of the dataset obtained by data acquisition. This FT could be performed rapidly with the extensively used algorithm of fast Fourier transform (FFT). The depth-wise information could be extracted from each A-scan, i.e., axial scan and also the B-scan was obtained from the A-scan to see the structure of sample. The maximum penetration depth achieved with proposed system was 2.82mm for 1024 data points. First and second layer of leaf were getting at thickness of 1mm and 1.6mm, respectively. A-scans for Human fingertip gave its first, second and third layer was at a thickness of 0.75mm, 0.9mm and 1.6mm, respectively. A-scans for foam sheet gave its first, second and third layer was at a thickness of 0.6mm, 0.75mm, and 0.85mm, respectively.

Simulation of subway tunnel construction process based on fi-nite element analysis

Simulation of subway tunnel construction process based on fi-nite element analysis

Optical image encryption using different twiddle factors in the butterfly algorithm of fast Fourier transform

We studied the flow of data in the paths of a fast Fourier transform (FFT) and inverse FFT in the butterfly algorithm in order to establish a relationship between the forward and backward data. We show that an FFT image can be encrypted by modifying the data of a path of FFT and decrypted by restoring the data during the inverse FFT. For encryption, the data in a path of FFT are shuffled by enforcing right bit shifts on the bit representation of the data index. Moreover, the phases of the data are changed by multiples of a fixed fraction of 2π; the data index is used as the multiplication constant, and the divisor for the fraction is used as an encryption key. The path number and number of right bit shifts are two other encryption keys. The three integers constitute a set of encryption keys for two-dimensional images. Four different images were encrypted with two sets of encryption keys. The decryptions were performed with partially correct keys in order to test the robustness of our method against brute attacks. The correlation between the original and decrypted images was evaluated using the correlation coefficient.

Three Dimensional Measurement System for the Dynamic Deformation of Aero-engine Blade Profile

In order to solve the technical problem of three-dimensional profile measurement of aero-engine blades with high speed rotation, an optical dynamic measurement system for aero-engine blades was developed. Firstly, the system is calibrated by the algorithm of spatial truncation phase calibration to establish the index between truncation phase and spatial coordinates. During the measurement, the deformation map of the rotating measured blade is obtained by synchronous projection and snapshot. By using the fast Fourier algorithm, the truncation phase is obtained, and then the profile information of blade can be obtained through the truncation phase and spatial coordinate index. Through the design and construction of the blade simulation platform and dynamic measurement experiment system, the three-dimensional profile data of the blade at different rotating speeds are obtained, compared and analyzed, and then the overall profile deformation law is discovered, which verifies the effectiveness and feasibility of the algorithm. The system can obtain the dynamic profile information of the whole blade completely, and provide innovative technical means for blade design verification and performance analysis.

Computing Maximal Lyndon Substrings of a String

There are two reasons to have an efficient algorithm for identifying all right-maximal Lyndon substrings of a string: firstly, Bannai et al. introduced in 2015 a linear algorithm to compute all runs of a string that relies on knowing all right-maximal Lyndon substrings of the input string, and secondly, Franek et al. showed in 2017 a linear equivalence of sorting suffixes and sorting right-maximal Lyndon substrings of a string, inspired by a novel suffix sorting algorithm of Baier. In 2016, Franek et al. presented a brief overview of algorithms for computing the Lyndon array that encodes the knowledge of right-maximal Lyndon substrings of the input string. Among those presented were two well-known algorithms for computing the Lyndon array: a quadratic in-place algorithm based on the iterated Duval algorithm for Lyndon factorization and a linear algorithmic scheme based on linear suffix sorting, computing the inverse suffix array, and applying to it the next smaller value algorithm. Duval’s algorithm works for strings over any ordered alphabet, while for linear suffix sorting, a constant or an integer alphabet is required. The authors at that time were not aware of Baier’s algorithm. In 2017, our research group proposed a novel algorithm for the Lyndon array. Though the proposed algorithm is linear in the average case and has O(nlog(n)) worst-case complexity, it is interesting as it emulates the fast Fourier algorithm’s recursive approach and introduces τ-reduction, which might be of independent interest. In 2018, we presented a linear algorithm to compute the Lyndon array of a string inspired by Phase I of Baier’s algorithm for suffix sorting. This paper presents the theoretical analysis of these two algorithms and provides empirical comparisons of both of their C++ implementations with respect to the iterated Duval algorithm.

Aliasing error of the exp⁡(β1−z2) kernel in the nonuniform fast Fourier transform

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel ϕ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new “exponential of semicircle” kernel ϕ ( z ) = e β 1 − z 2 , for z ∈ [ − 1 , 1 ] , zero otherwise, whose Fourier transform ϕ ˆ is unknown analytically. We place this kernel on a rigorous footing by proving an aliasing error estimate which bounds the error of the one-dimensional NUFFT of types 1 and 2 in exact arithmetic. Asymptotically in the kernel width measured in upsampled grid points, the error is shown to decrease with an exponential rate arbitrarily close to that of the popular Kaiser–Bessel kernel. This requires controlling a conditionally-convergent sum over the tails of ϕ ˆ , using steepest descent, other classical estimates on contour integrals, and a phased sinc sum. We also draw new connections between the above kernel, Kaiser–Bessel, and prolate spheroidal wavefunctions of order zero, which all appear to share an optimal exponential convergence rate.

Application of Nonuniform Fourier Transform to Solving Ultrasonic Tomography Problems with Antenna Arrays

In ultrasonic tomography that employs the technique of digital focusing with an antenna (DFA), the problem of increasing the inspection performance is inextricably linked with the problem of increasing the speed of producing synthesized images. This problem can be effectively solved by concurrently using several approaches aimed at increasing the image production speed. In the framework of this paper, we consider the algorithm of nonuniform fast Fourier transform (NUFFT). The NUFFT can be used in the existing computationally efficient spatiotemporal processing algorithms with calculations in the frequency domain and is capable of correctly reconstructing synthesized images with sparse and non-uniform antenna arrays (AAs). The experimental data obtained indicate the ability of the NUFFT-based algorithm to restore synthesized images with high resolution.