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Shah, Yogendra Prasad. "Applications of Fourier Series and Fourier Transformation." Cognition 2, no. 1 (2019): 145–56. http://dx.doi.org/10.3126/cognition.v2i1.55605.
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This paper investigates into the application of fourier transformation and series, which converts time domain signal to frequency domain signals, at which signals can be analyzed. Unlike Laplace transform, Fourier Transforms does not have full S plane, it just have the frequency j2πf plane. Fourier Transforms helps to analyze spectrum of the signals, helps in find the response of the LTI systems. (Continuous Time Fourier Transforms is for Analog signals and Discrete time Fourier Transforms is for discrete signals). Discrete Fourier Transforms are helpful in Digital signal processing for making convolution and many other signal manipulations. Overall, the paper will conclude the impact of Fourier Transforms in life.
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Fischer, Jens. "Four Particular Cases of the Fourier Transform." Mathematics 6, no. 12 (2018): 335. http://dx.doi.org/10.3390/math6120335.
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In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.
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Ponomareva, O. V., V. A. Alekseev, and A. V. Ponomarev. "A New Method to Construct Algorithms for Fast Discrete Fourier Transform of Finite Complex and Real Signals Based on Thesecond Type Parametric Discrete Fourier Transforms." Intellekt. Sist. Proizv. 22, no. 1 (2024): 78–84. http://dx.doi.org/10.22213/2410-9304-2024-1-78-84.
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The article develops a simple, efficient and effective method for fast discrete Fourier transform, which allows to calculate Fourier coefficients (bins) independently at positive and negative frequenciesfor finite complex and real signals.The algebraic and matrix forms of the discrete Fourier transform and the structure of its basis - the basis of exponential Fourier functions - are briefly considered.The main section of the article discusses generalizations of the discrete Fourier transform in the form of parametric discrete Fourier transforms.Two types of parametric discrete Fourier transforms have been studied, that have a parameter θ in a frequency variable or a parameterθ in a time variable. The structure and properties of the bases of these transformations being the bases of parametric discrete exponential functions were analyzed and investigated.Based on parametric discrete Fourier transforms of the second type, a new method for constructing algorithms for fast discrete Fourier transforms of complex and real signals has been developed and described in detail.In order to verify the obtained theoretical results, a step-by-step testing of a new method for constructing algorithms for fast discrete Fourier transform of finite complex and real signals was carried out.Testing of a new method for constructing algorithms for fast discrete Fourier transform of finite complex and real signals has fully confirmed the validity of the obtained results. For finite complex signals, the result obtained is (until the corresponding practical problem appears) of a theoretical nature.For finite real signals, the obtained result has theoretical and important practical significance.Since they have a redundant characterdue to the property of Hermitian symmetry of the finite real signalspectra.They can only be calculated at positive or negative frequencies.This allows to reduce the required amount of memory and the number of basic operationsfor finite real signals.
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Hanna, M. T., and S. A. Mansoori. "The discrete time wavelet transform: its discrete time Fourier transform and filter bank implementation." IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 48, no. 2 (2001): 180–83. http://dx.doi.org/10.1109/82.917787.
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Jani, A., and V. H. Makwana. "Modified discrete Fourier transform algorithm for protection of shunt compensated distribution line." Electrical Engineering & Electromechanics, no. 2 (March 5, 2023): 63–68. http://dx.doi.org/10.20998/2074-272x.2023.2.10.
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Introduction. The response time of the relay plays vital role when fault occurs on the line. Various algorithms are adopted to increase the sampling rate of the relay which, in turn, improves the response time. Methods. Discrete Fourier transform and modified discrete Fourier transform are the two algorithms used to calculate the fundamental frequency phasor of the signal required by the relay to initiate trip command. It is known that discrete Fourier transform takes four to five cycles to produce the fundamental frequency phasor but it fails to deal with the decaying DC component. On the other hand, modified discrete Fourier transform improves the response time by removing the decaying DC component along with the other harmonics in just one cycle and a few samples. The aim of this paper is to cover discrete Fourier transform and modified discrete Fourier transform algorithms to analyze the performance of the three overcurrent and one earth fault relaying scheme for different types of faults occurring in the distribution system. Methodology. The concept of three overcurrent and one earth fault scheme is also explained in this paper for protection of shunt-compensated distribution system. The scheme is designed for variable power factor. MATLAB/Simulink is used as the software tool to validate the results obtained for various types of faults occurring in the system. The results are represented graphically to illustrate the time of response of the protection scheme when shunt compensators are connected at the receiving end of distribution network.
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Ponomarev, Vladimir, Olga Ponomareva, Alexey Ponomarev, and Natalya Smirnova. "Sliding signal processing in telecommunication networks based on two-dimensional discrete Fourier transform." ITM Web of Conferences 30 (2019): 04013. http://dx.doi.org/10.1051/itmconf/20193004013.
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A method of vertical sliding processing of two-dimensional discrete signals in the spatial frequency domain is proposed — a method of fast vertically sliding two-dimensional discrete Fourier transform. The mathematical representation of the two-dimensional discrete Fourier transform in algebraic and matrix form is considered. An effective method of vertically sliding two-dimensional discrete Fourier transform is proposed. The algorithm developed in the framework of the proposed method allows calculating the coefficients (bins) of this transformation in real time.
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Singh, Pushpendra. "Novel Fourier quadrature transforms and analytic signal representations for nonlinear and non-stationary time-series analysis." Royal Society Open Science 5, no. 11 (2018): 181131. http://dx.doi.org/10.1098/rsos.181131.
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The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well known and widely used mathematical formulations for modelling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose 16 Fourier quadrature analytic signal (FQAS) representations with following properties: (1) real part of eight FQAS representations is the original signal, and imaginary part of each representation is FCQT of real part; (2) imaginary part of eight FQAS representations is the original signal, and real part of each representation is FSQT of imaginary part; (3) like the GAS, Fourier spectrum of all FQAS representations has only positive frequencies; however, unlike the GAS, real and imaginary parts of FQAS representations are not orthogonal. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set Fourier intrinsic band functions. This study also proposes new formulations of the FDM using discrete cosine transform with GAS and FQAS representations, and demonstrates its efficacy for improved time-frequency-energy representation and analysis of many real-life nonlinear and non-stationary signals.
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Wei, Shu Ang, Li Gao, Zhi Ye Sun, and Shi Jue Zheng. "A Mobile Data Minging Algorithm Based on Discrete Fourier Transform by Genetic Algorithm." Advanced Materials Research 108-111 (May 2010): 1452–57. http://dx.doi.org/10.4028/www.scientific.net/amr.108-111.1452.
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In the mobile environment, considering resource constraint, the frequent disconnect, synchronous data flow, the cost of communication, mobility and so on, combined with discrete Fourier transform (Discrete Fourier Transform, DFT) algorithm to facilitate time-domain and frequency domain conversion advantages as well as the genetic algorithm’s (Genetic Algorithm, GA) good global search capability ,this paper proposes a mobile data mining model which is based on the combination of Discrete Fourier Transform and Genetic Algorithm (DFTGA).
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Wen, Cong Xi, Shan Gao, Xiang Ning Xiao, Yong Hai Xu, and Shun Tao. "Forecast of Power Quality Index Based on the Discrete Fourier Decomposition and AR Model." Advanced Materials Research 732-733 (August 2013): 1420–26. http://dx.doi.org/10.4028/www.scientific.net/amr.732-733.1420.
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A method of autoregressive (AR) based on discrete Fourier analysis is proposed to forecast the indexes of unbalance factor and total harmonic distortion. The discrete Fourier transform of the index sequence is analyzed and the low frequency components are extracted. Autoregressive method is applied to forecast each low frequency component. Through inverse discrete Fourier transform, the forecasting low frequency components are inverted to forecasting sequence of the index in time domain. Actual data is used to test this method and the results show that discrete Fourier analysis is possible to reduce the influence of high frequency noise and that the AR method based on Fourier transform can effectively forecast the indexes of unbalance factor and total harmonic distortion.
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Fernández, Carmen, Antonio Galbis, and Josep Martínez. "Localization Operators and an Uncertainty Principle for the Discrete Short Time Fourier Transform." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/131459.
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Localization operators in the discrete setting are used to obtain information on a signalffrom the knowledge on the support of its short time Fourier transform. In particular, the extremal functions of the uncertainty principle for the discrete short time Fourier transform are characterized and their connection with functions that generate a time-frequency basis is studied.
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Gera, A. E. "The relationship between the Z-transform and the discrete-time Fourier transform." IEEE Transactions on Automatic Control 44, no. 2 (1999): 370–71. http://dx.doi.org/10.1109/9.746268.
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A., Jani, and H. Makwana V. "Modified discrete Fourier transform algorithm for protection of shunt compensated distribution line." Electrical Engineering & Electromechanics, no. 2 (March 5, 2023): 63–68. https://doi.org/10.20998/2074-272X.2023.2.10.
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<strong><em>Introduction. </em></strong><em>The response time of the relay plays vital role when fault occurs on the line. Various algorithms are adopted to increase the sampling rate of the relay which, in turn, improves the response time. <strong>Methods</strong>. Discrete Fourier transform and modified discrete Fourier transform are the two algorithms used to calculate the fundamental frequency phasor of the signal required by the relay to initiate trip command. It is known that discrete Fourier transform takes four to five cycles to produce the fundamental frequency phasor but it fails to deal with the decaying DC component. On the other hand, modified discrete Fourier transform improves the response time by removing the decaying DC component along with the other harmonics in just one cycle and a few samples. The <strong>aim</strong> of this paper is to cover discrete Fourier transform and modified discrete Fourier transform algorithms to analyze the performance of the three overcurrent and one earth fault relaying scheme for different types of faults occurring in the distribution system. <strong>Methodology</strong>. The concept of three overcurrent and one earth fault scheme is also explained in this paper for protection of shunt-compensated distribution system. The scheme is designed for variable power factor. MATLAB/Simulink is used as the software tool to validate the results obtained for various types of faults occurring in the system. The <strong>results</strong> are represented graphically to illustrate the time of response of the protection scheme when shunt compensators are connected at the receiving end of distribution network.</em>
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Somasekhar, B., Ch Mohana Krishna, and Y. Murty. "Investigations on wavelet and Fourier transform based channel estimation in MIMO-OFDM system." International Journal of Engineering & Technology 7, no. 2.21 (2018): 228. http://dx.doi.org/10.14419/ijet.v7i2.21.12178.
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In this paper channel estimation methods for MIMO-OFDM system are investigated based on Fourier Transform and Wavelet Transform. The channel estimation algorithm based on Discrete Fourier Transform (DFT) cause energy leakage in multipath channel with non-sample-spaced time delays. Discrete Cosine Transform (DCT) based channel estimator can mitigate the drawback of Discrete Fourier Transform based channel estimator, when the non-sample spaced path delays are available in multipath fading channels. Wavelet based systems provide better spectral efficiency because of no cyclic prefix requirement, with narrow side lobes and also exhibit improved BER performance. Simulation results reveal that the DWT based transform outperforms the conventional DFT and DCT based channel estimator in terms of bit error rate and mean square error.
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Collar, Steven, and Garth Frazier. "Acoustic array signal processing with the Stockwell transform." Journal of the Acoustical Society of America 151, no. 4 (2022): A36. http://dx.doi.org/10.1121/10.0010576.
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The Stockwell transform is a wavelet-like transformation based on a Fourier kernel weighted by a symmetric, frequency-scaled, time-shifted window function. Thus, it is suitable for analysis of non-stationary waveforms and transients in particular. In its discrete-time, orthogonal basis realization known as the discrete orthogonal Stockwell transform (DOST), it is possible to transform an N-point sequence in O(N log N) operations. In this presentation, we show how to perform direction-of-arrival and transient waveform estimation in the Stockwell basis in a manner that is similar to traditional multi-channel frequency-domain (discrete Fourier transform) techniques. This enhances detection of multiple transients within the same data frame sequence.
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Chen, Jiao, Tianhong Pan, Tianqing Pu, et al. "Predicting GHS toxicity using RTCA and discrete-time Fourier transform." Journal of Bioinformatics and Computational Biology 14, no. 01 (2016): 1650004. http://dx.doi.org/10.1142/s0219720016500049.
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In order to promote the acceptance of cell-based toxicity testings, the accuracy of cytotoxicity test must be determined when compared to in vivo results. Traditional methods of cytotoxicity analysis, such as LC[Formula: see text] (concentration where 50% of the cells are killed) can be problematic since they have been found to vary with time. Technological advances in cytotoxicity testing make it easy to record the dynamic data on changes in cell proliferation, morphology, and damage. To effectively and reasonably analyze the dynamic data, we present a new in vitro toxicity assessed method using the discrete-time Fourier transform (DTFT) which maps the measured cell index from the time domain to the frequency domain. The direct current (DC) component of the DTFT is extracted as a feature which reflects the intensity of cytotoxicity. The smaller the value, the higher the cytotoxicity. Then, a novel toxicity index, as expressed in terms of DC[Formula: see text], is calculated. Results generated with selected test chemicals are compared favorably with data obtained from The Interagency Coordinating Committee on the Validation of Alternative Method (ICCVAM) report concerning the prediction of acute systemic toxicity in rodents. The method can be applied with the standard and high throughput to estimate acute rodent oral toxicity which reduces the number of animals required in subsequent pharmacological/toxicological studies.
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Assous, Said, and Laurie Linnett. "High resolution time delay estimation using sliding discrete Fourier transform." Digital Signal Processing 22, no. 5 (2012): 820–27. http://dx.doi.org/10.1016/j.dsp.2012.05.001.
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Chakraborty, Avijit, and David Okaya. "Frequency‐time decomposition of seismic data using wavelet‐based methods." GEOPHYSICS 60, no. 6 (1995): 1906–16. http://dx.doi.org/10.1190/1.1443922.
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Spectral analysis is an important signal processing tool for seismic data. The transformation of a seismogram into the frequency domain is the basis for a significant number of processing algorithms and interpretive methods. However, for seismograms whose frequency content vary with time, a simple 1-D (Fourier) frequency transformation is not sufficient. Improved spectral decomposition in frequency‐time (FT) space is provided by the sliding window (short time) Fourier transform, although this method suffers from the time‐ frequency resolution limitation. Recently developed transforms based on the new mathematical field of wavelet analysis bypass this resolution limitation and offer superior spectral decomposition. The continuous wavelet transform with its scale‐translation plane is conceptually best understood when contrasted to a short time Fourier transform. The discrete wavelet transform and matching pursuit algorithm are alternative wavelet transforms that map a seismogram into FT space. Decomposition into FT space of synthetic and calibrated explosive‐source seismic data suggest that the matching pursuit algorithm provides excellent spectral localization, and reflections, direct and surface waves, and artifact energy are clearly identifiable. Wavelet‐based transformations offer new opportunities for improved processing algorithms and spectral interpretation methods.
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John Rathinaraj, Joshua David, and Gareth H. McKinley. "Gaborheometry: Applications of the discrete Gabor transform for time resolved oscillatory rheometry." Journal of Rheology 67, no. 2 (2023): 479–97. http://dx.doi.org/10.1122/8.0000549.
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Oscillatory rheometric techniques such as small amplitude oscillatory shear (SAOS) and, more recently, medium amplitude oscillatory shear and large amplitude oscillatory shear (LAOS) are widely used for rheological characterization of the viscoelastic properties of complex fluids. However, in a time-evolving or mutating material, the build-up or breakdown of microstructure is commonly both time- and shear-rate (or shear-stress) dependent, and thixotropic phenomena are observed in many complex fluids including drilling fluids, biopolymer gels, and many food products. Conventional applications of Fourier transforms for analyzing oscillatory data assume the signals are time-translation invariant, which constrains the mutation number of the material to be extremely small. This constraint makes it difficult to accurately study shear-induced microstructural changes in thixotropic and gelling materials, and it is becoming increasingly important to develop more advanced signal processing techniques capable of robustly extracting time-resolved frequency information from oscillatory data. In this work, we explore applications of the Gabor transform (a short-time Fourier transform combined with a Gaussian window), for providing optimal joint time-frequency resolution of a mutating material’s viscoelastic properties. First, we show using simple analytic models and measurements on a bentonite clay that the Gabor transform enables us to accurately measure rapid changes in both the storage and/or loss modulus with time as well as extract a characteristic thixotropic/aging time scale for the material. Second, using the Gabor transform we demonstrate the extraction of useful viscoelastic data from the initial transient response following the inception of oscillatory flow. Finally, we consider extension of the Gabor transform to nonlinear oscillatory deformations using an amplitude-modulated input strain signal, in order to track the evolution of the Fourier–Tschebyshev coefficients of thixotropic fluids at a specified deformation frequency. We refer to the resulting test protocol as Gaborheometry (Gabor-transformed oscillatory shear rheometry). This unconventional, but easily implemented, rheometric approach facilitates both SAOS and LAOS studies of time-evolving materials, reducing the number of required experiments and the data postprocessing time significantly.
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TCHIGIRINSKY, JU L., D. V. KRAINEV, T. D. NGUYEN, and N. V. CHIGIRINSKAYA. "USE OF STANDARD EQUIPMENT OF A LATHE TO ANALYZE THE VIBRATION BACKGROUND OF THE CUTTING PROCESS." IZVESTIA VOLGOGRAD STATE TECHNICAL UNIVERSITY, no. 1(284) (January 2024): 25–28. http://dx.doi.org/10.35211/1990-5297-2024-1-284-25-28.
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Predictive maintenance is a proactive maintenance strategy that uses condition monitoring tools to detect signs of damage, abnormalities, and equipment performance. The most commonly used method for rotating machines is vibration analysis. The most commonly used method for condition analysis of rotating parts is vibration analysis. Most signal analysis tools today use the fast Fourier transform (FFT), which is a special case of the generalized discrete Fourier transform and converts a vibration signal from its time domain representation to an equivalent frequency domain representation. This article focuses on the use of discrete Fourier transform for machining signal analysis.
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Wang, Yutian, Fanglin Chen, Songnian Fu, et al. "Nonlinear Fourier transform assisted high-order soliton characterization." New Journal of Physics 24, no. 3 (2022): 033039. http://dx.doi.org/10.1088/1367-2630/ac5a86.
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Abstract Nonlinear Fourier transform (NFT), based on the nonlinear Schrödinger equation, is implemented for the description of soliton propagation, and in particular focused on propagation of high-order solitons. In nonlinear frequency domain, a high-order soliton has multiple eigenvalues depending on the soliton amplitude and pulse-width. During the propagation along the standard single mode fiber (SSMF), their eigenvalues remain constant, while the corresponding discrete spectrum rotates along with the SSMF transmission. Consequently, we can distinguish the soliton order based on its eigenvalues. Meanwhile, the discrete spectrum rotation period is consistent with the temporal evolution period of the high-order solitons. The discrete spectrum contains nearly 99.99% energy of a soliton pulse. After inverse-NFT on discrete spectrum, soliton pulse can be reconstructed, illustrating that the eigenvalues can be used to characterize soliton pulse with good accuracy. This work shows that soliton characteristics can be well described in the nonlinear frequency domain. Moreover, as a significant supplement to the existing means of characterizing soliton pulses, NFT is expected to be another fundamental optical processing method besides an oscilloscope (measuring pulse time domain information) and a spectrometer (measuring pulse frequency domain information).
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Wang, Zhuoran. "Review on application of fractional Fourier transform in LinearFrequencyModulation signal and communication system." Theoretical and Natural Science 41, no. 1 (2024): 100–107. http://dx.doi.org/10.54254/2753-8818/41/20240504.
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Traditional Fourier transform often apply to analyze and process stationary signals, however, it is weak for time-varying non-stationary signals, and fractional Fourier transform (FRFT) can better solve such problems. The FRFT can be comprehended as the expressive methods on the fractional Fourier domain constituted by the spinning coordinate axis of the signal anticlockwise about the origin at arbitrarily Angle in the time-frequency plane. In this paper, the improved fractional Fourier transform is combined with other calculation methods to achieve high precision estimation of chirp signal parameters. And the communication system built on weighted fractional Fourier transform and discrete fractional Fourier transform is studied and simulated respectively, which verifies the feasibility and improves the anti-jamming and anti-interception ability of the communication system.
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Li, Kai, and Wei Nai. "Rapid Extraction of the Fundamental Components for Non-Ideal Three-Phase Grid Based on an Improved Sliding Discrete Fourier Transform." Electronics 11, no. 12 (2022): 1915. http://dx.doi.org/10.3390/electronics11121915.
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In order to make an effective extraction of the fundamental components for a non-ideal three-phase grid, an improved sliding discrete Fourier transform (ISDFT) has been proposed in this paper. Firstly, the non-ideal signal characteristics are studied in detail, which reveals that there are not only typical harmonic components, but also double frequency components, that exist in dq coordinates when the three-phase grid voltages are unbalanced. Then, the structure form of the conventional sliding discrete Fourier transform (SDFT) has been redesigned to form the ISDFT algorithm, in which a special offset link is introduced to reduce the extraction time while the effectiveness is guaranteed. The experimental results show that this proposed ISDFT is suitable for types of non-ideal signals extraction and can keep a nice dynamical and steady performance in cases of grid or load disturbance. For the average extraction time, ISDFT is saving about 44.56% more of the time than SDFT and about 65.32% more than discrete Fourier transform (DFT).
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Grindlay, J. "On an application of a generalization of the discrete Fourier transform to short time series." Canadian Journal of Physics 79, no. 5 (2001): 857–68. http://dx.doi.org/10.1139/p01-054.
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A generalization of the discrete Fourier transform (DFT) is discussed. This generalization or GDFT provides a smooth interpolation between the points of the DFT. The GDFT of a sinusoidal function in a finite time window is (a) described in detail and (b) shown to coincide (aside from a simple scaling constant) with the corresponding Fourier transform, provided that certain conditions are satisfied by the sinusoidal parameters. The sinusoidal GDFT is proposed as a tool to investigate, (independently of any Fourier transform connection) the sinusoidal nature of time series. The method is applied successfully to the case of a specific trajectory of the Hénon and Heiles model. PACS Nos.: 02.30, 05.45
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Naumović, M. B. "Interrelationships between the one-sided discrete-time Fourier transform and one-sided delta transform." Electrical Engineering (Archiv fur Elektrotechnik) 83, no. 3 (2001): 99–101. http://dx.doi.org/10.1007/s002020000063.
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Boronakhin, A. M., A. V. Bolshakova, D. M. Klionskiy, D. Yu Larionov, and R. V. Shalymov. "Techniques for Accelerometer Reading Processing on Railway Transport Using Wavelet Transform." Journal of the Russian Universities. Radioelectronics 27, no. 1 (2024): 6–16. http://dx.doi.org/10.32603/1993-8985-2024-27-1-6-16.
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Introduction. Safety issues of railway transport are inevitably connected with the condition of railway tracks and railway wheels. Various defects, such as irregularities of the railway track, may lead to emergencies and incidents. Therefore, it is important to measure and calculate short and impulse irregularities. A joint analysis of vibration acceleration signals is needed in order to study the types and size of railway track irregularities.Aim. Development of an algorithm for irregularity search based on accelerometer readings with the vertical measurement axis.Materials and methods. The research encompassed wavelet transformation and wavelet-based signal processing including discrete-time signal processing and continuous wavelet processing. In addition, time-frequency analysis based on Fourier transform and continuous wavelet scalogram was used. These methods provide for time-frequency localization for irregularity detection and measurement.Results. Algorithms for vibrational signal processing using the continuous and discrete-time wavelet transform are proposed. The results show that the discrete wavelet transform is effective for multiresolution and multiband analysis, and continuous wavelet transform and wavelet scalogram allows extraction of irregularities and determination of their parameters. The relative error for irregularity depth was improved by 18 %, and the absolute error for irregularity length determinations was reduced by 7 times.Conclusion. Application of the discrete Fourier transform and Fourier spectrogram provides for fine resolution in the frequency domain. However, separation of signal components in the time-frequency domain is impeded. The continuous-time wavelet transform ensures sufficient resolution in the low-frequency domain for component localization and visualization of irregularities.
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Offelli, C., and D. Petri. "Weighting effect on the discrete time Fourier transform of noisy signals." IEEE Transactions on Instrumentation and Measurement 40, no. 6 (1991): 972–81. http://dx.doi.org/10.1109/19.119777.
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Olkkonen, H. "Running discrete Fourier transform for time-frequency analysis of biomedical signals." Medical Engineering & Physics 17, no. 6 (1995): 455–58. http://dx.doi.org/10.1016/1350-4533(94)00001-p.
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Antoni, J. "Leakage-free identification of FRF's with the discrete time Fourier transform." Journal of Sound and Vibration 294, no. 4-5 (2006): 981–1003. http://dx.doi.org/10.1016/j.jsv.2005.12.037.
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Keller, D. M. "Periodic functions and the discrete Fourier transform: a time-domain view." IEEE Transactions on Education 34, no. 1 (1991): 36–38. http://dx.doi.org/10.1109/13.79877.
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BLOUGH, J. R. "DEVELOPMENT AND ANALYSIS OF TIME VARIANT DISCRETE FOURIER TRANSFORM ORDER TRACKING." Mechanical Systems and Signal Processing 17, no. 6 (2003): 1185–99. http://dx.doi.org/10.1006/mssp.2002.1500.
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Lalwani*, Natasha, Aishna Sharma, and Dr Mani Roja M. Edinburgh. "Biometric Identification using Human Ear." International Journal of Engineering and Advanced Technology 9, no. 1 (2019): 4893–98. http://dx.doi.org/10.35940/ijeat.a2027.109119.
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Biometrics refers to the metrics of the human characteristics which has gained much popularity in recent times. It is a form of identification and access control. Widely used forms of biometrics are facial recognition, finger print recognition, iris recognition, etc. but the drawback is that most of these features change over time. The human ear is a cogent source of data to classify biometrically since its attributes do not change substantially as time progresses. This paper explores the field of ear biometric wherein the database images are re-sized to 128 x 256 pixels and then converted to grayscale image. Various transforms viz. Discrete Cosine Transform, Discrete Fourier Transform, Discrete Wavelet Transform are then applied to extract the features. The coefficients of the test image are compared with the coefficients of the registered database image. On comparison, Euclidean distance classifier is used to recognize the test image from the database. The database used consists of 25 subjects with 6 images per person out of which the initial 4 images are used to train the model, and the remaining 2 for testing. The outputs of various transforms were compared and the best accuracy obtained is 86% using Discrete Wavelet Transform.
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Sharkova, S. B., and V. A. Faerman. "Wavelet transform-based cross-correlation in the time-delay estimation applications." Journal of Physics: Conference Series 2142, no. 1 (2021): 012019. http://dx.doi.org/10.1088/1742-6596/2142/1/012019.
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Abstract The article discusses the application of wavelet analysis for the time-frequency time-delay estimation. The proposed algorithm is wavelet transform-based cross-correlation time delay estimation that applies discrete time wavelet transform to filter the input signal prior to computation of cross-correlation function. The distinguishing feature of the algorithm that it uses the variation of continuous wavelet transform to process the discrete signals instead of dyadic wavelet transform that is normally applied to the case. Another feature that the implication of convolution theorem is used to compute coefficients of the wavelet transform. This makes possible to omit redundant discrete Fourier transforms and significantly reduce the computational complexity. The principal applicability of the proposed method is shown in the course of a computational experiments with artificial and real-world signal. So the method demonstrated expected selectivity for the signals localized in the different frequency bands. The application of the method to practical case of pipeline leak detection was also successful. However, the study concluded that this method provides no specific advantages in comparison with the conventional one. In the future, alternative applications in biological signal processing will be considered.
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Jabeen, Dur-e., M. Ghazanfar Monir, M. Rafiullah, Faiza Waqqas, and Habib Shaukat. "Implementation and analysis of symmetrical signals in the sequency domain." International Journal of ADVANCED AND APPLIED SCIENCES 10, no. 5 (2023): 195–202. http://dx.doi.org/10.21833/ijaas.2023.05.023.
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Every transform has unique attributes and traits that are crucial to reducing computing costs and offering simple solutions. Many different frequency domain transformations, for instance, have properties that can be used in a variety of signal processing applications and analyses. Some of the Complex Hadamard Transform's variants' sequencies can be compared to those of the Discrete Fourier Transforms. It is proven the characteristics of the Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform symmetry. These qualities are crucial for signal analysis and image processing. Due to duplicate spectra across the origin, it reduces computational complexity, makes an analytical analysis for symmetric signals simpler, and needs less storage. Its analysis shows that the Discrete Fourier Transform and this Complex Hadamard Transform version exhibit similar symmetry tendencies. By using elementary signals in the time domain to connect the positive and negative sequencies with their associated phasor conceptions, sequency domain spectra are used to highlight these properties. As a result of image representation, relative spectra are represented in related domains. Transform can be used to extract and analyze various aspects from a wide range of medical images.
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Ramadhyagita, Irsya Luthfiah, Annisa Annisa, Faishal Kamindra, and Farhan Muhammad Rizky. "Kajian Discrete Fourier Transform untuk Menganalisis Sinyal Arbitrer." Mitra Pilar: Jurnal Pendidikan, Inovasi, dan Terapan Teknologi 1, no. 1 (2022): 7–16. http://dx.doi.org/10.58797/pilar.0101.02.
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Abstract This study aims to examine the Discrete Fourier Transform using arbitrary signals. Signal processing is a mathematical operation performed on a signal to obtain the required information. In this case, a transformation occurs. The Fourier transform is a popular method often used to change the time form to the frequency form intended to facilitate analysis. The Fourier transform is divided into two parts: the Continuous Fourier and the Discrete Fourier. The program developed by the researcher, DFT, analyzes arbitrary signals created in two programs. The first program is a function of the DFT and the second is the main program used to generate DFT graphs. In this project, researchers will analyze arbitrary signals decomposed into sine waves. This program uses two main libraries, namely matplotlib, and NumPy. In this case, we have used the DFT method and plotted the results of the calculations performed. Now the researcher will analyze the process that the researcher did while using the DFT method and plot the results. Using DFT, arbitrary signals can be arranged into a series of sinusoids, each with a different frequency. The DFT method can arrange signals into a series of sinusoids with different frequencies. In the DFT amplitude spectrum, the signal frequency is shown as a vertical bar, with the height being the signal amplitude in the time domain. DFT can convert a sequence of equally spaced signals into information about the frequencies of all sine waves needed to sum the time domain signals. Abstrak Penelitian ini bertujuan untuk mengkaji Discrete Fourier Transform dengan menggunakan sinyal arbitrer. Pengolahan sinyal adalah suatu operasi matematik yang dilakukan terhadap suatu sinyal sehingga diperoleh suatu informasi yang dibutuhkan. Dalam hal ini terjadi suatu transformasi. Transformasi Fourier merupakan salah satu metode popular yang sering digunakan untuk mengubah bentuk waktu bentuk frekuensi yang ditujukan untuk mempermudah analisis. Transformasi Fourier dibagi menjadi 2 bagian yaitu Fourier Kontinu dan Fourier Diskret. Program yang dikembangkan oleh peneliti yaitu, DFT untuk menganalisis sinyal arbitrer yang dibuat dalam dua program. Program pertama adalah fungsi dari DFT dan program kedua adalah program utama yang digunakan untuk memunculkan grafik DFT. Dalam project ini peneliti akan menganalisis sinyal arbitrer yang sudah diuraikan menjadi gelombang sinus. Program ini menggunakan dua library utama, yaitu matplotlib dan numpy. Pada case kali ini, telah menggunakan metode DFT dan membuat hasil plot dari perhitungan yang dilakukan. Sekarang peneliti akan menganalisis proses yang peneliti lakukan selama menggunakan metode DFT dan membuat hasil plot. Sebelum membahas hasil perhitungan, mari kembali membahas metode DFT itu sendiri. Dengan menggunakan DFT, dapat disusun sinyal arbitrer menjadi serangkaian sinusoid dan masing-masing akan memiliki frekuensi yang berbeda.Metode DFT dapat menyusun sinyal menjadi serangkaian sinusoid yang memiliki frekuensi yang berbeda-beda. Dalam spektrum amplitude DFT, frekuensi sinyal ditampilkan sebagai batang vertikal dengan ketinggiannya adalah amplitude sinyal dalam domain waktu. DFT dapat mengubah urutan sinyal dengan jarak yang sama menjadi informasi tentang frekuensi semua gelombang sinus yang diperlukan untuk menjumlahkan sinyal domain waktu.
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Puryear, Charles I., Oleg N. Portniaguine, Carlos M. Cobos, and John P. Castagna. "Constrained least-squares spectral analysis: Application to seismic data." GEOPHYSICS 77, no. 5 (2012): V143—V167. http://dx.doi.org/10.1190/geo2011-0210.1.
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An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continuous wavelet transform, the method has greatly improved temporal resolution, particularly at low frequencies.
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Ponomareva, O. V., A. V. Ponomarev, and N. V. Smirnova. "Algorithms for Direct and Inverse Parametric Fast Fourier Transform." Informacionnye Tehnologii 28, no. 1 (2022): 9–19. http://dx.doi.org/10.17587/it.28.9-19.
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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.
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Priyadarshini, M. S., Mohit Bajaj, Shwetank Avikal, and Pradeep Vishnuram. "Conception of Voltage Interruption Signal using Continuous Wavelet, Discrete Wavelet, and Wavelet Packet Analysis." E3S Web of Conferences 564 (2024): 07001. http://dx.doi.org/10.1051/e3sconf/202456407001.
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This paper focuses on conception of voltage interruption signal in different domains. According to IEEE Standard 1159, Interruption is categorised under short-duration voltage variations in the description of classification of ‘Power quality Disturbances’. Power quality disturbances affect the quality of electric power supplied by utilities to power consumers. An interruption represents loss of voltage for a duration of time from 0.5 cycles to a duration less than 1 minute. This variation in supply voltage will affect the connected load. This acts as a significant challenge in terms of stability and reliability of power supply. Signal processing techniques of Fourier transform (FT), Short-Time Fourier transform (STFT), continuous wavelet transform (CWT), discrete wavelet transform (DWT) and Wavelet Packet Transform (WPT) are applied to the voltage signals in MATLAB. This approach results in analysis and obtaining characteristics of this disturbance of interruption. This work focuses on extracting information from voltage signal resulting in detection of power quality issues in different perspectives. Continuous wavelet transform provides a time- frequency representation of signals. Discrete wavelet transform is based on multi-resolution analysis resulting in breaking down a signal into different frequency bands. For finer resolution, wavelet packet transform is used. These different transforms result in better understanding of the disturbance present in the signal. MATLAB platform is used to implement FT, STFT, CWT, DWT, WPT on interruption disturbance. Based on the same approach, conception of different signals time domain signals having practical relevance and physical significance can be analyzed to extract information in different perspectives.
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Nikonov, Nikolay Sergeevich, Igor Ivanovich Borzenkov, and Igor Leonidovich Lebedinsky. "DEVELOPMENT OF A MEASUREMENT SYSTEM AND SOFTWARE PRODUCT TO COLLECT AND ANALYSE ELECTRICITY QUALITY PARAMETERS." Bulletin of the National Technical University "KhPI". Series: Energy: Reliability and Energy Efficiency, no. 1 (2) (July 2, 2021): 86–90. http://dx.doi.org/10.20998/2224-0349.2021.01.12.
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In the real operating conditions of electrical networks, there are modes of operation characterised by deviations of their parameters from their nominal values. Of particular importance for the operation of electrical equipment are variations in the amplitude and frequency of the supply voltage. The permissible and limit deviations of these parameters are regulated in accordance with national standards. To calculate the main power quality parameters it is not sufficient to know only the methods of their calculation. Additional algorithms are needed to determine fundamental harmonic voltages and frequencies. Such a method is the discrete Fourier transform. This algorithm is designed for signal analysis. However, this algorithm was not widely used in calculating Fourier coefficients in modern software packages. The reason is that it takes much time and computer resources to determine the Fourier coefficients which reduces the attractiveness of this approach. For this reason, it is advisable to use the fast Fourier transform algorithm. This algorithm uses the periodicity properties of the trigonometric function, which allows reducing the number of multiplication operations. The results of using the fast Fourier transform algorithm are similar to the discrete Fourier algorithm, but the number of operations required for calculation is several times less. At the same time, fast and discrete Fourier transform algorithms can give quite a significant error in determining the frequency estimate. This deviation is related to multiplicity of time between signal measurements and its period. If the period of the analogue signal is a multiple of the sampled signal measurement distance, an additional Quin method must be used to reduce the error in determining the frequency of the main signal. In this regard, the development of algorithms and software complex for automated measurement systems of electrical power quality indicators using digital data acquisition and processing devices in real time is an urgent task.
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Keefe, C. Dale, and Melvin B. Comisarow. "Exact Interpolation of Apodized, Magnitude-Mode Fourier Transform Spectra." Applied Spectroscopy 43, no. 4 (1989): 605–7. http://dx.doi.org/10.1366/0003702894202418.
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A procedure is developed for the exact interpolation of apodized, magnitude-mode Fourier transform (FT) spectra. The procedure gives the true center frequency, i.e., the location of the continuous peak, from just the largest three discrete intensities in the discrete magnitude spectrum. The procedure is applicable for the peaks in the apodized magnitude spectrum of a time signal of the form f( t) = cos( ωt) exp(– t/τ). There are no restrictions on the value of the damping ratio T/τ. The procedure is demonstrated for the sine-bell and Hanning windows and is generalizable to other windows which consist of a sum of constants and sine/cosine terms. This includes the majority of commonly used windows.
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Javid, Tariq, Muhammad Faris, and Pervez Akhtar. "Integrated representation for discrete Fourier and wavelet transforms using vector notation." Mehran University Research Journal of Engineering and Technology 41, no. 3 (2022): 175–84. http://dx.doi.org/10.22581/muet1982.2203.18.
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Many mathematical operations are implemented easily through transform domain operations. Multiple transform domain operations are used independently in large and complex applications. There is a need to develop integrated representations for multiple transform domain operations. This paper presents an integrated mathematical representation for the discrete Fourier transformation and the discrete wavelet transformation. The proposed combined representation utilizes the powerful vector notation. A mathematical operator, called the star operator, is formulated that merges coefficients from different transform domains. The star operator implements both convolution and correlation processes in a weighted fashion to compute the aggregated representation. The application of the proposed mathematical formulation is demonstrated successfully through merging transform domain representations of time-domain and image-domain representations. Heart sound signals and magnetic resonance images are used to describe transform-domain data merging applications. The significance of the proposed technique is demonstrated through merging time-domain and image-domain representations in a single- stage that may be implemented as the primary processing engine inside a typical digital image processing and analysis system.
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Revathi, T., T. M. Rajalaxmi, R. Sundara Rajan, and Wilhelm Passarella Freire. "Deep quaternion Fourier transform for salient object detection." Journal of Intelligent & Fuzzy Systems 40, no. 6 (2021): 11331–40. http://dx.doi.org/10.3233/jifs-202502.
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Salient object detection plays a vital role in image processing applications like image retrieval, security and surveillance in authentic-time. In recent times, advances in deep neural network gained more attention in the automatic learning system for various computer vision applications. In order to decrement the detection error for efficacious object detection, we proposed a detection classifier to detect the features of the object utilizing a deep neural network called convolutional neural network (CNN) and discrete quaternion Fourier transform (DQFT). Prior to CNN, the image is pre-processed by DQFT in order to handle all the three colors holistically to evade loss of image information, which in-turn increase the effective use of object detection. The features of the image are learned by training model of CNN, where the CNN process is done in the Fourier domain to quicken the method in productive computational time, and the image is converted to spatial domain before processing the fully connected layer. The proposed model is implemented in the HDA and INRIA benchmark datasets. The outcome shows that convolution in the quaternion Fourier domain expedite the process of evaluation with amended detection rate. The comparative study is done with CNN, discrete Fourier transforms CNN, RNN and masked RNN.
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Ashtari Jafari, Mohammad. "Comparative Application of Time-Frequency Methods on Strong Motion Signals." Advances in Civil Engineering 2021 (July 31, 2021): 1–14. http://dx.doi.org/10.1155/2021/9933078.
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Real-world physical signals are commonly nonstationary, and their frequency details change with time and do not remain constant. Fourier transform that uses infinite sine/cosine waves as basis functions represents frequency constituents of signals but does not show the variations of the signal frequency contents over time. Multiresolution demonstration of the time-frequency domain may be achieved by the techniques that can support adjustable resolution in time and frequency. Earthquake strong motion signals are nonstationary and indicate time-varying frequency content due to the scattering from the source to the site. In this paper, we applied short-time Fourier transform, S-transform, continuous wavelet transform, fast discrete wavelet transform, synchrosqueezing transform, synchroextracting transform, continuous wavelet synchrosqueezing, filter bank synchrosqueezing, empirical mode decomposition, and Fourier decomposition methods on the near-source strong motion signals from the 7 May 2020 Mosha-Iran earthquake to study and compare the frequency content of this event estimated by these methods. According to the results that are examined by Renyi entropy and relative error, synchroextracting performed better in terms of energy concentration, and the Fourier decomposition method revealed the lowest difference between the original and reconstructed records.
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Yi, Hua, Peichang Ouyang, Tao Yu, and Tao Zhang. "An algorithm for Morlet wavelet transform based on generalized discrete Fourier transform." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 05 (2019): 1950030. http://dx.doi.org/10.1142/s0219691319500309.
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Continuous wavelet transform (CWT) is a linear convolution of signal and wavelet function for a fixed scale. This paper studies the algorithm of CWT with Morlet wavelet as mother wavelet by using nonzero-padded linear convolution. The time domain filter, which is a non-causal filter, is the sample of wavelet function. By making generalized discrete Fourier transform (GDFT) and inverse transform for this filter, we can get a geometrically weighted periodic extension of the filter when evaluated outside its original support. From this extension of the time domain filter, we can get a causal filter. In this paper, GDFT-based algorithm for CWT, which has a more concise form than that of linear convolution proposed by Jorge Martinez, is constructed by using this causal filter. The analytic expression of the GDFT of this filter, which is essential for GDFT-based algorithm for CWT, is deduced in this paper. The numerical experiments show that the calculation results of GDFT-based algorithm are stable and reliable; the running speed of GDFT-based algorithm is faster than that of the other two algorithms studied in our previous work.
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Sakhare, Umesh. "Implementation of Generalized Discrete Fourier Transform for CDMA on Real Time System." International Journal of Computer Applications 90, no. 14 (2014): 7–11. http://dx.doi.org/10.5120/15786-4467.
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Tzu-Hsien Sang. "The Self-Duality of Discrete Short-Time Fourier Transform and Its Applications." IEEE Transactions on Signal Processing 58, no. 2 (2010): 604–12. http://dx.doi.org/10.1109/tsp.2009.2032038.
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Pham, Dinh Tuan, and Dominique Guégan. "Asymptotic normality of the discrete Fourier transform of long memory time series." Statistics & Probability Letters 21, no. 4 (1994): 299–309. http://dx.doi.org/10.1016/0167-7152(94)00023-9.
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Suwarno, Djoko Untoro. "A visual approximation of a Fourier transforms operation using GeoGebra." ITM Web of Conferences 61 (2024): 01013. http://dx.doi.org/10.1051/itmconf/20246101013.
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Many engineering fields use the Fourier Transform. The purpose of the Fourier Transform is to decompose periodic signals into fundamental frequencies and harmonic frequencies. The Fourier transform operation in ordinary textbooks is mathematically analytical, this often makes it difficult for students who study it. This paper describes the mathematical operations that occur in the Fourier transform. Discrete Fourier transform operations consist of conversion operations of continuous-time signals into data sequences, exponential complex number multiplication operations, and integration operations of number data sequences. Visualization of Fourier transform operations using the GeoGebra application. This research obtains visualization results of Fourier transform operations. The resulting visualization includes the effect of increasing signal frequency and the integral limit. The influence of the greater integral limit causes the peak of the frequency spectrum to become larger. The integral result of the pulse signal is in the form of a sinc function and corresponds to the illustration in the textbook. The limitation of the results of this research is that the Fourier transform visualization was carried out at low frequencies
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Xu, Zhaoyang. "Analysis of application of FFT in quantum field." Theoretical and Natural Science 10, no. 1 (2023): 289–95. http://dx.doi.org/10.54254/2753-8818/10/20230363.
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The algorithm concept of fast Fourier transform, Heisenberg uncertainty, and the combination of the two are expounded. Fast Fourier Transform (FFT) is not a new discrete Fourier Transform, but a fast algorithm for Discrete Fourier Transform (DFT). For a long time, the DFT was not really used because of the large amount of computation, even if it was used by computers, it was difficult to deal with problems in real time. It was not until when Cooley and Key first proposed a fast algorithm for DFT operations, and later, when fast algorithms by G. Sullivan and. Key appeared in succession, that fundamental changes took place. People began to realize some inherent laws of DFT operation, and thus quickly developed and improved a set of high-speed and effective operation methods, which is now commonly known as the fast Fourier transform FF11 algorithm. Specifically, the position of a particle and its momentum cannot be determined simultaneously in a quantum mechanical system. It focuses on the proof and application of the Heisenberg inequality by TTF. Due to the different ways and methods of practical application, the practical significance and practical value of the application are for readers' reference. Besides, the space is limited and my knowledge is limited. Finally, the development direction of FFT in quantum physics is summarized.
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Samiee, Kaveh, Peter Kovacs, and Moncef Gabbouj. "Epileptic Seizure Classification of EEG Time-Series Using Rational Discrete Short-Time Fourier Transform." IEEE Transactions on Biomedical Engineering 62, no. 2 (2015): 541–52. http://dx.doi.org/10.1109/tbme.2014.2360101.
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Ponomareva, O. V., V. V. Khvorenkov, and N. V. Ponomareva. "A New Effectivemethod to Determine the Discrete Finite Real Signalenvelopes Based on The Parametric Discrete Fourier Transform of the Second Type." Intellekt. Sist. Proizv. 22, no. 1 (2024): 85–92. http://dx.doi.org/10.22213/2410-9304-2024-1-85-92.
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The article shows that expanding the applicationscope of digital signal processing (DSP) systems, increasing the scale of tasks and problems solved by such systems, led to the need to develop a theory,improve DSP methods and algorithms, including those based on discrete finite Fourier and Hilbert transforms (DFT) and (DPG). DFT and DPG, due to their properties, the emergence of the fast Fourier transform (FFT) algorithm (Cooley J.W., Tukey J.W., 1965), have found the widest application in DSP systems. It is shown that DFTs, along with their advantages, also have fundamental disadvantages, thatreveal in the time and frequency domains in a number of negative effects, and the calculations of DFTs are accompanied by a number of difficulties. The paper briefly examines the fundamentals of the DSP theory in parametric Fourier bases. Parametric discrete Fourier transforms (DFT-P) are the two generalizations of the classical DFT. At the same time, introducing a parameter into the DFT-P allows you to “control” the properties of the unitary transformation within the frequency or time domain. The article considers two types of mathematically timeequivalent descriptions of discrete finite real (DFR) signals: in the form of a spectrum (the sum of discrete harmonic components) and in the form that uses the instantaneous parameters of the DFD signal: instantaneous amplitude, instantaneous phase and envelope. From the information descriptionviewpoint, instantaneous parameters provide a more complete representation and informationidentification about the properties and states of the objects, phenomena, processes and systems under study. DFT and DPG transformations play an important role in describing DPD signals. The article, for example, shows that the DPG is the only linear operator that allows you to determine the instantaneous parameters of the DPD signalunambiguously, subject to the fulfillment of completely understandable requirements. In this work, a new effective method for determining envelopes based on parametric Fourier transforms of the second type has been developed. The theoretical results obtained in the article are confirmed by mathematical modeling.