Application Of Fast Fourier Transform Research Articles

Hierarchical FFT-LSTM-GCN based model for nuclear power plant fault diagnosis considering spatio-temporal features fusion

As safety-critical infrastructure, nuclear power plants (NPPs) require enhanced safety measures and risk minimization. To achieve this goal and to aid operator decision-making while reducing human error, various fault diagnosis (FD) methods have been proposed. Among these, deep learning-based approaches have demonstrated significant success in FD because of their ability to effectively extract information about machine degradation. However, most existing methods primarily focus on temporal features while neglecting spatial features. To leverage both temporal and spatial features effectively and achieve high diagnostic accuracy, we propose a hierarchical deep learning based model that comprises the fast Fourier transform (FFT), long short-term memory networks (LSTM) and graph convolutional networks (GCN). The application of FFT to sensor sequences effectively mitigates their volatility. The use of GCN enables automated extraction of intricate spatial features from multi-sensor data, while LSTM is adept at directly extracting temporal features from historical input data. To validate the proposed model, we conducted three experiments using data simulated by the personal computer transient analyzer (PCTRAN), and the results demonstrate that the diagnostic accuracy of the proposed hierarchical FFT-LSTM-GCN model surpasses that of any single model for NPP FD.

Application of Fast Fourier Transform to the Speech Signals Scrambling

The article is devoted to the closure of speech signals by implementing a fast Fourier transform in a bandpass scrambler. The most radical measure to prevent the interception of speech signals is the use of cryptographic security methods. The first technical security systems began to be developed immediately after the invention of the telephone. The main goal when developing speech transmission systems is to preserve those characteristics that are most important for the listener to perceive. Communication security in the transmission of voice messages is based on the use of a large number of different methods of blocking signals that change the characteristics of speech in such a way that it becomes illegible and unrecognizable to attackers who listen to or intercept the closed voice message. The choice of closure methods depends on the specific application and the technical characteristics of the transmission channel. That is, technical channels of information leakage are a source of information for technical intelligence, which extracts information using technical means. Therefore, the problem of protecting information from technical intelligence is of particular relevance. The object of the study is the transmission of voice messages. The subject of the research is real-time speech closing systems. The purpose of the paper is to use fast Fourier transform to close speech signals in a bandpass scrambler. To do this, it is necessary to assess the capabilities of technical intelligence to intercept information of limited distribution; analyze types of speech signal closing systems.
 The paper presents a block diagram of a combined scrambler, an algorithm for rearranging input signal values using the fast Fourier transform, and hashing functions that are used to obtain a speech signal from a combined scrambler. It has been determined that "through-window" scramblers are the most reliable, and for greater levels of closure, combination scramblers should be used. The task set for further development of the topic is the development of a software application for the operation of streaming data encryption methods in real time.

Development of a facile method to compute collagen network pathological anisotropy using AFM imaging

Type I collagen, a fundamental extracellular matrix (ECM) component, is pivotal in maintaining tissue integrity and strength. It is also the most prevalent fibrous biopolymer within the ECM, ubiquitous in mammalian organisms. This structural protein provides essential mechanical stability and resilience to various tissues, including tendons, ligaments, skin, bone, and dentin. Collagen has been structurally investigated for several decades, and variation to its ultrastructure by histology has been associated with several pathological conditions. The current study addresses a critical challenge in the field of collagen research by providing a novel method for studying collagen fibril morphology at the nanoscale. It offers a computational approach to quantifying collagen properties, enabling a deeper understanding of how collagen type I can be affected by pathological conditions. The application of Fast Fourier Transform (FFT) coupled with Atomic Force Microscope (AFM) imaging distinguishes not only healthy and diseased skin but also holds potential for automated diagnosis of connective tissue disorders (CTDs), contributing to both clinical diagnostics and fundamental research in this area. Here we studied the changes in the structural parameters of collagen fibrils in Ehlers Danlos Syndrome (EDS). We have used skin extracted from genetically mutant mice that exhibit EDS phenotype as our model system (Col1a1Jrt/+ mice). The collagen fibrils were analyzed by AFM based descriptive-structural parameters, coupled with a 2D Fast Fourier Transform(2D-FFT) approach that automated the analysis of AFM images. In addition, each sample was characterized based on its FFT and power spectral density. Our qualitative data showed morphological differences in collagen fibril clarity (clearness of the collagen fibril edge with their neighbouring fibri), D-banding, orientation, and linearity. We have also demonstrated that FFT could be a new tool for distinguishing healthy from tissues with CTDs by measuring the disorganization of fibrils in the matrix. We have also employed FFT to reveal the orientations of the collagen fibrils, providing clinically relevant phenotypic information on their organization and anisotropy. The result of this study can be used to develop a new automated tool for better diagnosis of CTDs.

Analysis of application of FFT in quantum field

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.

Predicting respiration rate in unrestrained dairy cows using image analysis and Fast Fourier Transform

The respiratory rate (RR) is commonly employed for identifying animals experiencing heat-stress conditions and respiratory diseases. Recent advancements in computer vision algorithms have enabled the estimation of the RR in dairy cows through image-based approaches, with a primary focus on standing positions, thermal imaging, and deep learning techniques. In this study, our objective was to develop a system capable of accurately predicting the RR of lying Holstein cows under unrestrained conditions using RGB and infrared night vision (IR) images. Thirty lactating cows were continuously recorded for 12 h per day over a 3-d period, capturing at least one 30-s video segment of each cow during lying time. A total of 95 videos were manually annotated with rectangular bounding boxes encompassing the flank area (region of interest; ROI) of the lying cows. For future applications, we trained a model for ROI identification using YOLOv8 to avoid manual annotations. The observed RR was determined by visual counting of breaths in each video. To predict the RR, we devised an image processing pipeline involving: (1) capturing the ROI for the entire video, (2) reshaping the pixel intensity of each image channel into a 2D object and calculating its per-frame mean, (3) applying Fast Fourier Transform (FFT) to the average pixel intensity vector, (4) filtering frequencies specifically associated with respiratory movements, and (5) executing inverse FFT on the denoised data and identifying peaks on the resulting plot, with the count of peaks serving as the predicted RR per minute. The evaluation metrics, Root Mean Squared Error of Prediction (RMSEP), and R2 yielded values of 8.3 breaths/min (17.1% of the mean RR) and 0.77, respectively. To further validate the method, an additional data set comprising pre-weaned dairy calves was utilized, consisting of 42 observations from 25 calves. The RMSEP and R2 values for this data set were 13.0 breaths/min and 0.73, respectively. The model trained to identify the ROI exhibited a precision of 100%, a recall of 71.8%, and an F1 Score of 83.6% for bounding box detection. These are promising results for the implementation of this pipeline in future studies. The application of FFT to signals acquired from both RGB and IR images proved to be an effective and accurate method for computing the RR of cows in unrestrained conditions.

A Hybrid Human Activity Recognition Method Using an MLP Neural Network and Euler Angle Extraction Based on IMU Sensors

Inertial measurement unit (IMU) technology has gained popularity in human activity recognition (HAR) due to its ability to identify human activity by measuring acceleration, angular velocity, and magnetic flux in key body areas like the wrist and knee. It has propelled the extensive application of HAR across various domains. In the healthcare sector, HAR finds utility in monitoring and assessing movements during rehabilitation processes, while in the sports science field, it contributes to enhancing training outcomes and preventing exercise-related injuries. However, traditional sensor fusion algorithms often require intricate mathematical and statistical processing, resulting in higher algorithmic complexity. Additionally, in dynamic environments, sensor states may undergo changes, posing challenges for real-time adjustments within conventional fusion algorithms to cater to the requirements of prolonged observations. To address these limitations, we propose a novel hybrid human pose recognition method based on IMU sensors. The proposed method initially calculates Euler angles and subsequently refines them using magnetometer and gyroscope data to obtain the accurate attitude angle. Furthermore, the application of FFT (Fast Fourier Transform) feature extraction facilitates the transition of the signal from its time-based representation to its frequency-based representation, enhancing the practical significance of the data. To optimize feature fusion and information exchange, a group attention module is introduced, leveraging the capabilities of a Multi-Layer Perceptron which is called the Feature Fusion Enrichment Multi-Layer Perceptron (GAM-MLP) to effectively combine features and generate precise classification results. Experimental results demonstrated the superior performance of the proposed method, achieving an impressive accuracy rate of 96.13% across 19 different human pose recognition tasks. The proposed hybrid human pose recognition method is capable of meeting the demands of real-world motion monitoring and health assessment.

Accelerated Variant of an EMD-Based-R Peaks Detection Algorithm Involving FFT-Based Time-Domain Down-Sampling and up-Sampling

As most of developed empirical mode decomposition (EMD) based R-peaks detection algorithms consume a considerable time of calculation caused by the large length of the input ECG signal, the design of a new technique that allows the acceleration of such methods becomes necessary. Accordingly, a new variant of an EMD-based strategy for R-peaks localization is presented. The new accelerated variant is constituted of three essential parts. The first step is the length reduction of the input signal by means of the truncation in the Fast Fourier Transform (FFT) domain followed by the application of the inverse FFT guaranteeing a suitable time-domain down-sampling. Consequently, the new input signal of a reduced length preserves all medical information contained initially in the original lengthy signal. The second part is dedicated to identify the QRS complex using EMD-based R-peaks detection. This latter comprises a low-pass filter, Empirical Mode Decomposition (EMD) and the Hilbert transform, Finally, the third phase is the time-domain up-sampling using the FFT, the zero padding and the Inverse Fast Fourier Transform (IFFT) to obtain a resulting processed signal which has the same length as the original signal. Next, as a post-processing step, final R-peaks refined localization is achieved. It is noticeable that the new variant ensures same results, in term of accuracy, as the standard method; however, a significant speed-up ratio of 6.95:1 is reported. Additionally, to more prove the effectiveness of the suggested strategy, it has been applied to accelerate two other efficient algorithms and satisfactory speed up ratios of, 7.20:1 and 4.23:1, respectively have been reached.

MODELING OF A DIGITAL COMMUNICATION SYSTEM WITH INTERFERENCE-RESISTANT CODING OVER DELAYED MULTIPATH CHANNELS FOR A FIXED WIRELESS ACCESS SYSTEM


 
 
 In this study, an analysis of the performance of a digital communication system was conducted to improve the efficiency of the communication channel. The system was de- signed and evaluated in the MatLab/Simulink environment, specifically for the transmission of binary data in a multipath channel with static fading. One of the main advantages of this system is its ability to provide high noise immunity, even in the presence of noise, interference, and signal delays. The performance of the system was investigated in terms of bit error rate (BER) over both an additive white Gaussian noise (AWGN) channel and a multipath Rayleigh fading channel. The study also considered the impact of inter-symbol interference (ISI) and explored different parameters for a static channel, resembling a wireless LAN system based on IEEE 802.11 (as defined by Rec ITU-R M.1225). This study uses a solution to suppress inter-symbol interference using cascading coding (convolutional turbo codes with Reed Solo- mon code), Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT), amplifiers at the transmitter side and normalizer at the receiver side, by regulating the amplification ratios of transmitter and receiver sides. Modeling of a digital communication system for a wide-band fixed wireless LAN system (for Indoor office and Outdoor to indoor systems) has been performed. The modeling outcomes show that the applied method provides a good per- formance improvement in channels with ISI with static fading.
 The application of cascading coding (turbo code together with RS code), amplifiers, and FFT on the transmitter side, and IFFT on the receiver side will eliminate the effects of inter-symbol interference on digital signals in a multipath channel with static fading (as defined by Rec. ITU-R M.1225 for internal and external transmission systems) for wireless fixed systems.
 
 

Efficient simulation of non-stationary non-homogeneous wind field: Fusion of multi-dimensional interpolation and NUFFT

The stochastic wave approach is a new realization of the spectral representation method and has been widely used for the simulation of stationary non-homogeneous or non-stationary homogeneous wind fields. However, it inevitably creates many unnecessary simulation points due to the invoking of the Fast Fourier Transform (FFT) and suffers from low efficiency in non-stationary non-homogeneous scenarios due to the incapability to accelerate the summation of trigonometric functions. In this study, an efficient approach is developed for the simulation of non-stationary non-homogeneous wind field. Central to this approach is a multi-dimensional interpolation used to decouple the evolutionary spectra accompanied by the application of non-uniform FFT (NUFFT) to expedite the simulation. A surrogate model that accurately approximates the 3D evolutionary spectrum related to height, frequency, and time is established by the 3D reduced Hermite interpolation, which enables a successful separation of the frequency components. Then, the NUFFT is applied to speed up the summation of trigonometric functions over wavenumbers with a well-designed non-uniformly distributed sampling points, which effectively reduces the segments used in the Fourier transform. Since there is no required mapping between the wavenumbers and the locations of simulation points in NUFFT, a significant number of unnecessary simulation points in the FFT scenario is avoided. Moreover, the wind samples at different locations can be obtained by directly altering the query points of NUFFT, thus the efficiency of the enhanced approach is almost independent of the number of simulation points. The numerical example in simulating the non-stationary non-homogeneous wind field of a long-span cable-stayed bridge demonstrates the efficiency of the developed approach and validates the effectiveness of the simulated wind samples in view of evolutionary spectrum and coherence function.

Analysis of Circular Saw Tooth Marks Profile on Material Machined Surface After Filtering with Fast Fourier Transform (FFT)

The article presents theoretical analysis of machined surface roughness after sawing on circular saw and implementation of fast Fourier transform (FFT) as a possible simple filtering method for filtering out just the saw blade and saw tooth influence on the surface roughness. Surface roughness profile is represented as a signal that can be obtained as a sum of complex periodic signals that represent theoretical profile of tooth marks and lateral movement of tooth due to saw lateral movement and signals that represent structural roughness of wood combined with machining roughness, represented as a Gaussian noise. The application of FFT based filtering on such a signal can be effectively used to extract the main frequency components due to tool influence on total surface signal and the time domain of filtered signals display can then be obtained by use of the inverse Fourier transform. In order to test the theoretical assumptions, the machining tests in sawing of solid oak wood (Quercus robur L.) and medium density fiberboard (MDF) was conducted. Machined surface roughness was measured and analyzed in accordance with theoretical assumptions. It was concluded that a combination of discrete Fourier transform of surface roughness profile and standard roughness parameters can give a more complete representation of machined surface roughness after sawing with circular saws and that filtering of surface roughness profile signal with FFT filter can be used as a simple and effective method in quantifying tool influence on machined surface roughness after sawing on circular saw in varying machining conditions and on different workpiece material.

Computation of the probability of ultimate ruin and some other actuarial quantities under the classical risk model via Fast Fourier Transform

The Probability of ultimate ruin under the classical risk model is obtained as a solution of an integro -differential equation involving convolutions and we have used Fast Fourier Transform (FFT) to obtain the approximate values of the probability of ultimate ruin from this integro -differential equation under the situation when the claim severity is modelled by the Mixture of 3 Exponentials and the Weibull distribution. Another application of FFT in ruin theory is shown by means of applying it to obtain the quantiles of the aggregate claim distribution under these claim severity distributions. Extension of the application of FFT is shown by using it to obtain the first moment of the time to ruin under the classical risk model for these distributions. The distributions which have been used are such that one is light tailed and the another is heavy tailed so that a comparison can be made between them on the precision of the actuarial quantities obtained through FFT. FFT has been found to be efficient in obtaining these actuarial quantities when used in conjunction with certain modifications like exponential tilting to control the aliasing error.

A QOS ESTIMATION ALGORITHM FROM CALLER RINGTONE ANALYSIS IN GSM NETWORK

Call Setup Time (CST) is one of the key performance indicators (KPIs) that Mobile Network Providers (MNPs) are mostly appertain. It has been established that long CST usually severely affects the user experience. Owing to the limitations associated with gleaning the CST data from MNPs, this paper provides the development of QoS estimation algorithm from various CST parameters. The algorithm involves the determination of CST; Inter-Burst time; Intra-Burst time and Call duration in time domain. The caller Frequency content was also determined by the application of fast Fourier Transform before computing the Mean Square Error (MSE). The eventual QoS rating is done after the computation of the MSE from various individual parameters. Four hourly data consisting of 10 sets each were collected three times in a week for four weeks for each MNP’s for creating Caller Ringtone dataset and testing the developed algorithm. Performance analysis of the system in accurately determining: CST; Intraburst time; Interburst time and Call durations were carried out. Results obtained shows that the proposed technique accurately computes these parameters and maximum error obtained was to the value of 10%. Furthermore, the QoS obtained shows an error margin of less than 5 % was observed when the developed technique was compared to the ground truth. Thus, the proposed algorithm was able to compute the QoS using Caller Ringtone only, thus independent of MNP.

Fast control plane for flexible and scalable optical interconnects.

The growth of data centers (DC) and high-performance computing (HPC) requires higher bandwidth, but traffic usually occurs between a small number of computing nodes, and the location of the communication bottleneck dynamically changes as the application runs. Therefore, the traditional static network that solves the communication bottleneck by providing excessive bandwidth cannot meet the demand of high performance and low cost at the same time. The reconfiguration of optical interconnects greatly improves the flexibility of the network, which can allocate unutilized bandwidth to node pairs with dense communication and improve resource utilization. However, this flexibility relies on a fast control plane to achieve efficient interaction between devices in the network. We made improvements in traffic collection, topology calculation, and optical switch configuration, and built an experimental platform to evaluate our control plane. The flexibility of optical interconnects shows a good acceleration effect when running applications that solve large-scale problems, and the experimental results show that a proper reconfiguration cycle can reduce the completion time of 3-D Fast Fourier Transform application by up to 53%.

NUMERICAL SIMULATION OF TRI-LAYERED MATERIALS UNDER CONTACT LOAD

Various biomedical components, such as dental crowns and hip prostheses, data processing devices, and other numerous mechanical components that transmit load through a mechanical contact, may benefit from a tri-layer design. The coating may be optimized for wear protection and corrosion prevention, whereas the intermediate layer provides increased adhesion between the outer layer and the substrate, and confines the crack propagation. The solution to the contact problem involving tri-layered materials can be pursued numerically with the finite element or the boundary element methods, but semi-analytical techniques benefitting from the efficiency of the fast Fourier transform (FFT) technique have also been successfully applied. At the heart of the FFT-assisted approach lie the frequency response functions (FRFs), which are analytical solutions for fundamental problems of elasticity such as the Boussinesq and Cerruti problems, but expressed in the frequency domain. Considering recent efforts and results in application of FFT to convolution calculations in contact problems, the displacement arising in a tri-layer configuration is computed in the frequency domain, and the contact problem is subsequently solved in the space domain using a state-of-the-art algorithm based on the conjugate gradient method. The method relies on the FRFs derived in the literature for tri-layered materials, and the efficiency and accuracy of computations in the frequency domain is assured by using the Discrete Convolution Fast Fourier Technique (DCFFT) with influence coefficients derived from the FRFs. The computer program reproduces well-known results for bi-layered materials. Numerical simulations are performed for various configurations in which the elastic properties of the layers, as well as the frictional coefficient, are varied. By using the newly advanced simulation technique, design recommendations may be advanced for the optimal configuration of tri-layered materials under contact load.

Exploring the fundamentals of fast Fourier transform technique and its elementary applications in physics

Fast Fourier transform (FFT) is a powerful and versatile tool used in engineering science. As expected, papers related to the methods and applications of FFT are galore in literature. However, most of them are too advanced for comprehension by the undergraduates. In this work, the nitty-gritty of FFT algorithm is discussed in a manner so as to equip the reader with this ubiquitous computational tool with less time investment. The target readers of the paper are students pursuing physics at the high school or introductory undergraduate level. The aim is to illustrate the use of FFT technique as a simulation tool in physics and motivate the students to use it with ease for solving realistic problems, that often lack analytical solutions. Most of the scientific computing softwares available today have ready to use built-in FFT modules. Broadly speaking, most of these modules are known to be developed on the basis of a common FFT algorithm which was first proposed by Cooley et al (1969 IEEE Trans. Educ. 12 27–34; 1987 Mikrochim. Acta 93 33–45). An attempt is made to simplify the explanation of this not-so-straightforward algorithm. The built-in FFT module of SCILAB (2012 Scilab: Free and Open Source Software for Numerical Computation) (an open source software for scientific computation) has been used to simulate some simple problems from introductory undergraduate level topics like oscillations and wave optics.

FFT Implementation using Modified Booth Multiplier and CLA

In the field of digital signal and image processing the Fast Fourier Transform (FFT) is one of the rudimentary operations. Telecommunication, Automotive, Hearing devices, Voice recognition systems are some of the applications of Fast Fourier Transform. DFT is implemented using FFT which is a type of algorithm that computes DFT in a fast and efficient manner. This project concentrates on the development of the Fast Fourier Transform (FFT), based on Decimation In Time (DIT) domain, Radix2 algorithm, using VHDL as a design entity.The objective of this project is to establish an efficient design that computes FFT in a faster way. In this project FFT is implemented using modified booth multiplier and CLA and simulated on Xilinx ISE. K

Comparison of manual and automatic approaches for characterisation of morphology and nanostructure of soot particles

This study is an attempt to compare the manual and automatic approaches for analysis of the morphology and nanostructure of soot particles. The automated methodology is composed of two parts: soot particle analysis (morphology characterisation) and soot fringe analysis (nanostructure characterisation) such as fringe length, fringe tortuosity and fringe separation distance. In this study the application of Fast Fourier Transform (FFT) is introduced, which omits the manual selection of fringe pairs for measurement of fringe separation distance and aids fast computation. The parameters which effect the performance of automated approach were identified and optimised. The automated approach was applied to analyse images from different fuels. Using the optimised parameters, the performance of the automatic approach was tested against results obtained by the manual approach. The statistical comparison between two approaches showed similar results but with better accuracy for automated approach (lower standard error) and thus validates the functionality of the proposed methodology. The application of FFT in computation of fringe separation distance reduces the computation time and avoids biases of operator in selecting fringe pairs.

On the use of fast Fourier transform for optical layer thickness determination

Thin film thickness determination with a reflectometer is a fast and pretty cheap method that can be applied on many thin and thick films that are transparent or semitransparent in the considered spectral range. For evaluation of the reflectance spectrum either nonlinear regression analysis is used for very thin films or fast Fourier transform (FFT) for films thicker than approximately 1 µm. Using FFT for layer thickness determination there are some special facts to consider in contrast to the common use of FFT in signal processing. First of all, the sampling points are in general not equidistant as the wavelengths of the used spectrometer are not equidistant. Next the number of sampling points may be different from a power of 2. The reason is that the measured spectral range may be restricted by the user. And finally the analogues in layer thickness determination to the independent parameters “time” and “frequency” in signal processing are not independent of each other. The reason is the dispersion of the refractive index of the layer material that causes unwanted Moire effects in the reflectance spectrum. All these deviations lead to additional sources for errors in the thickness determination beyond those error sources that are well-known from common FFT applications in signal processing. In this paper we discuss how these deviations affect the thickness determination and present a solution for the main problem caused by the dispersion of the refractive index. Further improvements of the FFT result in combination with a grid search algorithm and a nonlinear regression are presented and discussed. Finally we present with structured samples and highly doped semiconductors two specific applications of FFT on layer thickness determination.

An FFT-Based CLEAN Deconvolution Method for Interferometric Microwave Radiometers With Spatially Variable Beam Pattern

A modification to the extended CLEAN deconvolution method for passive microwave interferometric radiometers is presented to account for the interferometer beam patterns that vary depending on the source position within the interferometer field of view. This method is developed for the newly proposed multisatellite rotating microwave interferometric radiometer concept for geostationary atmospheric sounding, which is found to have a spatially varying beam pattern. The proposed method is implemented by the piecewise application of Fast Fourier Transform (FFT)-based convolution, i.e., the image is divided into discrete segments, where the beam pattern is assumed to be constant within each segment. This approach allows the proposed method to exploit the speed advantage of the FFT. Applying this method to the proposed multisatellite interferometer shows a twofold improvement in the radiometric accuracy of the restored brightness temperature map, as compared with the constant beam pattern CLEAN.

50 Years of FFT Algorithms and Applications

The fast Fourier transform (FFT) algorithm was developed by Cooley and Tukey in 1965. It could reduce the computational complexity of discrete Fourier transform significantly from $$O(N^2)$$ to $$O(N\log _2 {N})$$ . The invention of FFT is considered as a landmark development in the field of digital signal processing (DSP), since it could expedite the DSP algorithms significantly such that real-time digital signal processing could be possible. During the past 50 years, many researchers have contributed to the advancements in the FFT algorithm to make it faster and more efficient in order to match with the requirements of various applications. In this article, we present a brief overview of the key developments in FFT algorithms along with some popular applications in speech and image processing, signal analysis, and communication systems.