Computation Of Discrete Fourier Transform Research Articles

Cooley-Tukey FFT Algorithm based on GDFT for Phasor Estimation by PMU under Power Quality Disturbances

Due to the high penetration of renewable energy sources, such as photovoltaic panels and wind turbines, in addition to the use of different electric power supplies in the power grid, there are major disturbances in the forms of electric waves. These variations and disturbances must be monitored and controlled for the efficient management of transmission and distribution of electrical energy, safety, and electrical protection systems. Nowadays, Phasor Measurement Unit (PMU) technology is an essential tool to develop the supervision, protection, and control of the electrical power grid. PMUs measure the amplitude and angle of current and voltage waveforms on a Coordinated Universal Time (UTC) time scale and speedily measure the fundamental frequencies and their rates of change using fast and accurate estimation algorithms. This paper presents a phasor estimation using a Modified Cooley-Tukey Fast Fourier Transform algorithm based on the Generalized Discrete Fourier Transform (DFT) used in PMUs, using simulations in MATLAB. This algorithm was utilized to accelerate and simplify the computation of DFTs. To validate the performance under waveform disturbances, several tests with different waveforms and disturbances were simulated and interpreted according to the standard and compared with DFT.

The Discrete Fourier Transform Over the Binary Finite Field

The novel methods for binary discrete Fourier transform (DFT) computation over the finite field have been proposed. The methods are based on a binary trace calculation over the finite field and use the cyclotomic DFT. The direct DFT computational complexity has been reduced due to using the binary trace function over the finite field and the functions of trace which are stored in small tables. The computational complexity of the inverse DFT has been reduced due to representation of elements in the finite field with respect to the normal basis. The proposed methods can be used for encoding/decoding subfield subcodes, especially for binary Bose–Chaudhuri–Hocquenghem (BCH) codes. The computational complexity of the direct/inverse DFT computation methods is the smallest of all known methods.

Theoretical and experimental insight into plasma-catalytic degradation of aqueous p-nitrophenol with graphene-ZnO nanoparticles

This paper aims to investigate plasma-catalytic degradation of p-nitrophenol (PNP) with graphene-ZnO nanoparticles based on theoretical and experimental perspects. The graphene-ZnO nanoparticles were firstly prepared by hydrothermal process, which were then systematically characterized with scanning electron microscope (SEM), transmission electron microscope (TEM), X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), Raman spectrum, UV–Vis spectrophotometer, X-ray photoelectron spectroscopy (XPS), and nitrogen adsorption desorption analyzer. Combined with the analysis of Discrete Fourier Transform (DFT), the morphology, structure, specific surface area, absorption light range and electron transfer of the prepared catalyst are described in detail. The prepared graphene-ZnO nanocomposites displayed larger specific surface area and wider light absorption range. The charge could be transferred from ZnO to graphene. The catalytic degradation performace was further examined. Results indicated that degradation rate of PNP was enhanced from 68.5% to 80.2% with ZnO, which could futher improved to 97.6% with graphene (6%)-ZnO nanocomposites. The prepared samples exhibited good stability. Active species including OH, O2–, HO2, 1O2, h+ and e- played crucial role in PNP degradation. The degradation pathways were proposed based on DFT calculation and relevant literatures. Lastly, ECOSAR algorithm illustrated that the toxicity of PNP can be relieved after plasma-catalytic treatment.

Insights into water film DBD plasma driven by pulse power for ibuprofen elimination in water: performance, mechanism and degradation route

• IBP elimination using water film DBD plasma by pulse power was investigated. • The impact of various parameters on IBP elimination was inspected. • OH, electron, 1 O 2 and · O 2 – participation in IBP decomposition. • The degradation route was proposed based on DFT calculation and LC-MS. • The toxicity of actual wastewater alleviated after water film DBD treatment. Development of efficient abatement technology for drugs and personal care products (PPCPs) has important scientific and practical significance to restrain its discharge to the natural water environment. This research aims to investigate ibuprofen (IBP) elimination in water film dielectric barrier discharge (DBD) plasma system driven by pulse power. With raising input power, the removal rate was improved. When input was 95 W, the removal rate and kinetic constant were up to 96.1% and 0.090 min −1 , respectively. Besides, heightening frequency, duty ratio and solution flowrate would promote the degradation of IBP. IBP were more favorably degraded at lower IBP concentration and neutral pH condition. Active species including ·OH, electron, 1 O 2 and · O 2 – dominated decomposition of IBP. The generation of O 3 , · OH and H 2 O 2 in deionized water and IBP solution was compared. UV–Vis spectrum and three dimensional fluorescence spectroscopy illustrated that IBP molecules could be decomposed effectively after DBD plasma treatment. During IBP degradation, COD and TOC was also relieved with extension of treatment time. The possible degradation routes of IBP in water film DBD plasma system were proposed based on LC-MS and Discrete Fourier transform (DFT) simulation. Based on the identified intermediates, their potential toxicity was evaluated with the help of US-EPA-TEST software. Water film DBD plasma technology was also suitable for actual wastewater treatment. The comparison of other technologies for IBP degradation was carried out finally. This work can provide exploratory platforms for comprehensively understanding and enhancing IBP degradation in DBD plasma technology.

A fast and precise power system frequency estimation method without iterations

The conventional power system frequency estimation methods based on the Discrete Fourier Transform (DFT) normally require iterative calculations to eliminate the errors due to spectral leakage in off-nominal frequencies. It cannot provide both fast response and good accuracy under a system disturbance. In this paper, a fast and precise frequency estimation method is presented. The theoretical analysis shows the angle difference of the positive sequence voltage is exactly linear to the frequency deviation. Based on this angle characteristic, this method is able to precisely calculate the frequency deviation from angle differences between the continuous DFT calculations without the need for any iteration. Even if there is only one single phase voltage input available, the simulated positive sequence voltages can be derived by constructing the remaining two phase-voltages utilizing an appropriate phasor shift. To effectively restrain the harmonic interferences, a two-cycle Hanning window is applied as a filter. One method to detect the phase angle jump is proposed to avoid transient frequency errors by using the 2nd order derivative of the frequency. The simulation compares the frequency estimation performance between the proposed non-iterative method and the conventional iterative method. The simulation results show the frequency response is faster and the accuracy is reliable within the range 40–70 Hz, even if there are harmonic interferences or three-phase unbalances. This proposed frequency estimation method is easy to implement, and it can be widely applied in the intelligent electronic devices within the power system.

Multibeam Digital Array Receiver Using a 16-Point Multiplierless DFT Approximation

An N-element array with receivers subject to an N-point spatial fast Fourier transform (FFT) leads to N directionally orthogonal radio frequency (RF) beams. FFTs are fast algorithms for computing the N-point discrete Fourier transform (DFT) at reduced complexity. The brute-force computation of a DFT requires O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) multiplications while an FFT provides the same computation at O(N log N) multiplications. The digital chip area and power consumption of the DFT computation are still dominated by the multipliers required by the FFT used. In this paper, an approximation to the 16-point DFT is proposed which maintains mathematical properties close to the ideal 16-point DFT to obtain 16 RF beams by computing an approximate spatial DFT in every clock cycle at significantly lower area and power in the digital realization. The proposed approximation can be implemented using FFT-like fast algorithms that are multiplierless, thereby further reducing the digital chip area and power consumption associated with multiplication in a conventional FFT approach to zero. A 16-beam beamformer employing a 16-element linear array of patch antennas, direct-conversion receivers, and a Xilinx Virtex6 field-programmable gate array-based real-time digital back-end clocked at 240 MHz are described as an example realization of 16 complex-valued (IQ) receive-mode RF beams, centered at 2.4 GHz with 120 MHz of bandwidth per beam.

FFAST: An Algorithm for Computing an Exactly $ k$ -Sparse DFT in $O( k\log k)$ Time

It is a well-known fact that the Discrete Fourier Transform (DFT) $ \vec {X}$ of an arbitrary $ n$ -length input signal $ \vec {x}$ , can be computed from all the $ n$ time-domain samples in $O( n\log n)$ operations via a Fast Fourier Transform (FFT) algorithm. If the spectrum $ \vec {X}$ is $ k$ -sparse (where $ k\ll n$ ), can we do better? We show that asymptotically in $ k$ and $ n$ , when $ k$ is sub-linear in $ n$ (precisely, $ k= O( n^{ \delta }$ ), where $0 \leq \delta ), and the support of the non-zero DFT coefficients is uniformly random, the fast fourier aliasing-based sparse transform (FFAST) algorithm, proposed in this paper, computes, with asymptotically high probability, the $k$ non-zero DFT coefficients of $\vec {X}$ from $O( k)$ samples of $ \vec {x}$ in $O( k\log k)$ arithmetic operations. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when $ \delta , which essentially covers a wide range of sub-linear sparsity cases, the sample cost is less than $4 k$ . Although, in this paper we assume that the samples of the signal $ \vec {x}$ observed by the FFAST are noise-free, a noise-robust extension of the FFAST is provided in a companion (Part II) paper [1] . Our approach is based on filter-less sub-sampling of the input signal using a set of carefully chosen uniform sub-sampling patterns guided by the Chinese Remainder Theorem (CRT) . The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by sub-sampling. Specifically, the sub-sampling operation on the time-domain signal $ \vec {x}$ is designed to create aliasing patterns of the non-zero coefficients of the spectrum $ \vec {X}$ to “look like” parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT $ \vec {X}$ is equivalent to decoding of an appropriate sparse-graph code . The sparse-graph codes further permit a fast peeling-style decoding. Consequently, the resulting DFT computation is low in both the sample and the decoding complexity. We analytically connect our proposed CRT-based aliasing framework to random sparse-graph codes, and analyze the performance of our algorithm using density evolution techniques from coding theory. We provide extensive simulation results, that are in tight agreement with our theoretical findings.

An Exact and Fast Computation of Discrete Fourier Transform for Polar and Spherical Grid

Numerous applied problems of two-dimensional (2-D) and 3-D imaging are formulated in continuous domain. They place great emphasis on obtaining and manipulating the Fourier transform in polar and spherical coordinates. However, the translation of continuum ideas with the discrete sampled data on a Cartesian grid is problematic. There exists no exact and fast solution to the problem of obtaining discrete Fourier transform for polar and spherical grids in the literature. In this paper, we develop exact algorithms to the above problem for 2-D and 3-D, which involve only 1-D equispaced fast Fourier transform with no interpolation or approximation at any stage. The result of the proposed approach leads to a fast solution with very high accuracy. We describe the computational procedure to obtain the solution in both 2-D and 3-D, which includes fast forward and inverse transforms. We find the nested multilevel matrix structure of the inverse process, and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number.

An Accurate and Stable Sliding DFT Computed by a Modified CIC Filter [Tips &amp;amp; Tricks

The sliding discrete Fourier transform (SDFT) is a popular algorithm used in nonparametric spectrum estimation when only a few frequency bins of an M-point discrete Fourier transform (DFT) are of interest. Although the classical SDFT algorithm described in [1] is computationally efficient, its recursive structure suffers from accumulation and rounding errors, which lead to potential instabilities or inaccurate output. Duda [2] proposed a modulated SDFT (mSDFT) algorithm, which has the property of being guaranteed stable without sacrificing accuracy, unlike previous approaches described in [1], [3], and [4]. However, all of these conventional SDFT methods presume DFT computation on a sample-by-sample basis. This is not computationally efficient when the DFT needs only to be computed every R(R g 1) samples. To address such cases when R-times downsampling is needed, Park et al. [5] proposed a hopping SDFT (HDFT) algorithm. Recently, Wang et al. [6] presented a modulated HDFT (mHDFT) algorithm, which combines the HDFT algorithm with the mSDFT idea to maintain stability and accuracy at the same time. In parallel, Park [7] updated the HDFT algorithm with its guaranteed stable modification called gSDFT, which exists only for certain M and L relationships.

Accurate efficient carrier estimation for single-shot digital holographic imaging.

Off-axis digital holograms are commonly processed using the Fourier transform method. This method requires estimation of the carrier frequency peak in the two-dimensional Fourier domain. The accuracy of peak location information using the standard fast Fourier transform (FFT) routines is limited by the pixel size, thus giving rise to a residual carrier component in the reconstructed image. In this Letter, we use an efficient two-step process for cross-term peak location to sub-pixel accuracy where the standard FFT-based peak finding procedure is followed by a high-resolution discrete Fourier transform (DFT) calculation at the nominal peak location. It is shown that the residual tilt artifacts due to a small change in the magnitude of the carrier frequency or due to the rotation of the fringe pattern are uniformly removed with this method. For a digital hologram of size M×N pixels, the high-resolution DFT calculation with an up-sampling factor of α requires an additional computational effort of O(MNα) which is marginal when α≪M,N and, thus, provides a practical single-shot method for accurate demodulation of digital holograms.

Error compensation of IQ modulator using two-dimensional DFT

It is important to precisely set and keep the phase and amplitude of an rf signal in the accelerating cavity of modern accelerators, such as an X-ray Free Electron Laser (XFEL) linac. In these accelerators an acceleration rf signal is generated or detected by an In-phase and Quadrature (IQ) modulator, or a demodulator. If there are any deviations of the phase and the amplitude from the ideal values, crosstalk between the phase and the amplitude of the output signal of the IQ modulator or the demodulator arises. This causes instability of the feedback controls that simultaneously stabilize both the rf phase and the amplitude. To compensate for such deviations, we developed a novel compensation method using a two-dimensional Discrete Fourier Transform (DFT). Because the observed deviations of the phase and amplitude of an IQ modulator involve sinusoidal and polynomial behaviors on the phase angle and the amplitude of the rf vector, respectively, the DFT calculation with these basis functions makes a good approximation with a small number of compensation coefficients. Also, we can suppress high-frequency noise components arising when we measure the deviation data. These characteristics have advantages compared to a Look Up Table (LUT) compensation method. The LUT method usually demands many compensation elements, such as about 300, that are not easy to treat. We applied the DFT compensation method to the output rf signal of a C-band IQ modulator at SACLA, which is an XFEL facility in Japan. The amplitude deviation of the IQ modulator after the DFT compensation was reduced from 15.0% at the peak to less than 0.2% at the peak for an amplitude control range of from 0.1V to 0.9V (1.0V full scale) and for a phase control range from 0 degree to 360 degrees. The number of compensation coefficients is 60, which is smaller than that of the LUT method, and is easy to treat and maintain.

Separation of adjacent interharmonics using maximum energy retrieving algorithm

It is known that flickers and interharmonics have an inherent relationship, and the magnitude of a voltage can fluctuate as the signal contains interharmonics. In particular, flicker or modulation will occur when the interharmonic is close to a harmonic (or fundamental) frequency. Current spectrum separation approaches can detect harmonics and main interharmonics effectively, but they may not be applicable for the situation of adjacent interharmonics from either divergence effect, sensitive spectral leakage or other drawbacks. For this reason, this study develops a strategy of maximum energy retrieving (MER) method to distinguish and identify those interharmonics which are adjacent to fundamental or harmonics in power systems. Based on the appropriate selection of sampling window length using discrete Fourier transform (DFT), neighbouring dispersed energy can be collected to retrieve its original amplitude value. The frequency component can be thus determined simultaneously when the total collected energy reaches the maximum. The MER implementation can be converged within only several iteration loops where parallel DFT computation is required before the convergent process. The numerical example is used to verify the effectiveness of the proposed approach in term of reliability and accuracy.

Efficient Design of Bit Parallel Multiplication based Radix-4 Multipath Delay Commutator (R4mdc) FFT

This paper presents the Fast Fourier Transform (FFT) techniques for converting a signal in the time domain to the frequency domain. The advent of Fast Fourier Transform (FFT) method has greatly extended our ability to implement the Fourier methods on digital computers. FFT is an algorithm that speeds up the calculation of Discrete Fourier Transform (DFT). Pipelined FFT have features like simplicity, modularity and high throughput. Radix-2 Multi-Path Delay Commutator (R2MDC) FFT is the best frequency transformation architecture for FFT calculation. However it requires more delay elements to provide desired frequency transformation functions. To overcome this problem, Radix-4 Multi-Path Delay Commutator (R4MDC) has been designed in this paper. In addition, structure of Bit Parallel Multiplication (BPM) has been modified to alleviate the performance of twiddle factor multiplier of FFT architecture. Modified BPM structure has utilized only little hardware to perform the twiddle factor multiplication. Finally, Modified BPM structures have been incorporated into R4MDC FFT structure for alleviating the performances of frequency transformation process. Design of proposed methods has been validated by using ModelSim 6.3C and Synthesis results has been evaluated by using Xilinx 12.4i (Family: Spartan 3, Device: Xc3s200, Package: PQ208, Speed: -5) design tool.

Unified commutation-pruning technique for efficient computation of composite DFTs

An efficient computation of a composite length discrete Fourier transform (DFT), as well as a fast Fourier transform (FFT) of both time and space data sequences in uncertain (non-sparse or sparse) computational scenarios, requires specific processing algorithms. Traditional algorithms typically employ some pruning methods without any commutations, which prevents them from attaining the potential computational efficiency. In this paper, we propose an alternative unified approach with automatic commutations between three computational modalities aimed at efficient computations of the pruned DFTs adapted for variable composite lengths of the non-sparse input-output data. The first modality is an implementation of the direct computation of a composite length DFT, the second one employs the second-order recursive filtering method, and the third one performs the new pruned decomposed transform. The pruned decomposed transform algorithm performs the decimation in time or space (DIT) data acquisition domain and, then, decimation in frequency (DIF). The unified combination of these three algorithms is addressed as the DFTCOMM technique. Based on the treatment of the combinational-type hypotheses testing optimization problem of preferable allocations between all feasible commuting-pruning modalities, we have found the global optimal solution to the pruning problem that always requires a fewer or, at most, the same number of arithmetic operations than other feasible modalities. The DFTCOMM method outperforms the existing competing pruning techniques in the sense of attainable savings in the number of required arithmetic operations. It requires fewer or at most the same number of arithmetic operations for its execution than any other of the competing pruning methods reported in the literature. Finally, we provide the comparison of the DFTCOMM with the recently developed sparse fast Fourier transform (SFFT) algorithmic family. We feature that, in the sensing scenarios with sparse/non-sparse data Fourier spectrum, the DFTCOMM technique manifests robustness against such model uncertainties in the sense of insensitivity for sparsity/non-sparsity restrictions and the variability of the operating parameters.

Simplified approach to DFT computation for nonprogrammable scientific calculators

Fourier analysis is an important tool used as it is or it’s different variants in many fields of sciences and engineering. It’s importance is due to it’s simplicity with which it expands a given function in terms of circular or complex exponents. Further it is quite versatile to handle many functions of practical interest, specifically, the functions with several mathematical disabilities that are hard to be handled with tools like Taylor series. Discrete Fourier Transform (DFT) is a form of Fourier analysis where the discrete function and it’s transform are both of finite length. This processing requires lot many computations. Here in this work a simplified and non programmable calculator based scheme is presented with which one can easily determine the DFT of the given function by feeding in the DFT equation once and a few presses of the calculator keys. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11681BIBECHANA 12 (2015) 13-19

Wavelet Cesptral Coefficients for Isolated Speech Recognition

The study proposes an improved feature extraction method that is called Wavelet Cepstral Coefficients (WCC). In traditional cepstral analysis, the cepstrums are calculated with the use of the Discrete Fourier Transform (DFT). Owing to the fact that the DFT calculation assumes signal stationary between frames which in practice is not quite true, the WCC replaces the DFT block in the traditional cepstrum calculation with the Discrete Wavelet Transform (DWT) hence producing the WCC. To evaluate the proposed WCC, speech recognition task of recognizing the 26 English alphabets were conducted. Comparisons with the traditional Mel-Frequency Cepstral Coefficients (MFCC) are done to further analyze the effectiveness of the WCCs. It is found that the WCCs showed some comparable results when compared to the MFCCs considering the WCCs small vector dimension when compared to the MFCCs. The best recognition was found from WCCs at level 5 of the DWT decomposition with a small difference of 1.19% and 3.21% when compared to the MFCCs for speaker independent and speaker dependent tasks respectively. DOI: http://dx.doi.org/10.11591/telkomnika.v11i5.2510 Full Text: PDF

Approximate frequency analysis in structural dynamics

Frequency analysis is often used in structural dynamics, and the precautions and requirements necessary to obtain accurate results in the frequency domain are well known. However there are applications where resources limitations mean that full accuracy has to be compromised; one example is structural health monitoring using wireless sensor systems with severe constraints on power consumption and processing power. This paper investigates two approximations that may be introduced into the Discrete Fourier Transform computation, namely the quantization of the kernel function (the mono-bit FFT) and the quantization of the time signal, and considers the errors introduced into the frequency spectrum. The paper demonstrates that surprising good results may be obtained and these approximate methods should be considered during the design of systems constrained by resource limitations.

On the structure of the matrix isolated water trimer

Infrared spectra of partially deuterated water trimers have been investigated. It is found that HDO(H(2)O)(2) has a single, bound OD stretching fundamental, (HDO)(2)H(2)O two bound OD stretches. (HDO)(3) has a single, bound OD stretch and (H(2)O)(3) has a pair of bound OH stretches. Ab initio and discrete Fourier transform (DFT) calculations predict that the water trimer has C(1) symmetry with six different, isoenergetic minima. These calculations consequently give three numerically different OD stretches for HDO(H(2)O)(2), six for (HDO)(2)H(2)O, three for (HDO)(3), and three bound OH stretches for (H(2)O)(3). The connection between the observations and the pseudorotation of the trimer is discussed with the help of Wales' pseudorotation model. It is found that pseudorotation is sufficiently fast to average the effective symmetry of the A(3) trimer to C(3h) and to eliminate the difference between the different ab initio minima for A(2)B. The only exception is (H(2)O)(3) where the splitting between the different bound OH stretches is largest. Here a doublet is observed due to incomplete averaging. DFT calculations indicate that the D-bonded form of HDO(H(2)O)(2) is between 50 and 60 cm(-1) more stable than the H-bonded form. The energy difference is determined by differences in zero point vibration energy of intermolecular librations of the two forms. Attempts to measure the energy difference indicate that the energy difference is larger, of the order of 100 cm(-1).

A Software Agent for Speech Abiding Systems

Problem statement: In order to bring speech into the mainstream of business process an efficient digital signal processor is necessary. The Fast Fourier Transform (FFT) and the butter fly structure symmetry will enable the harwaring easier. With the DSP and software proposed, togetherly established by means of a system, named here as “Speech Abiding System (SAS)”, a software agent, which involves the digital representation of speech signals and the use of digital processors to analyze, synthesize, or modify such signals. The proposed SAS addresses the issues in two parts. Part I: Capturing the Speaker and the Language independent error free Speech Content for speech applications processing and Part II: To accomplish the speech content as an input to the Speech User Applications/Interface (SUI). Approach: Discrete Fourier Transform (DFT) of the speech signal is the essential ingredient to evolve this SAS and Discrete-Time Fourier Transform (DTFT) links the discrete-time domain to the continuous-frequency domain. The direct computation of DFT is prohibitively expensive in terms of the required computer operations. Fortunately, a number of “fast” transforms have been developed that are mathematically equivalent to the DFT, but which require significantly a fewer computer operations for their implementation. Results: From Part-I, the SAS able to capture an error free Speech content to facilitate the speech as a good input in the main stream of business processing. Part-II provides an environment to implement the speech user applications at a primitive level. Conclusion/Recommendations: The SAS agent along with the required hardware architecture, a Finite State Automata (FSA) machine can be created to develop global oriented domain specific speech user applications easily. It will have a major impact on interoperability and disintermediation in the Information Technology Cycle (ITC) for computer program generating.

Efficient computation of DFT of Zadoff-Chu sequences

An important property of a Zadoff-Chu (ZC) sequence is derived, namely that the discrete Fourier transform (DFT) of a ZC sequence is a time-scaled conjugate of the ZC sequence, multiplied by a constant factor. This result has many practical applications. For example, it can be used to generate 3GPP LTE access preambles more efficiently than the standard suggests as it allows the DFT of a ZC sequence of prime length P to be computed with P instead of PlogP arithmetic operations.