Trigonometric Operations Research Articles

A Novel Coordinate Rotation Digital Computer Method for Energy and Latency Saving by Trigonometric Operations Spatial Locality Principle

In this work, we propose an approximate and energy-efficient CORDIC method, based on a trigonometric function spatial locality principle derived from benchmarks profiling. Successive sine/cosine computation requests cover more than 50% when the absolute phase difference is at most ten degrees. Consequently, this property suggests an optimized circuit implementation, both iterative or a succession of microrotation modules, where the last CORDIC requires fewer iterations, reducing the latency and the total energy budget at the same precision of two separate and independent instances. Thus, this simple design strategy allows significant area and energy dissipation in general-purpose VLSI architectures, but it introduces also dramatically optimizations in applicationspecific embedded systems used in the area of signal processing and radio frequency communication. In this contribution, we introduce a method, the hardware overhead and the energy budget per single cycle. Simulation results show the total energy saving in considered benchmarks is 40% in pipelined and iterative general purposes CORDIC. Furthermore, our application-specific systems (fast Fourier transform and digital oscillators for radiofrequency down conversions) show remarkable cycle savings when the successive sine/cosine computation requests are more than 70%. Finally, in this work, we extend the proposed approach to whichever phase difference less than 26.56° , as a variable for the second CORDIC number of angle rotations.

Analysing of the Proficiency in Basic Mathematical Operations of Physics Teacher Candidates

Bu çalışma, fizik öğretmeni adaylarının temel matematiksel işlem yapabilme yeterliklerini incelemeyi amaçlamaktadır. Bu amaçla, araştırmacı tarafından açık uçlu 20 maddeden oluşan bir ölçek geliştirilmiştir. Ölçek maddeleri; Temel İşlem Bilgisi, Köklü Sayılarda Temel İşlemler, Üslü Sayılarda Temel İşlemler, Logaritmik Sayılarda Temel İşlemler ve Temel Trigonometrik İşlemler olmak üzere temel matematiğin beş farklı konusunda dörder maddeden oluşturulmuştur. Maddeler kolay hesaplanabilir işlemlerden oluşmaktadır. Öğretmen adaylarından hesap makinesi kullanmadan ölçeğe yanıt vermeleri istenmiştir. Çalışma 98 öğretmen adayının katılımıyla gerçekleştirilmiştir. Ayrıca öğretmen adaylarından üçü ile görüşme yapılmıştır. Araştırma sonuçları, öğretmen adaylarının Temel İşlem Bilgisi bölümünde yeterli ancak diğer bölümlerde yetersiz olduklarını göstermiştir. Yapılan işlem hatalarının yanı sıra, işlem önceliğini belirleme, bazı temel işlem bilgilerinin yetersizliği ve özellikle üslü sayılarda ve trigonometrik işlemlerde temel işlemlerde sorunlar belirlenmiştir. Ayrıca fizik öğretmeni adaylarının matematiksel işlem yeterliklerinin cinsiyetten bağımsız olduğu, ancak öğrenim yılı, üniversiteye giriş puanı ve güncel not ortalaması ile kısmen anlamlı olarak pozitif yönde ilişkili olduğu görülmüştür.

A Simple Modulation Scheme With Zero Common-Mode Voltage and Improved Efficiency for Direct Matrix Converter-Fed PMSM Drives

For the direct matrix converter (DMC) applied to permanent-magnet synchronous motor (PMSM) drives, a novel modulation scheme is proposed in this article, which could reduce the control complexity and improve the efficiency while achieving zero common-mode voltage (CMV). Only three counterclockwise or clockwise rotating vectors are adopted to operate the DMC. The proposed scheme has the inherent ability to generate sinusoidal input currents without explicit control in the modulation. The duty cycle expressions for the three vectors are quite simple without any trigonometric operations. The switching pattern is fixed, saving the need for sector and sequence determination. Therefore, compared with the conventional modulation scheme which also uses only rotating vectors, the proposed scheme could reduce the execution time by over 80%. In addition, it achieves higher efficiency benefiting from the reduced switching actions. Its validity is verified by experimental results on a 2.4-kW DMC-fed PMSM drive prototype.

VCO‐less PLL control‐based voltage‐source converter for power quality improvement in distributed generation system

This paper presents the voltage-controlled oscillator (VCO) less phase-locked loop (PLL)-based control of voltage source converter. It is used in the distributed power generation system for improved power quality. This control extracts the fundamental component from the sensed load current using three phases to alpha–beta transformation without using VCO, unlike synchronous reference frame–PLL, where fundamental component extraction needs two transformations with the use of VCO. Owing to the absence of VCO in the proposed method, the computational burden of trigonometric operations is very less that results in a faster dynamic response under variable conditions. This characteristic makes it suitable for the distributed power generation system, where fast changes occur frequently. In this study, wind-based distributed generation system has been taken for study, which comprises of self-excited induction generator feeding non-linear load with varying electrical conditions. Using the proposed control, the reactive power compensation, harmonics reduction and load balancing are carried out in the system. In addition to it, neutral current compensation is also accomplished with the help of star–delta transformer connected at the load end. The system performances are investigated under steady state as well as in transient state and results are found satisfactory.

An Alternative Carrier-Based Implementation of Space Vector Modulation to Eliminate Common Mode Voltage in a Multilevel Matrix Converter

The main aim of the paper is to find a control method for a multilevel matrix converter (MMC) that enables the elimination of common mode voltage (CMV). The method discussed in the paper is based on a selection of converter configurations and the instantaneous output voltages of MMC represented by rotating space vectors. The choice of appropriate configurations is realized by the use of space vector modulation (SVM), with the application of Venturini modulation functions. A multilevel matrix converter, which utilizes a multilevel structure in a traditional matrix converter (MC), can achieve an improved output voltage waveform quality, compared with the output voltage of MC. The carrier-based implementation of SVM is presented in this paper. The carrier-based implementation of SVM avoids any trigonometric and division operations, which could be required in a general space vector approach to the SVM method. With use of the proposed control method, a part of the high-frequency output voltage distortion components is eliminated. The application of the presented modulation method eliminates the CMV in MMC what is presented in the paper. Additionally, the possibility to control the phase shift between the appropriate input and output phase voltages is obtained by the presented control strategy. The results of the simulation and experiment confirm the utility of the proposed modulation method.

Об основном свойстве комплексной операторной экспоненциальной функции комплексного операторного аргумента

Operator functions e^A, sin B, cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B, cos B, formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function e^Z of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions e^At , sin Bt, cos Bt, e^Zt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent e^Zt is given. The rule of differentiation of the function e^Zt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.

Airy functions and Riesz fractional trigonometric operators

ABSTRACTIn this paper, using the Riesz fractional derivative , we present the Mellin transforms of shifted Airy functions in terms of the Riesz fractional derivatives of Airy functions. We also show some integral addition formulas for the products of Airy functions in terms of the Riesz fractional trigonometric operators. Moreover, the fundamental solution of the Ostrovsky equation is given by the Riesz fractional trigonometric operators.

Fluctuations and a rigorous uncertainty relation of trigonometric operators of the phase and the number of photons of an electromagnetic field for general quantum superpositions of coherent states

The quantum-statistical properties of states of an electromagnetic field of general superpositions of coherent states of the form of N α,β(α⌤+e iξ β⌤ are investigated. Formulas for the fluctuations (variances) of Hermitian trigonometric phase field operators Ŝ ≡ cos φ, Ĉ ≡ sin φ (the so-called “Susskind–Glogower operators”) are found. Expressions for the rigorous uncertainty relations (Cauchy inequalities) for operators of the number of photons and trigonometric phase operators, as well as for operators Ŝ and Ĉ, are found and analyzed. The states of amplitude $${N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i\varphi }}\rangle + {e^{i\xi }}\left| {{{\sqrt {{n_\beta }e} }^{i\varphi }}\rangle } \right.} \right.} \right)$$ , φ = φα = φβ, and phase $${N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i{\varphi _\alpha }}}\rangle + {e^{i\xi }}\left| {{{\sqrt {ne} }^{i{\varphi _\beta }}}\rangle } \right.} \right.} \right)$$ , n = n α = n β, superpositions of coherent states are considered separately. The types of quantum superpositions of meso- and macroscales (n α, n β » 1) are found for which the sines and/or cosines of the phase of the field can be measured accurately, since, under certain conditions, the quantum fluctuations of these quantities are close to zero. A simultaneous accurate measurement of cosφ and sinφ is possible for amplitude superpositions, while an accurate measurement of one of these trigonometric phase functions is possible in the case of certain phase superpositions. Amplitude superpositions of coherent states with a vacuum state are quantum states of the field with a “maximum” level of the quantum uncertainty both in the case of a mesoscopic scale and in the case of a macroscopic scale of the field with an average number of photons n α/β ≈ 0, n β/α » 1.

A Demodulation-Based Technique for Robust Estimation of Single-Phase Grid Voltage Fundamental Parameters

This paper proposes a robust technique for the single-phase grid voltage fundamental amplitude, frequency, and phase angle estimation under distorted grid conditions. It is based on a demodulation method tuned at a fixed frequency. It does not have stability issue due to an open-loop structure, does not require real-time evaluation of trigonometric and inverse trigonometric functions, and also avoids the use of look-up table. It can provide accurate estimation of the single-phase grid voltage fundamental parameters under dc offset and harmonics. When compared with a frequency adaptive demodulation technique, the proposed one is less affected by dc offset, can provide faster frequency estimation, and also avoids interdependent loop, trigonometric, and inverse trigonometric functions operation. Simulation and experimental results are presented to verify the performance of the proposed technique.

On the Trigonometric Descriptions of Colors

<p class="1Body">An attempt is presented for the description of the spectral colors using the standard trigonometric tools in order to extract more information about photons. We have arranged the spectral colors on an arc of the circle with the radius R = 1 and the central angle θ = π/3 when we have defined cos (θ) = λ<sub>380</sub>/λ<sub>760</sub> = 0.5. Several trigonometric operations were applied in order to find the gravity centers for the scotopic, photopic, and mesopic visions. The concept of the center of gravity of colors introduced Isaac Newton. We have postulated properties of the long-lived photons with the new interpretation of the Hubble (Zwicky-Nernst) constant H<sub>0</sub> = 2.748… * 10<sup>-18</sup> kg kg<sup>-1 </sup>s<sup>-1</sup>, the specific mass evaporation rate (SMER) of gravitons from the source mass. The stability of international prototypes of kilogram has been regularly checked. We predict that those standard kilograms due to the evaporation of gravitons lost 8.67 μg kg<sup>-1</sup> century<sup>-1</sup>. The energy of long-lived photons was trigonometrically decomposed into three parts that could be experimentally tested: longitudinal energy, transverse energy and energy of evaporated gravitons. We tested the properties of the long-lived photons with the experimental data published for the best available standard candles: supernovae Type Ia. There was found a surprising match of those experimental data with the model of the long-lived photons. Finally, we have proposed a possible decomposition of the big G (Newtonian gravitational constant) and the small kappa κ (Einsteinian gravitational constant) in order to get a new insight into the mysterious gravitational force and/or the curvature concept.</p>

Low-complexity audio CODEC for resource-scarce embedded system with fixed-point arithmetic

This work presents a low-complexity audio coder-decoder (CODEC) based on fixed-point arithmetic to save the usage of system resources in a low-end embedded system. To reduce time complexity and memory usage, a simplified discrete wavelet transform (DWT) with only integer operations is developed. The simplified DWT reduces the computation that is required for the trigonometric operations in the fixed-point system. Moreover, separating the even and odd orders of wavelet coefficients halves the computation time of the original DWT. Additionally, a tri-mode zeroes recording algorithm (TZRA) is proposed. The proposed TZRA utilizes different encoding modes with their corresponding bit data structures to record the locations of zero wavelet coefficients, and determines the optimal encoding mode that uses the fewest bits. To further improve the compression performance, a dynamic wavelet level selecting algorithm is developed to decide the wavelet level for compressing each of frame. This algorithm can dynamically select the optimal wavelet level with the fewest bits by using the determined mode from the TZRA. The experimental results herein reveal that the encoder in the proposed CODEC can multiply computation by factors of 164 and 112 and reduce the size of the execution file by approximately 95.61 and 98.42% relative to baseline audio compression CODECs. Such results indicate that the efficiency when used in real consumer products with fixed-point arithmetic.

Design and Implementation of a Generic CORDIC Processor and its Application as a Waveform Generator

Background: With the advent in hand held mobile computing devices, the demand for high performance compact processors is increasing. In this work a processor is designed with hardwired instructions for elementary mathematical functions like sine, cosine, sinh, cosh, division and multiplication. Methods: The processor employs Coordinate Rotation Digital Computer (CORDIC) algorithm for efficient hardware implementation of the above mentioned instructions. The parallel and pipelined implementation of the processor is carried out. The pipelined processor is configured as waveform generator. The novelty of this work is the integration of both trigonometric and hyperbolic operations in the same processor. Findings: ASIC Implementation is carried out with 40nm technology libraries. The parallel processor so designed operates at maximum frequency of 24.23 MHz and pipelined processor operates at maximum frequency of 261.36 MHz. Conclusion: This increase in operating frequency is achieved at the cost of increased silicon area and optimal power dissipation. The waveform generator generates sine, cosine waves of 3.5 MHz and sine hyperbolic, cosine hyperbolic waves and exponential waves of 7.9 MHz. The limitation being the waveform generator generates waves of constant frequency. Additional circuit is required in generating waves of different frequencies.

New Trigonometric Basis Possessing Exponential Shape Parameters

Four new trigonometric Bernstein-like basis functions with two exponential shape pa- rameters are constructed, based on which a class of trigonometric Bezier-like curves, anal- ogous to the cubic Bezier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bezier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bezier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bezier-like curves are closer to the given control polygon than the cu- bic Bezier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape pa- rameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C 2 FC 3 continuous for a non-uniform knot vector, and C 3 or C 5 continuous for a uniform knot vector. A new class of trigonometric Bezier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bezier-like patch is developed. The conditions for G 1 continuous joining two trigonometric Bezier-like patches over triangular domain are deduced.

Design and Simulation of Quadrature Phase Detection in Electrical Capacitance Volume Tomography

Capacitance measurement accuracy is affected by phase conformity between signal and reference. This work describes phase detection scheme which is necessary for phase synchronization in tomography application. Core processing for calculating phase and amplitude of the detected signal was built on FPGA (Field-Programmable Gate-Array) platform. Phase shift demodulation algorithm employs IP core provided by Xilinx FPGA. Direct digital synthesizer (DDS), multiplier, accumulator, and CORDIC (coordinate rotation digital computer) modules were used as excitation-reference signal generator, signal multiplication, accumulation, and conversion to polar coordinate in order to conduct trigonometric operation respectively. Hardware design was emulated on MATLAB-Xilinx System Generator to observe its performance. Phase detection range 0-114.58 o and mean absolute error 0.58 o have been achieved. Data processing rate solely at digital signal stage was approximately 100data/s suitable for 32-channel electrical capacitance volume tomography (ECVT) system.

Xiangjiaba Reservoir Ecological Operation Preliminary Study Based on Genetic Algorithm

Traditional reservoir scheduling guarantees efficient utilization of water resources and maximizes hydropower station operation efficiency. However it doesn’t consider the surrounding ecological environment demand. Ecological operation which is based on Traditional reservoir scheduling makes the reservoir achieve long-term, sustainable operation and reduce adverse impacts on the ecological environment. Xiangjiaba reservoir is the ties of the downstream of jinsha river and the Yangtze river which is located in jinsha river basin ecological sensitive area. So the completion of the Xiangjiaba reservoir must result in unhealthy ecological environment impact. With traditional genetic algorithm (GA) encoding complexity and being easy to fall into local convergence limitation, put forward a kind of improved convergence of genetic algorithm, the improved genetic algorithm with real number coding and fitness function which is converted into a nonlinear trigonometric function selection operator. Select the adaptive crossover probability and mutation probability adjust, in order to improve the convergence of the algorithm. RVA method is used to calculate the IHA index upper and lower threshold value and the upper and lower threshold could value as the constraint conditions of ecological scheduling objective function model. The results show that the improved genetic algorithm converge makes the global optimal solution ability stronger and faster; RVA method is suitable for the jinshajiang river and results are more comprehensive and more reasonable.

Differential Evolution algorithms applied to Neural Network training suffer from stagnation

Large number of population-based Differential Evolution algorithms has been proposed in the literature. Their good performance is often reported for benchmark problems. However, when applied to Neural Networks training for regression, these methods usually perform poorer than classical Levenberg–Marquardt algorithm. The major aim of the present paper is to clarify, why? In this research, in which Neural Networks are used for a real-world regression problem, it is empirically shown that various Differential Evolution algorithms are falling into stagnation during Neural Network training. It means that after some time the individuals stop improving, or improve very occasionally, although the population diversity remains high. Similar behavior of Differential Evolution algorithms is observed for some, but not the majority of, benchmark problems. In the paper the impact of Differential Evolution population size, the initialization range and bounds on Neural Networks performance is also discussed.Among tested algorithms only the Differential Evolution with Global and Local neighborhood-based mutation operators performs better than the Levenberg–Marquardt algorithm for Neural Networks training. This version of Differential Evolution also shows the symptoms of stagnation, but much weaker than the other tested variants. To enhance exploitation in the final stage of Neural Networks training, it is proposed to merge the Differential Evolution with Global and Local neighborhood-based mutation operators algorithm with the Trigonometric mutation operator. This method does not rule out the stagnation problem, but slightly improves the performance of trained Neural Networks.

New Method of Givens Rotations for Triangularization of Square Matrices

This paper describes a new method of QR-decomposition of square nonsingular matrices (real or complex) by the Givens rotations through the unitary discrete heap transforms. This transforms can be defined by a different path, or the order of processing components of input data, which leads to different realizations of the QR-decomposition. The heap transforms are fast, because of a simple form of decomposition of their matrices. The direct calculation of the N-point discrete heap transform requires no more than 5(N-1) multiplications, 2(N-1) additions, plus 3(N-1) trigonometric operations. The QR-decomposition of the square matrix N × N uses about 4/3N3 multiplications and N(N-1)/2 square roots. This number varies depending on the path of the heap transform, and it is shown that “the optimal path” allows for significant reduction of number of operations in QR-decomposition. The heap transform and its matrix can be described analytically, and therefore, this transform can also be applied to the QR-decomposition of complex matrices.

Lp -general approximations by multivariate singular integral operators

Abstract In this article, we study the Lp , 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over RN , N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.

Adaptive Genetic Algorithm for Daily Optimal Operation of Cascade Reservoirs and its Improvement Strategy

For the specialty of cascade reservoirs optimization and the premature convergence of GA, several improvement strategies are presented in this paper. Firstly, solution space generation method is found application to generate feasible initial population. Secondly, chaos optimization is adopted to optimize initial population. Thirdly, new selective operators, trigonometric selective operators, are proposed to overcome the fitness requirement of non-negative and to maintain the diversity of population. Fourthly, adaptive probabilities of crossing and mutation are adopted in order to improve the convergence speed of GA. Besides, elitist strategy is used to ensure that the best individual can be remained in each generation. Furthermore, the performance of these proposed improvement strategies was checked against the historical improvement strategies by simulating optimal operation of Three Gorges cascade reservoirs premised on historical hourly inflows, and the comparison yields indications of superior performance. In these proposed improvement strategies, trigonometric selective operators are feasible and effective for optimizing operation of cascade reservoirs. These new selective operators could help GA to find a more excellent solution in the same algebra, and the performance of convergence speed is advanced. Adaptive probabilities of crossing and mutation have better performance than other improvement strategies, such as annealing chaotic mutation and simulated annealing of large probability of mutation, because this method realizes the twin goals of maintaining diversity in the population and advancing the convergence speed of GA.

Research on a New Digital Algorithm to Improve the Monitoring Accuracy of the Dielectric Loss Factor

Based on Hilbert transform and complex trigonometric operation, this paper presents a new digital algorithm to improve the monitoring accuracy of the dielectric loss factor, and the theoretical basis of the algorithm is also deduced. Simulation results show that, the new algorithm renders good application performance when there exists power frequency fluctuation or the sampled data length is not exactly an integer fold of a power cycle, which demonstrates the algorithm can effectively solve the issues incurred from asynchronous sampling.