Showing papers on "Inverse trigonometric functions published in 2015"

A Robust Technique for Single-Phase Grid Voltage Fundamental and Harmonic Parameter Estimation

Abstract: This paper reports a frequency-adaptive robust technique for the accurate estimation of the single-phase grid voltage fundamental and harmonic parameters. The technique is based on the discrete Fourier transform and a cascaded delayed signal cancellation strategy. There is no stability issue in the technique, since it does not contain any type of feedback loop. It can also be flexibly configured to estimate the parameters of the fundamental and/or one/multiple harmonic(s) from the grid voltage waveform distorted by various harmonics. Moreover, it does not require evaluation of trigonometric and inverse trigonometric functions for implementing on real-time digital signal processor. However, it needs computationally demanding high-order finite-impulse-response filters. The simulation and real-time experimental results are provided to verify the performance of the proposed technique.

Hardware Implementations of Fixed-Point Atan2

Abstract: The atan2 function computes the polar angle arctan(y/x) of a point given by its cartesian coordinates. It is widely used in digital signal processing to recover the phase of a signal. This article studies for this context the implementation of atan2 with fixed-point inputs and outputs. It compares the prevalent CORDIC shift-and-add algorithm to two multiplier-based techniques. The first one computes the bivariate atan2 function as the composition of two univariate functions: the reciprocal, and the arctangent, each evaluated using bipartite or polynomial approximation methods. The second technique directly uses piecewise bivariate polynomial approximations of degree 1 or 2. Each of these approaches requires a relevant argument reduction, which is also discussed. All the algorithms are last-bit accurate, and implemented with similar care in the open-source FloPoCo framework. Based on synthesis results on FPGAs, their relevance domains are discussed.

Turán type inequalities for generalized inverse trigonometric functions

Abstract: In this paper we study the inverse of the eigenfunction sinp of the one-dimensional p-Laplace operator and its dependence on the parameter p, and we present a Turan type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Turan type inequality for a series considered by Ramanujan, involving the digamma function

Convexity properties of generalized trigonometric and hyperbolic functions

Abstract: We study the power mean inequality for generalized trigonometric and hyperbolic functions with two parameters. The generalized p-trigonometric and (p, q)-trigonometric functions were introduced by Lindqvist and Takeuchi, respectively.

A proof of two conjectures of Chao-Ping Chen for inverse trigonometric functions

Abstract: In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873], using a method for proving inequalities of mixed trigonometric polynomial functions.

A trigonometric interval method for dynamic response analysis of uncertain nonlinear systems

Abstract: This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing approximation interval methods. We consider trigonometric approximation polynomials of three types: both cosine and sine functions, the sine function, and the cosine function. Thus, special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results. The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.

Inverse limits of families of set-valued functions

Abstract: In this paper, we investigate the inverse limits of two related parameterized families of upper semi-continuous set-valued functions. We include a theorem one consequence of which is that certain inverse limits with a single bonding function from one of these families are the closure of a topological ray (usually with indecomposable remainder). Also included is a theorem giving a new sufficient condition that an inverse limit with set-valued functions be an indecomposable continuum. It is shown that some, but not all, functions from these families produce chainable continua. This expands the list of examples of chainable continua produced by set-valued functions that are not mappings. The paper includes theorems on constructing subcontinua of inverse limits as well as theorems on expressing inverse limits with set-valued functions as inverse limits with mappings.

Inverse tangent based switching type reaching law for discrete time sliding mode control systems

Abstract: In this work the reaching law approach to the sliding mode control of discrete time systems is considered. We propose a reaching law, in which the convergence of the sliding variable to the vicinity of zero is governed by the inverse tangent function. First we analyze the case of the unperturbed system, and then we consider a second scenario with unknown disturbances and parameter uncertainties. We demonstrate, that for both cases the presented reaching law guarantees the quasi sliding motion of the representative point defined as crossing the sliding hyperplane in each successive control period while remaining inside an a priori known band around the hyperplane. When compared to the most popular, constant plus proportional reaching law, the proposed solution offers better robustness and faster convergence.

Oscillation criterion for discrete trigonometric systems

Abstract: In this paper, we investigate oscillation properties of discrete trigonometric systems whose coefficients matrices are simultaneously symplectic and orthogonal. The main result generalizes a necessary and sufficient condition of non-oscillation of trigonometric systems proved by M. Bohner and O. Doslý (J. Differential Equations 163 (2000), pp. 113–129) in the case when the block in the upper right corner of the coefficient matrix is symmetric and positive definite. Now, we present this oscillation criterion for an arbitrary trigonometric system. The obtained results are applied to formulate a necessary and sufficient condition for non-oscillation of even-order Sturm–Liouville difference equations.

Implementation on FPGA for CORDIC-based Computation of Arcsine and Arccosine

Abstract: This paper concentrates on the hardware implementation of CORDIC algorithm for computing inverse trigonometric functions like arcsine and arccosine. We improve the existing algorithm by changing the initial rotating vector of the iterations and modifying the judging condition of rotation direction. Due to the improvement, two iterations are saved and the drawback of the previous algorithm is corrected. In contrasting with the previous implementation, the improved algorithm consumes less hardware resources and its computing results are more accuracy.

Generalizing trigonometric functions from different points of view

Abstract: In this survey we shall explore (one definition of) generalized trigonometric functions from different standpoints and illustrate the roles they play in various branches of mathematics. We start from the analytic point of view and for each p is an element of (1, infinity) introduce a function sin(p)(-1) by an integral formula, which is just an extension of the well known integral representation of arcsin, and then use it to define generalized sine, cosine and tangent functions (labelled sin(p), cos(p) and tan(p) respectively). Numerous properties of these functions, such as an identity of Pythagorean type, are exhibited. Then we consider the unit circle in R-2 with the l(p) norm and define generalized trigonometric functions as is done in the standard case when the 12 norm is used. We show that these functions coincide with those introduced earlier. In the third section we consider the integral operator T : L-P (I) -> L-P (I) given by T f(x) = integral(x)(0) f(t)dt, where I = (0,1), and look at the problem of finding an extremal function (an element of the unit sphere of L-P(I) at which the norm of T is attained). It turns out that the extremal functions are given by cos(p). The following section deals with the Dirichlet eigenvalue problem for the p-Laplacian on a bounded interval: all eigenfunctions are expressible by means of sin(p) functions, which corresponds exactly to the classical situation when p = 2. After establishing a connection with approximation theory, we conclude with a review of other ways in which the classical trigonometric functions have been generalized. In the literature a variety of different definitions of generalized trigonometric functions can be found (see [1], [14], [15], [8]): all extend the classical functions and preserve some of their properties. It becomes clear that no single definition preserves all the classical properties and that which definition is adopted depends on the applications envisaged. Our focus on a particular choice reflects our research interests. For an earlier survey (in Czech) see [6].

On the asymptotics of cosine series in several variables with power coefficients

Abstract: The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G.H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to find the asymptotics of trigonometric series in several variables with the terms, having a form of “one minus the cosine” accurate to a decreasing power factor.

On trigonometric functions and norm in the generalized taxicab metric

Abstract: In this paper, the plane with the generalized taxicab metric is considered and the trigonometric functions and the norm are dened. Then, the cosine and the sine functions are developed by using the reference angle with respect to the generalized taxicab metric. It is shown that Schwarz's inequality is valid under restricted case of dTg -metric. Finally, the area of any triangle in the plane with the generalized taxicab metric is given as the geometrical interpretation that contains the trigonometric functions and norm.

QR-decomposition architecture based on two-variable numeric function approximation

Abstract: This paper presents a new approach for hardware-based QR-decomposition using an efficient computation scheme of the Givens-Rotation. In detail, the angle of rotation and its application to the Givens-Matrix are processed in a direct, straightforward manner. High-performance signal processing is achieved by piecewise approximation of the arctangent and sine function. In order to identify appropriate function approximations, several designs with varying constraints are automatically generated and analyzed. Physical and logical synthesis is performed in a 130 nm CMOS-technology. The application of our proposal in a multi-antenna mobile communication scenario highlights our work to be very efficient in terms of calculation accuracy and computation performance.

Generating and Mathematical Modelling of Discontinuous Impulse Waveforms

Abstract: Paper deals with mathematical modelling of impulse waveforms and impulse switching functions used in electrical engineering Impulse rectangular waveforms are created by periodical trigonometric functions with modulo π, so, the waveforms are discontinuous and strongly non-harmonic ones Impulse switching functions are investigated using directand inverse z-transformation The results make it possible to present those functions as infinite series expressed in pure numerical-, exponentialor trigonometric forms Theoretical derived waveforms are compared with simulation worked-out results Keywords-impulse switching function; Z-transform; inverse Z-transform; modelling and simulation; steady state operation; dynamical state model I MATHEMATICAL MODELLING OF NON-HARMONIC PERIODICAL DISCONTINUOUS FUNCTIONS It is known that periodical non-harmonic discontinuous function is possible to portray in compact closed form using Fourier infinite series [1]-[2] It yields for rectangular waveform, Figure 1b,c 4 1 2 1 sin 2 1 or 1 4 1 2 1 cos 2 1 , where t is the time, n 0, 1, 2, 3, F F 1 e T where F s is Laplace image of zero period; F s image of total Laplace function (t 0); T time period Inverse transform defined in complex form as 1 2 i F 1 e T d where i is imaginary unit √ 1, it is not so easy particularly for higher order systems Classical solution leads to results in Fourier series form, otherwise the Heaviside calculus is to be used One of the lesser known methods is using of FischerTurbar definition of arc tan for the main value ; based on a standardization of trigonometric function modulo π [3]-[4] So, increasing saw-tooth function with angular frequency ω can be expressed in closed form 2 arctan sin 1 cos , 1 as can be seen Figure 1a Similarly, for decreasing saw-tooth waveform 2 arctan sin 1 cos 2 Using addition (1) and (2) one obtains a rectangular waveform, Figure 1b 2 arctan sin 1 cos arctan sin 1 cos 3 The saw function (1) converges at t T 2 to 0 value, which is the half of the sum of the left limit and the right one, and it’s also valid for rectangular function (3), [6] Similarly, we can check the convergence of rectangular function at t π 2ω It yields [7] 2 4 1 2 1 sin 2 1 2 1 International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2015) © 2015 The authors Published by Atlantis Press 330

CAMs and high speed high precision data for trigonometric functions

Abstract: When high performance is required, the needed hardware implementation of trigonometric functions becomes often problematic. This paper generalizes and improves a CAM based arctangent architecture that has shown an exclusive appropriateness for some critical applications compared to the Look up Table based solution, the polynomial and the rational approximations. For more illustration, detailed design specifications and different sinus function implementation results are given.

Numerical Solution of second order hyperbolic telegraph equation via new Cubic Trigonometric B-Splines Approach

Abstract: This paper presents a new approach and methodology to solve the second order one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual finite difference scheme is used to discretize the time derivative. The cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, with a weighted scheme. The scheme is shown to be unconditionally stable for a range of values using the von Neumann (Fourier) method. Several test problems are presented to confirm the accuracy of the new scheme and to show the performance of trigonometric basis functions. The proposed scheme is also computationally economical and can be used to solve complex problems. The numerical results are found to be in good agreement with known exact solutions and also with earlier studies.

Rotating magnetic field sensor

Abstract: PROBLEM TO BE SOLVED: To reduce an error caused by the occurrence of magnetic anisotropy in a magnetic detection element.SOLUTION: A rotating magnetic sensor 1 includes detection circuits 10, 20, and 30 for generating signals S1, S2, and S3 and arithmetic circuits 61, 62, 63, and 64. The detection circuits include magnetic detection elements, and the signals S1 and S2 are set so that they are mutually different in modes of variation due to magnetic anisotropy. The third arithmetic circuit subjects the signals S1 and S2 to inverse trigonometric function computation to output a first inverse trigonometric function computation signal and a second inverse trigonometric function computation signal. The fourth arithmetic circuit computes a difference between the first and second inverse trigonometric function computation signals obtained at a set time and updates an amplitude variation term in an angular variation term based on first magnetic anisotropy on the basis of a maximum value and a minimum value of the difference. The first arithmetic circuit obtains correction information. The second arithmetic circuit uses the first signal and the correction information to computer a corrected detection value.

Estimates of the best m -term trigonometric approximations of classes of analytic functions

Abstract: In metric of spaces $L_{s}, \\ 1\\leq s\\leq\\infty$, we obtain exact in order estimates of best $m$-term trigonometric approximations of classes of convolutions of periodic functions, that belong to unit all of space $L_{p}, \\ 1\\leq p\\leq\\infty$, with generated kernel $\\Psi_{\\beta}(t)=\\sum\\limits_{k=1}^{\\infty}\\psi(k)\\cos(kt-\\frac{\\beta\\pi}{2})$, $\\beta\\in \\mathbb{R}$, whose coefficients $\\psi(k)$ tend to zero not slower than geometric progression. Obtained estimates coincide in order with approximation by Fourier sums of the given classes of functions in $L_{s}$-metric. This fact allows to write down exact order estimates of best orthogonal trigonometric approximation and trigonometric widths of given classes.

A Note on Integration of Trigonometric Functions

Abstract: In this paper, we derive equivalent equations of integration of secant and cosecant functions. Furthermore, we derive reduction formula for integration of product of integer powers of cosine and sine functions.

The Generalized Inverse of Distribution Functions

Abstract: The generalized inverse will be introduced for distribution functions of arbitrary real-valued random variables. The generalized inverse exists even if the ordinary inverse does not and they are identical if both exist. Further basic properties of the generalized inverse and connections to order relations are discussed. Also, the generalized inverse can be obtained from a sequence of ordinary inverses of suitable distribution functions; sufficient conditions therefore are stated.

A Method of Proving a Class of Trigonometric Inequalities Based on Approximations of the Sine and Cosine Functions by Maclaurin Polynomials

Abstract: In this article we consider a method of proving a class of inequalities of the form (1). The method is based on the precise approximations of the sine and cosine functions by Maclaurin polynomials of given order. By using this method we present new proofs of inequalities from the papers [7] and [10].

Electronic system for synthesising 3d surfaces

Abstract: The present invention relates to an electronic system that makes it possible to synthesise virtual 3D surfaces. Said system has a modular structure in which each module is made up of a set of mathematical operators. Said operators are used to implement an analytic expression that represents an implicit equation. The points that form the 3D surface to be synthesised are actually the solution points of said implicit equation. The operators implemented in said system make it possible to calculate polynomials with real exponents, direct trigonometric functions and inverse trigonometric functions. A very precise combination of all said operators has made it possible to synthesise a large variety of relatively complex 3D surfaces with the option of parametrising the deformations thereof.

An Improved 3D Location Algorithm Model Research Based on Directional TD-MUSIC

Abstract: In this paper, a TD-MUSIC spectrum estimation model based on directional antenna is proposed The model contains three directional antenna arrays with uniform line, by which the TD-MUSIC spectrum peak was calculated and analyzed The linear relationship between the amplitude of the signal source and the pitch angle was calculated by inverse trigonometric function The simulation results show that the pitch angle estimation of the signal source can be obtained by the improved 3D localization algorithm model fast and accurate

On Some Definite Integrals from Ramanujan's Notebooks

Abstract: Some definite integrals involving powers of inverse trigonometric functions that are found in the chapter 9 of Ramanujan notebooks (Part-I) are evaluated through elementary methods. Further, we present some identities connecting integrals with some infinite series and infinite products.