Showing papers on "Inverse trigonometric functions published in 2005"
TL;DR: A new integral algorithm for the integral of trigonometric functions utilizing omnipotence expressions tan x2=t had advantages of simple idea, excellent property for operation and powerful competence and would be useful for some calculating problems of engineering mathematics.
42 citations
TL;DR: In this article, a new upper bound for inverse sine is established, and it is shown that 3 and π,6 andπ( √ 2 + 1 ) in Shafer-Fink inequalities are optimal.
Abstract: In this paper, a new upper bound for inverse sine is established.We would point out that the numbers, 3 and π ,6 andπ( √ 2 + 1 ), in Shafer-Fink inequalities, are optimal.
32 citations
TL;DR: In this paper, the authors obtained the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions by taking an appropriate limit, and obtained explicit formuals for these functions when the multiplicity functions are even and positive integer-valued.
Abstract: By taking an appropriate limit, we obtain the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions . A more general class of Bessel functions is also investigated, which we shall call the \Theta-Bessel functions. Explicit formuals for the \Theta-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown.
29 citations
15 citations
TL;DR: Four newly developed algebraic approximations of the inverse sine function, defined for the full [-1,+1] input range, are presented, which are particularly suited to the determination of mechanical and electrical angles in sensors and instrumentation applications.
Abstract: This paper presents four newly developed algebraic approximations of the inverse sine function, defined for the full [-1,+1] input range. These approximations contain few terms, and have numerical coefficients with low number of significant figures. The maximum absolute errors of the formulas range between 4.07/spl times/10/sup -4%/ and 5.64/spl times/10/sup -2%/ of the maximum value (i.e., /spl pi//2) returned by the arc sine function. These approximations are particularly suited to the determination of mechanical and electrical angles in sensors and instrumentation applications. One of the proposed expressions has been implemented using analog electronic circuitry. The converter built was successfully tested and characterized using both a PC-based test rig, and a commercial resolver. The theory, computer simulation and some experimental results are given.
14 citations
26 Jun 2005
TL;DR: A domain-theoretic version of the inverse function theorem for a Lipschitz function is developed, in which the inversefunction is obtained as a fixed point of a Scott continuous functional and is approximated by step functions.
Abstract: We construct a domain-theoretic calculus for Lipschitz and differentiate functions, which includes addition, subtraction and composition. We then develop a domain-theoretic version of the inverse function theorem for a Lipschitz function, in which the inverse function is obtained as a fixed point of a Scott continuous functional and is approximated by step functions. In the case of a C/sup 1/ function, the inverse and its derivative are obtained as the least fixed point of a single Scott continuous functional on the domain of differentiable functions and are approximated by two sequences of step functions, which are effectively computed from two increasing sequences of step functions respectively converging to the original function and its derivative. In this case, we also effectively obtain an increasing sequence of polynomial step functions whose lower and upper bounds converge in the C/sup 1/ norm to the inverse function. A similar result holds for implicit functions, which combined with the domain-theoretic model for computational geometry, provides a robust technique for construction of curves and surfaces.
12 citations
11 citations
20 Jan 2005
TL;DR: In this article, a phase cosine wrapping function was proposed for interference phase measurement of slow-changing process by using phase-shifting technology in the initial state, the background and amplitude of interference fringes can be obtained, and a novel phase unwrapping method based on cosine function is proposed.
Abstract: In the interference phase measurement of slow-changing process, by use of the phase-shifting technology in the initial state, the background and amplitude of interference fringes can be obtained, and a novel phase unwrapping method based on cosine function is proposed. The holograms of the changing process are recorded, and then the phase cosine functions can be obtained by removing the background and amplitude from the holograms. The arccosine functions of phase cosine functions, which are called phase cosine wrapping function in this paper, can be unwrapped by utilizing the additional normal orientation information. The experimental analyses show that the residual noise and the phase-shifting errors have great influence on the accuracy of unwrapped phase. The tangent wrapping phase can’t be filtered by traditional method due to the π phase jumps, and the existing phase unwrapping algorithms are very complex. The phase-shifting errors can only influence the positions of phase jump points in the tangent wrapping phase. It is difficulty to optimize the tangent wrapping phase further. Compared with tangent wrapping phase, the phase cosine wrapping function is consecutive and can be filtered, and the unwrapping process is easier than that of tangent wrapping phase. The influence of phase-shifting errors on phase cosine function is not only positions but values of the wave crest and wave trough. The more precise the phase-shifting is, the closer the values of cosine function to ±1 at wave crest and wave trough are. The Experiment results show that cosine unwrapping method has the equivalent precision with tangent unwrapping method.
7 citations
14 Feb 2005
TL;DR: This article put forward a new way to acquire the values of inverse tan function, the proportion of the second order Bessel function to the first order bessel function and theportion of the fourth order Bespoke function toThe second orderBessel function by using of the virtue of DSP, high speed of data processing.
Abstract: With the development of the technique of the digital signal processor, the using of digital signal processor in the open loop IFOG (interferometric fiber optic gyroscope) signal processing brings a great improvement of the performance of IFOG, especially in the technique of digital filtering, acquisition the values of arcsine and inverse tan function. The traditional way to get the values of inverse tan function is linear interpolation technique in digital signal circuit. This article put forward a new way to acquire the values of inverse tan function, the proportion of the second order Bessel function to the first order Bessel function and the proportion of the fourth order Bessel function to the second order Bessel function by using of the virtue of DSP, high speed of data processing. The fundamental, second and fourth harmonic signals from photoelectrical detector are sampled and demodulated by analog circuit. Then the signals from analog circuit are sampled into DSP by a 24-bit analog-to-digital converter, which is used to improve the dynamical range of the system. Then DSP computes the values mentioned above by polynomial approach. The simulation in Mathematic and Microsoft Visual C++ 6.0 proved that the method of polynomial approach to the functions has higher precision compared to the method of linear interpolation. Meanwhile the designed circuit shows it can meet the demand of real time data process. The tested results of designed circuit prove it has a great improvement in the dynamic range and precision of IFOG.
5 citations
04 Dec 2005
TL;DR: In this paper, the authors presented three new models: (1) Saleh function in traditional orthogonal bandpass model to increase accuracy, (2) inverse tangent function to model RF power amplifier's AM-AM nonlinear behavior, and (3) inverse Tangent function in conventional band pass model.
Abstract: By carefully analysis measured data of power amplifier, this paper presents three new models Model A uses Saleh function in traditional orthogonal bandpass model to increase accuracy Model B uses an inverse tangent function to model RF power amplifier's AM-AM nonlinear behavior Model C uses inverse tangent function in traditional orthogonal bandpass model to model power amplifier's AM-AM and AM-PM nonlinearity Emulation shows that each model has its features, and model C gives the best performance
14 Nov 2005
TL;DR: In this article, the authors presented an algorithm to calculate the phase angle of the fundamental component (60 Hz) of an electrical signal using the look up table (LUT) and the discrete Fourier transform (DFT).
Abstract: Synchrophasor is the acronym of synchronized phasor and it is a phasor measured and calculated by sampling with a standard temporal reference. The utility of a synchrophasor is to measure the angle of phase between two electrical signals placed in remote nodes. In the article is expressed the computational algorithm to calculate the phasor of the fundamental component (60 Hz) of an electrical signal. This algorithm is implemented with the discrete Fourier transform (DFT) and a look up table (LUT) which in junction lets diminish the process time. It is also presented an algorithm which uses the same technique to obtain the inverse trigonometric function that is required to calculate the phase angle. The full algorithm presented in this paper was implemented in a DSP-Microcontroller0 fixed point from C2000 family. The Standard temporal reference is generated with a signal of 1 pulse per second (1PPS) from a Global Positioning System (GPS).
Journal Article•
TL;DR: Based on differential evolution(DE) algorithm, a method of fast implementation of the arc tangent function with high accuracy using assembly in fixed-point DSP is proposed and the research result has been applied to a satellite navigation and positioning system.
Abstract: Based on differential evolution(DE) algorithm, a method of fast implementation of the arc tangent function with high accuracy using assembly in fixed-point DSP is proposed. The validity and real-time behavior of this method is tested in a large scale. Compared with traditional methods, the approximating polynomial calculated by DE is of high accuracy and low order, and the real-time behavior of the arc tangent function designed by DE is much better than C library function. Besides, the idea of DE can be adopted in the effective implementation of other non-linear mathematical functions in a real-time environment. The research result has been applied to a satellite navigation and positioning system.
Patent•
27 Apr 2005
TL;DR: In this paper, a disk drive having servo control functions which can detect true off-track positions with high precision by applying a phase difference detecting system using servo burst patterns is presented.
Abstract: PROBLEM TO BE SOLVED: To provide a disk drive having servo control functions which can detect true off-track positions with high precision by applying a phase difference detecting system using servo burst patterns. SOLUTION: In a disk drive which makes positioning control of the head based on the servo burst patterns recorded on a disk medium 10, the servo processor 20 generates a tangent equivalent value using the burst values obtained from the servo burst patterns. Further, the servo processor 20 executes inverse tangent transformation by the inverse function approximate expression built in advance based on the relations between the off-track positions and the tangent equivalent value, and executes servo processing to detect the off-track positions. COPYRIGHT: (C)2007,JPO&INPIT
Journal Article•
TL;DR: In this article, a special case of the problem of continuous mixtures of generalized arc-sine distributions over their parameters is considered, where the uniform mixture of generalized arcsine distributions is linked by transformation to the Cauchy distribution.
Abstract: A single, tractable, special case of the problem of continuous mixtures of beta distributions over their parameters is considered. This is the uniform mixture of generalized arc-sine distributions which, curiously, turns out to be linked by transformation to the Cauchy distribution.
TL;DR: In this article, the authors show that the three cotangents for a single triangle ABC are governed as follows: (i) only one may be negative, (ii) the sum of two cotangsents is positive, and (iii) since cotZ?cotC + cot C cot A+ cotAcotB = 1, the three COTangents are dependent.
Abstract: cotA(cot£> + cot/?) + cotB(cot/?+ cotP) + cotC(cotP+ cotQ) > 2 with equality if, and only if, P = A, Q = B, R = C. We first demonstrate the simpler result cotA + cotB + cotC > V5 ([1]), observing that the three cotangents for a single triangle ABC are governed as follows: (i) only one cotangent may be negative, (ii) the sum of two cotangents is positive, (iii) since cotZ?cotC + cot C cot A + cotAcotB = 1, the three cotangents are dependent. We have
01 Jul 2005
18 Aug 2005
TL;DR: In this work, FPGA-based functional units for Cosine, Arctangent, and the Square Root functions are designed using bipartite tables and iterative algorithms for hardware rectangular-to-polar and polar- to-rectangular conversion macro-functions.
Abstract: Today FPGAs are used in many digital signal processing applications. In order to design high-performance area efficient DSP pipelines various arithmetic functions and algorithms must be used. In this work, FPGA-based functional units for Cosine, Arctangent, and the Square Root functions are designed using bipartite tables and iterative algorithms. The bipartite tabular approach was four to 12 times faster than the iterative approach but requires 8-40 times more FPGA hardware resources to implement these functions. Next, these functions along with the FPGA hardware multipliers and a reciprocal bipartite table unit are used for hardware rectangular-to-polar and polar-to-rectangular conversion macro-functions. These macro-functions allow for a 7-10 times performance improvement for the high-performance pipelines or an area reduction of 9-17 times for the low cost implementations. In addition, software tool to design FPGA based DSP pipelines using the Cosine, Sine, Arctangent, Square Root, and Reciprocal units with the hardware multipliers is presented.
Patent•
09 Jun 2005
TL;DR: In this article, a phase difference calculating circuit for high precision phase control in which the table size of a memory circuit for storing an inverse trigonometric function and the circuit scale associated with that can be reduced is presented.
Abstract: PROBLEM TO BE SOLVED: To provide a phase difference calculating circuit for use in high precision phase control in which the table size of a memory circuit for storing an inverse trigonometric function and the circuit scale associated with that can be reduced SOLUTION: A sequential signal processing circuit 102 calculates a phase rotation amount w (t) by LMS algorithm, and a real part output circuit 103 calculates the real part W r (t) of the w (t) An ROM 104 searches a phase θ based on the real part W r (t) and a table of arc cosine inverse trigonometric function storing 0-π rad A code determination circuit 105 determines the code data of the imaginary part w i (t) of the w (t) and a phase angle correcting circuit 106 corrects the phase θ based on the code data thus obtaining a desired phase difference θ r COPYRIGHT: (C)2005,JPO&NCIPI
19 Jun 2005
TL;DR: A new model of inverse sine function, more effective than the existing model in precision, simplicity, robustness, and the performance of an integrated wide-band FSK demodulator using the ISPD is described.
Abstract: The PLL ISPD is a new phase locked loop without using any filters, basis of an inverse sine phase detector (ISPD). In this paper, we proposed a new model of inverse sine function, more effective than the existing model in precision, simplicity, robustness. Besides, this model can be expressed to the order 1 or the order 3 according to the precision arranged by the foundry. The detail of the basic analysis and mathematical model of the ISPD PLL are presented and compared with the existing model. The digital design of the system involves the use of the SIMPLORER simulator for the hardware description language VHDL-AMS. We described the performance of an integrated wide-band FSK demodulator using the ISPD contrived to improve the characteristics of the system such as large bandwidth, high frequency, a wide keep range and seizure range, and a perfect reconstruction of an input signal.
TL;DR: In their haste to solve a particular problem, students (and their instructors) miss an opportunity to notice some interesting mathematics as mentioned in this paper, and they frequently run across a classic problem that goes something like this: There is a 6-foot tall picture on a wall, 2 feet above your eye level.
Abstract: There are times when, in their haste to solve a particular problem, students (and their instructors) miss an opportunity to notice some interesting mathematics. For example, when calculus students are introduced to the derivatives of inverse trigonometric functions, they frequently run across a classic problem that goes something like this: There is a 6-foot tall picture on a wall, 2 feet above your eye level. How far away should you sit (on the level floor) in order to maximize the vertical viewing angle 0? (See FIGURE 1.)
01 Jan 2005
TL;DR: In this article, the authors describe definitions of inverse hyperbolic functions and their main properties, as well as some addition formulas with hyper bolic functions, and their properties and properties.
Abstract: Summary. This article describes definitions of inverse hyperbolic functions and their main properties, as well as some addition formulas with hyperbolic functions.
TL;DR: In this paper, the cosine and sine operators in the Fock basis were introduced and their eigenfunctions were related in a similar way with arbitrary orthogonal polynomial sets on the intervall.
Abstract: The quantization of phase is still an open problem. In the approach of Susskind and Glogower so called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related with the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way with arbitrary orthogonal polynomial sets on the intervall [-1,1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We consider also the inverse arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states.
TL;DR: In this article, the authors consider the problem of finding a function f (x) such that the given trigonometric series is its Fourier series (then they can find the sum of the given series).
Abstract: 00 B. Given a trigonometric series Ek=l (Ak cos kx + Bk sin kx), find a function f (x) such that the given trigonometric series is its Fourier series (then we can find the sum of the given trigonometric series). Upon learning the theory of Fourier series, we know questions of type A are straightforward but questions of type B are not, for a given trigonometric series may not even be a Fourier series, as
TL;DR: In this article, the partial fractions decomposition of rational functions is used to integrate new combinations of trigonometric functions, which enable the integration of rational and non-rational functions.
Abstract: Parametric differentiation is used to derive the partial fractions decompositions of certain rational functions. Those decompositions enable us to integrate some new combinations of trigonometric functions.
01 Jan 2005
TL;DR: In this paper, the authors give several differentiable formulas of special functions, including rational functions, irrational functions, trigonometric functions, exponential functions, and logarithmic functions.
Abstract: Summary. In this article, we give several differentiable formulas of special functions. There are some specific composite functions consisting of rational functions, irrational functions, trigonometric functions, exponential functions or logarithmic functions.
TL;DR: In this paper, three major techniques are employed to calculate π: perimeter of polygons inscribed or circumscribed in a circle, calculus based methods using integral representations of inverse trigonometric functions, and modular identities derived from the transformation theory of elliptic integrals.
Abstract: Three major techniques are employed to calculate π. Namely, (i) the perimeter of polygons inscribed or circumscribed in a circle, (ii) calculus based methods using integral representations of inverse trigonometric functions, and (iii) modular identities derived from the transformation theory of elliptic integrals. This note presents a combination of the first two procedures, which allows the derivation of a family of series that may exhibit very fast convergence rates. The geometrical interpretation gives good insight into the acceleration method that is being implemented.