Showing papers on "Inverse trigonometric functions published in 1999"

Phase-evaluation methods in whole-field optical measurement techniques

Abstract: Many optical measurement techniques provide fringe patterns as their results. The decodification processes that employ one or several fringe patterns to automatically retrieve the phase are generally designated as phase-evaluation methods. In this work, an overview of these methods will be schematically presented. Their particular performances will be compared, stressing their main advantages and drawbacks. An important group of these methods employs the well-known phase-shifting algorithms as a tool for calculating the phase. In the general form of these algorithms, the principal value of the optical phase is computed by an inverse trigonometric function whose argument is a combination of phase-shifted intensity values, provided by the modulation of one or several fringe patterns. These algorithms will be also studied in the general context of the phase-evaluation methods.

Hardware implementation of a nonlinear processor

Abstract: Several advanced DSP algorithms, arising in applications such as wireless communications, computer graphics, computerized tomography, and speech compression, require extensive use of nonlinear functions. We discuss a new hardware approach to high-speed computation of nonlinear functions. With this approach all of the functions needed can be regularized into a single efficient algorithm. Further, highly reduced cycle implementations can be achieved. Specifically, for real arguments, a new result can be produced every cycle-in a pipelined mode. The underlying principle which has made the combined goals of high-speed and multi-functionality possible is significance-based polynomial interpolation of very small ROM tables. Considered are the following seven functions: arctangent, cosine, logarithm, reciprocal, reciprocal-square-root, sine, and square-root. Also presented is a theoretical development for error prediction, a tool for the selection of architectural parameters. Finally, the paper presents a novel technique, named here as 'microshaping', for avoiding overflows, thereby eliminating exception handling.

Algorithm (Method) and VLSI architecture for fast evaluation of trigonometric functions

Abstract: A fast CORDIC algorithm and the resulting VLSI architecture for the evaluation of trigonometric functions are disclosed. The new method employs signed digits to represent intermediate operands and requires a constant scaling or normalization factor which can be pre-computed and made available in read-only hardware for any desired target precision (i.e., word length). The speedup is achieved by performing CORDIC iterations in parallel in two separate modules. Each module executes a “double step” or two basic CORDIC rotations at every iteration cycle. Two angles arctan 2−2i and arctan 2−(2i+1) are used in each step i of the method. As a result, approximately {fraction (n/2)} steps (exactly ⌈ n + 3 2 ⌉ steps) are required to evaluate sine/cosine of n bit input argument up to n bits of accuracy. The VLSI architecture consists of two “zeroing” and two “rotator” modules, each consisting of signed digit adders, latches, shifters, etc. A novel decision block controls the iterations. The new ROM look-up table stores ⌈ n + 3 2 ⌉ values, each accurate to n+3 bits. For every pair of consecutive angles arctan 2−2i and arctan 2−(2i+1) for i=0, 1, 2, . . . , ⌈ n + 3 2 ⌉ only their sum and difference need be stored for the inventive double step branching CORDIC method. The result is a very fast VLSI architecture whose speedup to hardware-overhead ratio is close to 1. The double stepping method can be easily extended to the evaluation of inverse trigonometric, exponential as well as logarithm functions.

Schaum's Outline of Calculus

Abstract: Linear Coordinate Systems. Absolute Value. Inequalities. Rectangular Coordinate Systems. Lines. Circles. Equations and Their Graphs. Functions. Limits. Continuity. The Derivative. Rules for Differentiating Functions. Implicit Differentiation. Tangent and Normal Lines. Law of the Mean. Increasing and Decreasing Functions. Maximum and Minimum Values. Curve Sketching. Concavity. Symmetry. Review of Trigonometry. Differentiation of Trigonometric Functions. Inverse Trigonometric Functions. Rectilinear and Circular Motion. Related Rates. Differentials. Newton's Method. Antiderivatives. The Definite Integral. Area Under a Curve.The Fundamental Theorem of Calculus. The Natural Logarithm. Exponential and Logarithmic Functions. L'Hopital's Rule. Exponential Growth and Decay. Applications of Integration I: Area and Arc Length. Applications of Integration II: Volume. Techniques of Integration I: Integration by Parts. Techniques of Integration II: Trigonometric Integrands and Trigonometric Substitutions. Techniques of Integration III: Integration by Partial Fractions. Miscellaneous Substitutions. Improper Integrals. Applications of Integration II: Area of A Surface of Revolution. Parametric Representation of Curves.Curvature. Plane Vectors. Curvilinear Motion. Polar Coordinates.Infinite Sequences. Infinite Series. Series with Positive Terms. The Integral Test. Comparison Tests.Alternating Series. Absolute and Conditional Convergence. The Ratio Test. Power Series. Taylor and Maclaurin Series. Taylor's Formual with Remainder. Partial Derivatives.Total Differential. Differentiability. Chain Rules. Space Vectors.Surface and Curves in Space. Directional Derivatives. Maximum and Minimum Values. Vector Differentiation and Integration. Double and Iterated Integrals. Centroids and Moments of Inertia of Plane Areas.Double Integration Applied to Volume Under a Surface and the Area of A Curved Surface. Triple Integrals. Masses of Variable Density.Differential Equations of First and Second Order.

Trigonometric approximations for some Bessel functions

Abstract: Formulas are obtained for approximating the tabulated Bessel functions Jn(x), n = 0–9 in terms of trigonometric functions. These formulas can be easily integrated and differentiated and are convenient for personal computers and pocket calculators.

Joint cosine transforming and quantizing device and joint inverse quantizing and inverse cosine transforming device

Abstract: Each of the discrete cosine transforming (DCT) unit of a joint cosine transforming and quantizing device, and the inverse discrete cosine transforming (IDCT) unit of a joint inverse quantizing and inverse cosine transforming device includes a multiplication operation unit that is implemented in a look-up table device having a look-up table and an output multiplexer. The look-up table has a number of data fields, each of which contains products associated with a respective one of a corresponding number of coefficients used in multiplication-involving stages of a DCT or IDCT fast algorithm. The look-up table further has a number of outputs corresponding respectively to the data fields, and can be addressed in order to output the products, that correspond to the address data and that are stored in the data fields, at the outputs of the look-up table. The output multiplexer has data inputs connected to the outputs of the look-up table, and is operable so as to select one of the data inputs thereof and provide data at the selected one of the data inputs thereof to a data output of the output multiplexer.

Apparatus and method for implementing an inverse arctangent function using piecewise linear theorem to simplify

Abstract: The present invention discloses an apparatus and method for implementing an inverse arctangent function using piecewise linear theorem to simplify and used to transform a right-angled coordinate point X and Y to a phase angle θ of an inverse arctangent function. The present invention uses T-line combination to approach an inverse arctangent function between 0 degree to 45 degree, and determines which piecewise linear region the right-angled coordinate point is located at. A coefficient table stored in a memory is used to compute {circumflex over (θ)} which is the approximate value of the phase angle θ. The phase angle {circumflex over (θ)} between 45 degree to 360 degree can be mapped to the approximate phase angle {circumflex over (θ)} between 0 degree to 45 degree using linear combination.

On the bestm-term trigonometric and orthogonal trigonometric approximations of functions from the classesL Ψ β,ρ

Abstract: We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classesL Ψ β,ρ of functions of one variable in the spaceL q in the case 2

Baud timing synchronization system

Abstract: PROBLEM TO BE SOLVED: To accelerate estimating of an interpolation phase in a demodulation circuit, which acquires the identifying symbol value synchronous with the baud timing from the received signal that is asynchronously oversampled by the interpolation. SOLUTION: The output of a delay detection part 14', where a preamble series is received and the correlative arithmetic value set between the output series of a sinusoidal wave generation ROM 133a and a cosine wave generation ROM 133b are accumulated, and a sine wave component (A) and a cosine wave component (B) are outputted. A phase estimation means 133k accesses an inverse tangent ROM 133m, based on the ratio set between both components (A) and (B) and outputs the variable selection data, which decide the structure of an interpolation expression and the phase data. An interpolation coefficient ROM 134 outputs the coefficient data which are designated by the phase data to an interpolation part. An interpolation part constitutes a temporal change coefficient filter by the use of the coefficient data and the sample value which is designated by the variable selection data.

Surfaces singuli eres de fonctions m eromorphes

Abstract: Every non-constant meromorphic function in the complex plane has branches of the inverse that are denied in spherical discs of radii arbitrarily close to arctan p 8 7032 0 . This constant is best possible. The proof depends on the study of singular surfaces associated with meromorphic functions.

Trigonometric approximations for some bessel functions

Abstract: Formulas are obtained for approximating the tabulated Bessel functions Jn(x), n 0-9 in terms of trigonometric functions. These formulas can be easily integrated and differentiated and are convenient for personal computers and pocket calculators.

Automatic frequency control system with improved argument approximation

Abstract: An argument approximation method is used to estimate the frequency offset for an automatic frequency control (AFC) circuit. An angle of rotation can be calculated as arctan(Q/I), where Q is the imaginary component of the rotation and I is the real component of the angle. Rather than use the arctangent calculation, which is resource intensive, to calculate the angle of rotation, the angle of rotation is calculated as the sine of the angle of rotation, which is easily calculated from Q and I, and the sine approximation is offset by a correction factor of 1−cos (θ) in order to reduce approximation errors.

Decoding method and its device

Abstract: PROBLEM TO BE SOLVED: To reduce a calculation time required for decoding an image while keeping a high image quality by correcting processes of inverse cosine transform and inverse weighting so as to process only image information used for display SOLUTION: Applications of an inverse weighting(IW) 135 and an inverse cosine transform (IDCT) 140 are combined to obtain one function As a means of decoding an image of 1/4 size, a horizontal 4-point IW/ICDT processing is applied to coefficients up to a 4th column from a side of a first 4 rows corresponding to lower coefficients in a 8×8 matrix of transformation coefficients being part of a video frame A vertical 8-point IW/IDCT processing is applied to all coefficients of first 4 columns where the coefficients up to a 4th row from a side of each column corresponding to a high frequency are set to zero In the case that an input block consists of two 4×8 matrices consisting of sum and difference coefficients, the horizontal 4-point processing is applied to each row of each matrix and the vertical 4-point processing is applies to the sum of sum coefficients and difference coefficients

A unified systolic array algorithm for discrete inverse cosine and sine transforms

Abstract: This paper presents a new design approach for the inverse discrete sine and cosine transforms based on a new unified systolic array algorithm that can be used to obtain VLSI array architectures with very small differences in the hardware structure for the two transforms and improved performances. This new algorithm is based on using highly regular and modular computational structures as circular correlation and can be mapped into linear systolic arrays with a small number of I/O channels and low I/O bandwidth.

On the λ-means of some trigonometric series

Abstract: For functions λ differentiable on (0, 1) there are examined the means (2) of some trigonometric series (1). Section 2 contains a few basic estimates. In Section 3 the author presents two main theorems, which correspond to Propositions 1, 2 of [3].