Showing papers on "Inverse trigonometric functions published in 1976"
TL;DR: In this paper, the authors compare several transformations in terms of how close the actual variance of the transformed frequency corresponds to its nominal value, and suggest that the traditional Grewal-Smith inverse sine transformation not be used, and several alternatives are considered.
Abstract: In recent years osteologists have frequently used non-metric (dichotomous) cranial data to measure biological distance between skeletal samples of Homo sapiens. Applying methods used earlier by biologists, these workers begin by attempting to stabilize the variance of the measures used by transforming the observed trait frequencies using some type of inverse sine transformation. The frequently used Grewal-Smith transformation doesn't work well for small samples of the size often considered by osteologists. As a consequence the mean measure of divergence between populations determined by this method is strongly influenced by a bias which depends on sample size. This paper compares several transformations in terms of how close the actual variance of the transformed frequency corresponds to its nominal value. It is suggested that the traditional (Grewal-Smith) inverse sine transformation not be used, and several alternatives are considered.
144 citations
TL;DR: It is shown theoretically that the two-pair quantization scheme has a 10-bit superiority over other above-mentioned quantization schemes in the sense of theoretically assuring that a maximum overall log spectral deviation will not be exceeded.
Abstract: The topic of quantization and bit allocation in speech processing is studied using an L 2 norm. Closed-form expressions are derived for the root mean square (rms) spectral deviation due to variations in one, two, or multiple parameters. For one-parameter variation, the reflection coefficients, log area ratios, and inverse sine coefficients are studied. It is shown that, depending upon the criterion chosen, either log area ratios or inverse sine quantization can be viewed as optimal. From a practical point of view, it is shown experimentally that very little difference exists among the various quantization methods beyond the second coefficient. Two-parameter variations are studied in terms of formant frequency and bandwidth movement and in terms of a two-pair quantization scheme. A lower bound on the number of quantization levels required to satisfy a given maximum spectral deviation is derived along with the two-pair quantization scheme which approximately satisfies the bound. It is shown theoretically that the two-pair quantization scheme has a 10-bit superiority over other above-mentioned quantization schemes in the sense of theoretically assuring that a maximum overall log spectral deviation will not be exceeded.
117 citations
TL;DR: This paper contains a discussion of the properties of this one-condition g-inverse, an APL program for computing the inverse, and some examples demonstrating the use of the program.
Abstract: Rao (1962) suggested a method for calculating a generalized inverse of an arbitrary matrix and used the method to represent a solution to the set of normal equations obtained in the theory of least squares. This method is similar to that used for calculating the regular inverse of a non-singular matrix (namely, the sweep-out method) developed by Gauss. The resulting inverse is a one-condition g-inverse and possesses many properties which make its use more desirable than other proposed g-inverses.This paper contains a discussion of the properties of this one-condition g-inverse, an APL program for computing the inverse, and some examples demonstrating the use of the program. Finally we show that the four-condition unique Moore-Penrose inverse readily follows by the use of this program.
52 citations
TL;DR: In this paper, the application of matrix generalized inverses to the construction of inverse systems is discussed, with a focus on the use of generalized inverse systems in the context of control systems.
Abstract: (1976). On the application of matrix generalized inverses to the construction of inverse systems. International Journal of Control: Vol. 24, No. 5, pp. 733-739.
15 citations
Book•
01 Jan 1976
TL;DR: Inverse trigonometric functions have also been used in the context of trigonometric functions as discussed by the authors, including right triangle ratio and right triangle triangulation, in the plane geometry domain.
Abstract: Right Triangle Ratios. Trigonometric Functions. Graphing Trigonometric Functions. Identities. Inverse Trigonometric Functions Trigonometric Equations and Inequalities. Additional Topics: Triangles and Vectors. Polar Coordinates Complex Numbers. Appendix A: Comments on Numbers. Appendix B: Functions and Inverse Functions. Appendix C: Plane Geometry: Some Useful Facts. Selected Answers. Index.
11 citations
01 Apr 1976-Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
TL;DR: While acc uracy ove r the e ntire domain of definition rema in s o ur main co ncern , the authors have te nded toward me thod s that also e nsure e ffi c ie ncy, portability, and ease of programming and modifi ca tion.
Abstract: S ince Ihe sine, cosine, expone ntial int egral s a nd hype rbo li c s ine, a nd cos ine inl egra ls a re frequently enco unte red togethe r in phys ical proble ms and the ir ex pans ion s ha ve te rm s 111 co mmon , we have inco rpo rated these fun ction s into P a rt Ill. (For P a rt s I and II , see . I). While acc uracy ove r the e ntire domain of definition rema in s o ur main co ncern , we have te nded toward me thod s that also e nsure e ffi c ie ncy, portability, and ease of programming and modifi ca tion. The numbe r of te rm s in se ri es, the numbe r of conve rge nt s in an ite rative process, the s tarting argumen ts fo r d ifferent method s, are aU de termined by the program as a function of word le ngth, argume nts, accuracy des ired , e tc . More re ali sti c res ult s are re turned whe n erro r conditio ns a re encounte red. The proper analytic behavior of the fun c tion wiU always be re tained to furth e r e nsure correct limiting values, in particular of re lated fun ctions and fo r purposes of diffe re nti a tion a nd integration. In Pa11.s I and Il in add ition to the implementing ANS FORTRAN program , we had inc luded a driver (test) program and its results . S in ce e ithe r of these driver program s can be readi ly mod ifi ed to compute other fun ctions, we have omitted the drive r program and in place of it s resu lt s have included a table of co rrect results to 3.5 s ignificant figures cove rin g essentiall y the fun c tional ran ge of prese nt computers.
7 citations
TL;DR: The modification of Levinson's theorem for the case of non-local potentials suggested by many authors is shown to be incorrect as mentioned in this paper, which is a result of overlooking a simple property of the inverse tangent function.
Abstract: The modification of Levinson's theorem for the case of non-local potentials suggested by many authors is shown to be incorrect. The error is a result of overlooking a simple property of the inverse tangent function.
3 citations
01 Apr 1976
TL;DR: It is shown that overall, inverse sine quantization requires the minimum number of coding levels followed by log area quantization and then reflection coefficients, and with the exception of the first two coefficients, the differences among the methods are shown to be rather minimal.
Abstract: A comparison of reflection coefficient, log area and inverse sine quantization is presented. The approach makes use of analytically tractable results without numerical approximations or direct frequency domain calculation. By using parameters obtained from a speech data base, specific bit allocations are obtained which insure that a prescribed maximum spectral deviation is not exceeded for each analysis frame. It is shown that overall, inverse sine quantization requires the minimum number of coding levels followed by log area quantization and then reflection coefficients. With the exception of the first two coefficients, however, the differences among the methods are shown to be rather minimal.
TL;DR: In this paper, the form of reference mask functions and choice of c.c.d. clocking are considered for real implementations of an electro-optical correlator, and the prefiltering of reference functions (with an inverse sine filter) is described, as is a clocking system which utilises all receptor sites in an interlaced, interline transfer.
Abstract: The form of reference mask functions and choice of c.c.d. clocking are considered for real implementations of an electro-optical correlator. The prefiltering of reference functions (with an inverse sine filter) is described, as is a clocking system which utilises all receptor sites in an interlaced, interline transfer c.c.d.