Showing papers on "Trigonometric functions published in 1969"

A circular-harmonic computer analysis of rectangular dielectric waveguides

Abstract: This paper describes a computer analysis of the propagating modes of a rectangular dielectric waveguide. The analysis is based on an expansion of the electromagnetic field in terms of a series of circular harmonics, that is, Bessel and modified Bessel functions multiplied by trigonometric functions. The electric and magnetic fields inside the waveguide core are matched to those outside the core at appropriate points on the boundary to yield equations which are then solved on a computer for the propagation constants and field configurations of the various modes. The paper presents the results of the computations in the form of curves of the propagation constants and as computer generated mode patterns. The propagation curves are presented in a form which makes them refractive-index independent as long as the difference of the index of the core and the surrounding medium is small, the case which applies to integrated optics. In addition to those for small index difference, it also gives results for larger index differences such as might be encountered for microwave applications.

A table of integrals of the error functions.

Abstract: Tabulation of definite and indefinite integrals of products of error function with elementary and transcendental functions

Digital Walsh-Fourier Analysis of Periodic Waveforms

Abstract: A digital process is described for obtaining the Walsh-Fourier series of a periodic waveform, which requires at most two cycles of the waveform under measurement The first cycle of the periodic waveform is required for the determination of period The coefficients of the Walsh-Fourier series are obtained during the second cycle only, and they are available at the end of the cycle Given the Walsh-Fourier coefficients of the periodic wave, the individual sine and cosine components of its Fourier series may be obtained using conversion formulas Special features of the process are that there are no theoretical low-frequency limitations, and for an instrument with an internal clock whose frequency lies in the range 1 Hz to 1 MHz, the fundamental frequency component of a signal that can be analyzed would be in the range 1 cycle in 116 days to 60 Hz Also, whereas the digital processes required to obtain a Fourier series directly are complicated by the need to multiply sample values of voltage by sines and cosines, which are themselves functions of time, determination of Walsh-Fourier coefficients is achieved very simply by using gating circuits Generation of the required Walsh functions for a periodic signal of any fundamental frequency within the design range has been achieved

Electronic resolved sweep signal generator

Abstract: To display radar data on the face of a cathode-ray tube or other display device, it is necessary to generate product functions of the sine and the cosine components of the antenna position angle times the radar-sweep function. Both the sine product function and the cosine product function are electronically generated in respective multipliers. Each multiplier consists of a first differential amplifier pair coupled push-pull to a second differential amplifier pair. To minimize the effect of line voltage variation on the product functions, the sweep voltage is corrected prior to multiplication with the sine and cosine components in accordance with line voltage variation.

Approximation by Trigonometric Polynomials to Periodic Functions of Several Variables

Abstract: The present paper is, in spirit, a sequel to [5], but may be read independently of that paper. The underlying motive is to provide a satisfactory, and suitably quantitative, theory of “degree of approximation” for functions of several variables. In this section we discuss some ramifications of this general problem, and outline a program of investigation, of which some first steps are carried out in the following sections. Although, or perhaps because, our results are quite fragmentary, we hope they might stimulate interest in the present circle of questions.

On two functional equations for the trigonometric functions

Abstract: We consider the following cosine and sine functional equations: (1) (2) where f is an entire function of a complex variable z and x, y are complex variables [1; 2; 3].