A symposium of mathematical methods for radio engineers

A study of the properties of Fourier series with special reference to functions presenting discontinuities and to radio and pulse problems; summary survey of the methods of harmonic analysis; scope of these methods; extension to the calculation of a great number of harmonics and to the analysis of non-periodic curves. The Fourier series expansions are particularly applicable to functions having discontinuities, and the coefficients of these expansions can be expressed in terms of the discontinuities of the function and its derivatives. The discontinuity formulae thus obtained are very useful in that a great number of expansions can be written straight away. Such is the case of periodic pulses. The Fourier series can also be applied, by means of a limiting process, to the analysis of non-periodic functions such as a single pulse. In some cases the mathematical difficulties encountered when applying Fourier integrals are so avoided. As examples of application, the frequency characteristics of the unit step and of the impulse functions are calculated, and the response of an ideal low-pass filter to a rectangular pulse and to a double pulse is studied. A general survey of the methods of harmonic analysis is given. Methods of calculation are approximate since they start from a limited number of points of the curve to analyse. A considerable variety of methods have been suggested, the main tendency being to get the greatest accuracy with a limited number of points. The writer has studied closely all these methods and has found that they are closely correlated and can be reduced to a single one, the method using Lagrange´s equations with Runge´s schedules, and the writer´s corrective coefficients. Further studies by the writer, particularly on the quick calculation of a great number of harmonics such as 120 or 160, are mentioned.