Explicit form of f.m. distortion products with white-noise modulation

Abstract

When a frequency-modulated wave passes through a network whose phase or amplitude characteristics vary non-linearly with frequency, distortion terms appear as both frequency and amplitude modulation of the output wave. If the characteristics are expressed as power series, these distortion terms appear, to first order, as products of powers of time derivatives of the unwanted frequency modulation. When the frequency modulation may be simulated by a band of random noise (as in multiplex telephony carrying large numbers of channels), the spectra of the distortion products can, in principle, be described by simple algebraic functions of the characteristics (i.e. the minimum and maximum frequencies and the r.m.s. frequency deviation) of the modulating noise band.Except in certain special cases, the derivation of these algebraic formulae by straightforward analytical methods becomes prohibitively tedious for distortions of order much above the second. However, once the formulae are found, the insertion of numerical values for particular cases is straightforward. In the present paper it is shown how the problem can be reduced to the repetition of a number of standard operations which can be carried out using a digital computer. The technique is illustrated by application to fourth-order distortion appearing in the amplitude modulation, generated by terms in the amplitude characteristic up to sixth degree. Even in such an apparently simple case as this it appears from the literature that the closed form of the distortion formula has not hitherto been obtained. This example is of direct practical interest since, for example, the amplitude characteristic of a maximally-flat-amplitude triple-tuned circuit is of sixth degree in the region around the midband frequency. With a minor modification, the resulting formula also applies to fourth-order distortion appearing in the frequency modulation, owing to terms in the phase characteristic up to sixth degree.Use is made of a discontinuous contour integral applied to a similar, but somewhat simpler, case by Bennett;1 a closely analogous course can be followed using the more recently developed theory of generalized functions,2 but in this particular problem the contourintegral method is more economical.Formulae for the various orders of distortion in the top channel due to amplifier and discriminator characteristics are given in tabular form.