A General Solution of the Two‐Frequency Modulation Product Problem. I.

Abstract

1. Introduction. The purpose of thIS paper IS to present a general method for readIly obtammg apprOXImate numencal values of the amplItudes of the modulatIOn products which occur m the output when a two-frequency mput IS applied to an arbItrary modulator havmg a contmuous output versus input characteriStIC. a partial analytical solutIOn of the problem is also indIcated In solvmg the problem It is shown, m particular, that apprOXImate values of the modulation product amplitudes always can be determmed as simple lmear combmatIOns of the values of four new functions introduced by W. R. Bennett [1,2] and that the output can be approxImated umformly for all tIme by the convergent double FourIer senes havmg these apprOXImate values as coefficients. The approximate values of the modulation product amplItudes determined by the method gIven are shown to be capable of an arbItrmy degree of refinement, subject only to lImits of accuracy of determmatIOn of the four new functions mentioned, tables of which are to be given in Part II of the present paper, and an estimate of error is mdicated. The method yields exact values of the modulatIOn product amplitudes in the case of such modulators as the bIased Ideal rectifier or Ideal hmIter, whose output versus input characteristIC is a continuous polygonal functIOn, and IS an extension of the method employed by Bennett [2] in connection with those two devices Finally, the method will be applied in Part III to study in some detail a number of examples mcludmg among others the two partIcular modulators Just mentIOned.The problem IS formulated in §2 In §3 an apprOXImate solution IS derived. The coefficients in that solutIOn are reduced m §4 by the method of Bennett and are thus found to be expresslble m terms of the four new functIOns mentioned; several properties of these functions are noted and power senes expansions for them are obtamed in §5.

Multiple Fourier analysis for bang-bang controls with memory

“…for m + n = 0, 1, 2, 3. While the functions B\^n (h,k) for y = 0 and v= 1 with |/z| + A:^ 1 are simple polynomials in h and k. Other interesting, but for our purposes less useful, representations for the functions A™ a (h,k) for v = 0 and v = 1 include some simple algebraic expressions in terms of the complete elliptic integrals E(k) and K{k) for h = 0, and more sophisticated but less easily applied formulations for the Bennett functions in terms of Weber-Schafheitlin integrals, Schlomilch series, and Appel hypergeometric functions of two variables (see Bennett [1,2], Sternberg et al [3]- [10], Lampard [11], and for extensions to problems with more than two input frequencies, Feuerstein [12]).…”

Multiple Fourier analysis in rectifier problems. II

“…Formula (3.4) is based on the expansion of cos-1 (1 -x) in powers of x with x = 1 -h + fc cos v. We omit the details. 4. Power computations and concluding remarks.…”

“…when the device characteristic Y = Y(X) is not merely piecewise continuous but piecewise polygonal as well. The method is described in detail by Sternberg and Kaufman [4,5,6] for the continuous case and is readily extended to the piecewise continuous case without difficulty. Thus, the functions (1.1) have very broad applicability in all cross-talk problems in communications and control theory.…”

A General Solution of the Two‐Frequency Modulation Product Problem II. Tables of the Functions Amn(h, k)

“…and subsequently also to tabulate briefly the first six higher order functions Amn(h, k). These functions arise in the problem referred to in the title and the theory of their application and a number of their properties have been developed in Part I of this paper by the present writer with H. Kaufman [4] and in the earlier papers of W. R. Bennett [1], [2].…”