Multiple Fourier analysis in rectifier problems. II

Authors:
Robert L. Sternberg, Michael R. Sheets, Helen M. Sternberg, Abraham Shigematsu and Alice L. Sternberg

Journal:
Quart. Appl. Math. **32** (1974), 293-315

MSC:
Primary 78.42

DOI:
https://doi.org/10.1090/qam/434156

MathSciNet review:
434156

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Abstract: The nonlinear problem of the multiple Fourier analysis of the output from a cut-off power-law rectifier responding to a two-frequency input, reviewed in general in Part I of this study [1], is further scrutinized here for the special case of a zero-power-law device; i.e., a bang-bang device or a total limiter. Solutions for the modulation product amplitudes or multiple Fourier coefficients as in Part I appear as Bennett functions, and line graphs of the first fifteen basic functions for the problem are given. The new functions studied, being based on a discontinuous device, then, together with the functions studied in Part I, provide approximate solutions to the two-frequency modulation product problem for an arbitrary piecewise continuous nonlinear modulator, and the solution for this general problem is outlined. Finally, numerical tables of the zeroth-kind functions graphed have been prepared and are available separately in the United States and Great Britain. As before, the entire theory is based on the original multiple Fourier methods introduced by Bennett in 1933 and 1947.

**[1]**Robert L. Sternberg, Jerome S. Shipman, and Shirley Rose Zohn,*Multiple Fourier analysis in rectifier problems*, Quart. Appl. Math.**16**(1958), 335–360. MR**0099843****[2]**W. R. Bennett,*Bell System Tech. J*.**12**, 228 (1933)**[3]**W. R. Bennett,*Bell System Tech. J*.**26**, 139 (1947)**[4]**R. L. Sternberg and H. Kaufman,*A general solution of the two-frequency modulation product problem. I*, J. Math. Physics**32**(1954), 233–242. MR**0061479****[5]**Robert L. Sternberg,*A general solution of the two-frequency modulation product problem. II. Tables of the functions 𝐴_{𝑚𝑛}(ℎ,𝑘)*, J. Math. Physics**33**(1954), 68–79. MR**0061480****[6]**Robert L. Sternberg,*A general solution of the two-frequency modulation product problem. III. Rectifiers and limiters*, J. Math. and Phys.**33**(1954), 199–205. MR**0070453**, https://doi.org/10.1002/sapm1954331199**[7]**J. S. Shipman,*On Middleton’s paper “Some general results in the theory of noise through non-linear devices”*, Quart. Appl. Math.**13**(1955), 200–201. MR**0068773**, https://doi.org/10.1090/S0033-569X-1955-68773-3**[8]**Elliott W. Montroll,*Frequency spectrum of vibrations of a crystal lattice*, Proceedings of the symposium on special topics in applied mathematics, Northwestern University, 1953, 1954, pp. 46–73. MR**0063966**, https://doi.org/10.2307/2308448**[9]**E. Feuerstein,*Intermodulation products for 𝜈-law biased wave rectifier for multiply frequency input*, Quart. Appl. Math.**15**(1957), 183–192. MR**0092558**, https://doi.org/10.1090/S0033-569X-1957-92558-5**[10]**J. C. Hsu,*Integral representation of zero-memory nonlinear functions*, Bell System Tech. J.**41**(1962), 1813–1830. MR**0148502**, https://doi.org/10.1002/j.1538-7305.1962.tb00495.x**[11]**Paul Penfield Jr.,*Fourier coefficients of power-law devices*, J. Franklin Inst.**273**(1962), 107–122. MR**0132216**, https://doi.org/10.1016/0016-0032(62)90644-0**[12]**Cecil Hastings Jr.,*Approximations for digital computers*, Princeton University Press, Princeton, N. J., 1955. Assisted by Jeanne T. Hayward and James P. Wong, Jr. MR**0068915****[13]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[14]**E. W. Hobson,*The theory of a functions of a real variable and theory of Fourier's series*, 2nd ed.,**2**, Warren Press, Washington, D. C., 1950**[15]**L. Tonelli,*Serie trigonometriche*, Nicola Zanichelli, Bologna, 1928

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DOI:
https://doi.org/10.1090/qam/434156

Article copyright:
© Copyright 1974
American Mathematical Society