Multiple Fourier analysis in rectifier problems

Authors:
Robert L. Sternberg, Jerome S. Shipman and Shirley Rose Zohn

Journal:
Quart. Appl. Math. **16** (1959), 335-360

MSC:
Primary 78.00

DOI:
https://doi.org/10.1090/qam/99843

MathSciNet review:
99843

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Abstract: The non-linear problem of the multiple Fourier analysis of the output from a cut-off power law rectifier responding to a several-frequency input is reviewed for the one- and two-frequency problems and is briefly investigated for the three-frequency problem. The solutions for the modulation product amplitudes or multiple Fourier coefficients are obtained in exact although transcendental form. An account of the mathematical properties of these multiple Fourier coefficients or Bennett functions, including hypergeometric representations and power series expansions for them as well as recurrence relations satisfied by them, is given in the paper together with line graphs of the first ten basic functions for the one-frequency problem and of the first fifteen basic functions for the two-frequency problem. Further applications of the theory are also given to the computation of average output power with the aid of the multiple Fourier coefficients or Bennett functions studied in the paper, and the work is concluded with some brief remarks concerning the interpretation of the results in terms of the theory of almost periodic functions and the generalized Fourier series of Bohr under appropriate conditions. Numerical tables of the functions graphed have been prepared and are available separately in the United States and Great Britain for applications requiring great accuracy. Finally, the entire theory is based on the original method of the expansion of the rectifier output in multiple Fourier series introduced by Bennett in 1933 and 1947.

**[1]**W. R. Bennett,*Bell System. Tech. J*.**12**, 228 (1933)**[2]**W. R. Bennett,*Bell System Tech. J*.**26**, 139 (1947)**[3]**H. Kaufman,*Harmonic Distortion in Power-Law Devices*, Math. Mag.**28**(1955), no. 5, 245–250. MR**1570742****[4]**H. Kaufman,*Modulation Products in Power-Law Devices*, Math. Mag.**30**(1956), no. 1, 9–17. MR**1570831****[5]**D. G. Lampard,*Proc. Inst. Elect. Engrs., Pt. IV*,**100**, 3 (1953)**[6]**B. Salzberg,*Am. J. Phys*.**19**, 555 (1951)**[7]**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****[8]**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****[9]**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****[10]**R. L. Sternberg, J. S. Shipman, and W. B. Thurston,*Tables of Bennett functions for the two-frequency modulation product problem for the half-wave linear rectifier*, Quart. J. Mech. Appl. Math.**7**(1954), 505–511. MR**0067571**, https://doi.org/10.1093/qjmam/7.4.505**[11]**R. L. Sternberg, J. S. Shipman, and H. Kaufman,*Tables of Bennett functions for the two-frequency modulation product problem for the half-wave square-law rectifier*, Quart. J. Mech. Appl. Math.**8**(1955), 457–467. MR**0074946**, https://doi.org/10.1093/qjmam/8.4.457**[12]**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**[13]**S. O. Rice,*Mathematical analysis of random noise*, Bell System Tech. J.**24**(1945), 46–156. MR**0011918**, https://doi.org/10.1002/j.1538-7305.1945.tb00453.x**[14]**G. N. Watson,*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR**0010746****[15]**E. W. Hobson,*The theory of functions of a real variable and theory of Fourier's series*, 2nd ed. vol. 2, Chap. 8, Harren Press, Washington, D.C., 1950**[16]**L. Tonelli,*Serie trigonometriche*, Chap. 9, Nicola Zanichelli, Bologna, 1928**[17]**E. T. Whittaker and G. N. Watson,**A course of modern analysis**, American ed., Chap. 14, Macmillan Company, New York, 1946**[18]**Harald Bohr,*Almost Periodic Functions*, Chelsea Publishing Company, New York, N.Y., 1947. MR**0020163****[19]**Norbert Wiener,*The Fourier integral and certain of its applications*, dover Publications, Inc., New York, 1959. MR**0100201****[20]**Han Chang and V. C. Rideout,*A generalization of modulation spectra*, Quart. Appl. Math.**11**(1953), 87–100. MR**0052548**, https://doi.org/10.1090/S0033-569X-1953-52548-X

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DOI:
https://doi.org/10.1090/qam/99843

Article copyright:
© Copyright 1959
American Mathematical Society