QUARTERLY OF APPLIED MATHEMATICS 293
OCTOBER, 1974
MULTIPLE FOURIER ANALYSIS IN RECTIFIER PROBLEMS II*
BY
ROBERT L. STERNBERG and MICHAEL R. SHEETS (University of Rhode Island)
HELEN M. STERNBERG (University of Connecticut)
ABRAHAM SIIIGEMATSU (Naval Underwater Systems Center)
and ALICE L. STERNBERG (The Williams School)
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 Amnw(h, 7c) studied, being based on a discontinuous device,
then, together with the functions Amnn)(h, k) 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 Amnm (h, k) graphed have been
prepared and arc 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. Introduction and formulation of the problem. In Part I of this paper by
Sternberg et al. [1], the much-studied problem in theoretical electronics of the multiple
Fourier analysis of the output from a cut-off power-law rectifier responding to a several-
frequency input was studied in general, and the multiple Fourier coefficients or Bennett
functions
t <">
*- mn
(h, k) = [[ (cos u + k cos v — h)" cos mu du cos nv dv,
TV J J («
(R : cos u -f- k cos v > h, 0 < u < 7r, 0 < y < 7r, (i.i)
where the indices m, n take all integral values m, n > 0, and where v > 0, were examined
in some detail for the case in which v = 1. Here we study the zeroth-kind functions (1.1),
that- is to say the case v = 0, an apparently simple but actually quite complicated problem.
* Received May 8, 1973. The authors are indebted to the Computation Centers of the University of
Rhode Island, the University of Connecticut, and the Naval Underwater Systems Center in New London,
Connecticut for the computational and graphical work that went into the project. These computational
services were provided the authors by the several institutions named over a continuing period of time
without charge of any kind, and the authors wish to express their sincere thanks for this help.
294 R. L. STERNBERG ET AL.
For a cut-off power-law rectifier with output versus input characteristic Y = Y"(X; X0)
of the form F"(X; X0) = (X — X0)" if X > X0 and zero otherwise, the functions (1.1)
are the coefficients A±mn("\h, k) = Amn("\h, k) in the double Fourier series expansion
y(t) = \P"A00w{h, k) + P" X)* A±mJ"\h, lc) cos (co±mnt + <j>±mn) (1.2)
m ,n = 0
of the output y(t) = Y"[x(t)] X0] when the input X = x(t) has the form
x(t) = P cos (pt + d„) + Q cos (qt + dQ), P > Q > 0. (1.3)
In (1.1) we have h = X0/P and k = Q/P > 0, while in (1.2) the modulation product
frequencies and phase angles, co±„„ and <j>±mn, are given by the relations u±mn = mp ± nq
and <t>±mn = mdp ± ndQ . Finally, in (1.2) the asterisk on the summation sign indicates
that we sum only on all distinct arrangements of plus and minus signs, equivalent
arrangements being taken only with the plus signs and with the zero-order term, partic-
ularly, having been removed from the sum.
The nonlinear device in the problem is a biased linear rectifier when v = 1 and may
be described as a biased zero-one bang-bang device or total limiter when v = 0, as in
Bennett [2, 3], If we use appropriate linear combinations of functional values AmnU) (hi , k)
summed up with suitable coefficients for suitable choices of the h's, the corresponding
multiple Fourier coefficients B±mn in the double Fourier series expansion of the output
of a general nonlinear device with input (1.3) may be approximated to within an arbi-
trary e > 0 for all m, n when the output versus input characteristic Y = Y(X) is con-
tinuous; similarly, the B±m„ for a general device may be expressed exactly as linear sums
of functional values Amn(1' (/i,- , k) in the form
NU)
B±mn = P J2 giAmnw(hi , k) (1.4)
1 = 1
for suitable choices of the numbers <7, and hi when the device characteristic Y = Y(X)
is not only continuous, but also polygonal. Similarly, by a simple extension of the basic
technique both of these results can be extended, with the aid of the functions /lm„(0) (h, k),
to the case of an arbitrary nonlinear device with a piecewise continuous characteristic
Y = Y(X), approximate linear results being obtained in the one case and exact linear
expressions for the B±mn in terms of functional values Amnw(h*, k) and Amnw(hi , k)
in the form
N'U)
B±mn = P Z g*Aj"\h*, k)+P Z g,Ajl)(h, , k) (1.5)
t-i ,=1
being obtained in the other case, i.e. 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.
In addition to the foregoing, the functions ^4m„<1)(/(, k) occur also in various related
statistical information-processing problems as discussed, for example, by Shipman [7],
and the functions Amnw(h, k) have been independently applied to some unrelated
problems in crystallography by Montroll [8]. Similarly, if Feuerstein's results [9] and
MULTIPLE FOURIER ANALYSIS 295
Bennett's earlier results [1] are turned around, the Bennett functions AmnM(h, fc) may be
viewed generally as new special functions of mathematical physics in terms of which
many otherwise difficult generalized Weber-Schafheitlin integrals and Schlomilch series
or infinite integrals and sums involving Bessel function products may be directly eval-
uated.
Finally, since the appearance of Part I of this paper, a very interesting review article
by .Hsu [10] has appeared, and related problems in transistor circuits were studied in a
similar manner by Penfield [11[.
The interest and utility of Bennett functions thus appear to be gradually growing
and the basic case of the functions Amn'"' (//, fc) has been found to have special significance
of its own. Hence, we study these latter functions in some detail in the following.
Line graphs of the first fifteen functions Amn"" (h, k) are given in Figs. 1 through 15,
and six decimal tables with h and k varying in steps of one-tenth unit and with error
generally less than 1 X 10 ~6 units have been deposited in the Unpublished Mathematical
Tables file in the editorial offices of the journal Mathematics of Computation, in the
United States and in the Tables Collections of the Grace Library of the University
of Liverpool in Great Britain and with the authors.
2. Basic formulas and expansions for the zeroth-kind functions. In addition to
satisfying the numerous relations described in Part I for the Bennett functions Amn'"} (h, k)
in general, the zeroth-kind functions AmnW)(h, k) satisfy several special relationships and
have a few properties that are more or less unique. As usual, the variable h takes all real
values while the variable k ranges over the interval 0 < k < 1, and again, it is con-
venient to discuss the functions in terms of three cases defined, as in Part I, by the
conditions: (0) h > 1 + fc, (a) |/(| < 1 + k, and (°°) h < —1 — k where again, as before,
case (0) is trivial, the rectifier then being biased so strongly that Amn"" (h, fc) = 0 for
all m and n.
To begin with, since for v = 0 the kernel function in the integrals (1.1) defining the
functions AmnM (h, fc) reduces to unity, a first integration can always be carried out.
This leads, then, for the first several functions Amnf0>(h, fc) with h > 0, graphs of which
are shown in Figs. 1 through 15, to the formulas
2 ra
A0nw(h, fc) = ~2 / cos-1 (h — k cosv) cos nvdv,
TT J o
Alnm(h, k) = -2 [ [1 — (h — k cosw)2]1/2 cos nvdv,
IT J0
Aj°\h, k) = \ f [1 — (h — k cos v)2]1/2(h — fc cosv) cos nvdv, (2.1)
TT J o
Aj°\h, k) =-2 ( [1 -Qi-k cos y)T/2[!(/i - k cosy)2 - f] cos nvdv,
7T J o
A4n<0\h, fc) = -2 f [1 — (h — k cosv)2]W2m — k cosv)3 — (h — fc cosi>)] cos nvdv,
TT J o
where o = it if \h\ + fc < 1 and a = cos-1 [(h — 1)/fc] if 1 — fc < h < 1 + fc, and where
n takes all integral values n > 0. Except for the restrictions on h, these formulas hold
quite generally.
296 R. L. STERNBERG ET AL.
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