IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
465
Solving the Optimal PWM Problem for
Single-Phase Inverters
Dariusz Czarkowski, Member, IEEE, David V. Chudnovsky, Member, IEEE, Gregory V. Chudnovsky, and
Ivan W. Selesnick, Member, IEEE
Abstract—In this paper, the basic algebraic properties of the
optimal PWM problem for single-phase inverters are revealed.
Specifically, it is shown that the nonlinear design equations given
by the standard mathematical formulation of the problem can
be reformulated, and that the sought solution can be found by
computing the roots of a single univariate polynomial ( ), for
which algorithms are readily available. Moreover, it is shown that
the polynomials ( ) associated with the optimal PWM problem
are orthogonal and can therefore be obtained via simple recursions. The reformulation draws upon the Newton identities, Padé
approximation theory, and properties of symmetric functions. As
a result, fast ( log2 ) algorithms are derived that provide
the exact solution to the optimal PWM problem. For the PWM
harmonic elimination problem, explicit formulas are derived that
further simplify the algorithm.
Index Terms—Harmonic elimination, Newton identities, orthogonal polynomials, Padé approximation, pulsewidth modulation
(PWM), single-phase inverters, symmetric functions.
I. INTRODUCTION
T
HE PROBLEM of the optimal design of pulsewidth modulated (PWM) waveforms for single-phase inverters [1], [2]
is examined in this paper. PWM signals are used in power electronics, motor control and solid-state electric energy conversion
[2], [3]. The best voltage signal for these purposes is one with
a periodic time variation in which amplitudes of selected nonfundamental components of the signal have been controlled to
increase efficiency and reduce damaging vibrations. How can
the waveform be designed to most effectively accomplish this?
This paper describes a contribution to the theory and practice of
optimal PWM waveforms that addresses this question.
A PWM waveform consists of a series of positive and negative pulses of constant amplitude but with variable switching
instances as depicted in Fig. 1 (as in a power electronic PWM
full-bridge inverter). A typical goal is to generate a train of
pulses such that the fundamental component of the resulting
waveform has a specified frequency and amplitude (e.g., for a
speed control of an induction motor). Some of
constant
Manuscript received December 13, 1999; revised January 20, 2001. This
work was supported in part by the National Science Foundation under
CAREER Grant CCR-987452. This paper was recommended by Associate
Editor H. S. H. Chung.
D. Czarkowski and I. W. Selesnick are with the Department of Electrical
and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA
(e-mail: dcz@pl.poly.edu; selesi@taco.poly.edu).
D. V. Chudnovsky and G. V. Chudnovsky are with the Department of Mathematics, Polytechnic University, Brooklyn, NY 11201 USA.
Publisher Item Identifier S 1057-7122(02)03137-9.
Fig. 1. A three-level PWM waveform.
the proposed methods for PWM waveform design are: modulating-function techniques, space-vector techniques, and feedback methods [2]. These methods suffer, however, from high
residual harmonics that are difficult to control and from limitations in their applicability. A method that theoretically offers
the highest quality of the output waveform is the so-called programmed or optimal PWM [4]. Owing to the symmetries in the
PWM waveform of Fig. 1, only the odd harmonics exist. Assuming that the PWM waveform is chopped times per half a
cycle, the Fourier coefficients of odd harmonics are given by
(1)
and is the amplitude of the square
where
wave. Amplitudes of any harmonics can be set by solving
a system of nonlinear equations obtained from setting (1)
equal to prespecified values. In the harmonic elimination programmed PWM method, the fundamental component is set to
low-order harmonics are set to
a required amplitude and
zero. This is the most common approach in electric drives since
low-order harmonics are the most detrimental to motor performance. In other applications, like active harmonic filters or control of electromechanical systems, harmonics are set to nonzero
values. This task of designing a PWM waveform, the first
Fourier series coefficients of which match those of a desired
waveform has been the subject of many papers [1], [5]–[37].
Often, the Newton iteration method [5] or an unconstrained optimization approach [38] are used to solve the system of nonlinear
equations (1). Those methods are computationally intensive for
on-line calculations and the storage of off-line calculations leads
to high memory requirements. Recent results from the research
community [39], [40] show two approaches to real-time implementation of an approximate optimal PWM. One approach is to
fine tune conventional PWM techniques like regular-sampled
1057-7122/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
[39] or space-vector methods [41], [40] to approximate programmed PWM switching patterns. Another approach is to simplify the nonlinear harmonic elimination equations [26] in order
to obtain real-time approximate solutions using modern digital
signal processors.
algorithms for solving the
This paper develops
PWM harmonic elimination problem without any approximations in the problem statement. Since many PWM applications
allow for a computational time frame of a few milliseconds,
the developed algorithms will allow for real time generation of
switching patterns with of the order of hundreds.
Note that
polynomial. Let
where
, then
is the th Chebyshev
..
.
As the odd-indexed Chebyshev polynomials are odd polynomials, we can write
II. CONVERTING THE PWM PROBLEM
For the scope of this paper, a PWM waveform is a -periodic
that is binary-valued for
and has the
function
and
as shown
symmetries
can be written with the Fourier series as
in Fig. 1. As such,
With this notation, the PWM equations become
or
with
(6)
where
The optimal PWM problem, as it is considered here, is the deso that its first Fourier coeffisign of a PWM waveform
cients are equal to prescribed values. For a PWM waveform,
as shown in Fig. 1, we have
Therefore, the optimal PWM problem gives rise to the following
design equations [4]:
are the sums of powers of
. Equation (6) forms a set of
linear equations for
,
. Once the values
are obtained by solving the linear system (6), one has the fol, find the solution
lowing problem. Given
to the following system of nonlinear equations
(7)
(2)
..
.
..
.
(8)
(3)
, we have equations
Given the values
unknowns; we would like to find the
unknowns
and
, with
. We can
first simplify the equations as is done for example in [22].
for odd , and let
for even . Then
Let
(4)
..
.
(5)
Once are obtained, the original variables can be found by
,
for odd , and
for
letting
even . Due to the symmetry with respect to , any permutation
is also a solution set; likewise for . Yet it
of a solution set
is necessary to order appropriately such that
. Note that for odd ,
gives
. For even ,
gives
or
. This indicates how to obtain with the desired
let
;
ordering from : for those values of
let
.
and for those values of
However, the design equations (7) and (8) are nonlinear, so
is not so straightforward. In
obtaining the desired solution
the following sections, this nonlinear system of equations will
be closely examined and in Section III, a systematic procedure
is given to obtain the solutions.
CZARKOWSKI et al.: SOLVING THE OPTIMAL PWM PROBLEM FOR SINGLE-PHASE INVERTERS
A. Specialization to the Harmonic Elimination Problem
For the harmonic elimination PWM problem, which we will
focus on below, the Fourier coefficients of the PWM waveform
should match the Fourier coefficients of a pure sine wave.
appearing in (2) and (3) are given by
That is, the values
, and
for
. For this case, the values
depend on only and are given by
467
special cases this problem can be reduced to a manageable and
“exactly solvable” one. The classical cases of sum of powers
(Newton), Wronski and elementary symmetric functions are
the ones well described in the literature (see, e.g., [42] on
representation of symmetric groups and Frobenius and Schur
functions).
In Section III, we present a procedure to determine the polynomial
for the case where one is given the “sums of odd
powers,” as it appears in the optimal PWM problem.
III. GENERALIZING NEWTON’S IDENTITY
B. A Simpler Problem
It is useful to consider first the related, simpler problem
where contiguous sums of powers are known. Given the values
; find satisfying the following equations:
To develop a procedure to solve the optimal PWM problem,
it will be useful to examine more closely Newton’s identities.
The simple derivation of Newton’s identities will serve as our
as
starting point. First write the polynomial
(9)
..
.
then the logarithmic derivative is given by
(10)
It turns out that
are the roots of the polynomial
(11)
Expanding each term in the sum, one gets
where
(13)
where
etc. The coefficients
can be found as
are the sums of the root powers, with
Integrating (13) gives
..
.
.
(12)
etc. These are Newton’s Identities. When contiguous values
are known one can easily obtain
through these simple recan be obtained by taking the roots
cursive relations. Then
of (11) for which numerical algorithms are readily available.
these identities involve coefficients
beyond the
(For
highest power, which are implicitly defined as 0.) For the PWM
problem outlined above, we know only the first odd , for
. We do not know contiguous values ;
therefore Newton’s identities cannot be used. Our question is:
Does any such simple procedure exist for the harmonic elimination problem for PWM, where one is given only the odd ,
?
for
. The
Note that (7)–(10) are symmetric functions of
with given values
problem of finding the set of elements
in
, is in
of arbitrary symmetric functions
general a very complicated one because of nontrivial nature of
relations between symmetric functions of high degrees. Only in
Raising
gives
to the power of
and using the last equation
(14)
In order to generalize Newton’s procedure, we will obtain an
expression similar to (14), but having only odd . To this end,
note that
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
and that
is given by
The ? indicates unknown coefficients. Then, using (17) and (18),
can be determined up to and including
or
Writing out (16) gives
Then
Matching like powers of
gives
where
With the notation
(15)
one has
(16)
is the monic polynomial related to
by negating the
.
roots of
Equation (16) is the counter-part to (14). Likewise, by setting
like powers of equal, we can obtain equations that relate
and . However, in order to do this, we need to expand
into a power series of . Such a power series for
can be obtained using the following algorithm, described in [43,
Sec. 4.7]. Let
and
If
for
are known, then the values of
are given by
for
(17)
(18)
.
for
When the first
known for
odd values of
are known, then
are
. [
for
and
for
.] Therefore, using the
, and conserelations (17) and (18), we obtain for
quently, we can write out Equation (16), matching like powers
can be obtained.
of , to obtain linear equations from which
. That is,
For example, we write out the expressions for
we are given , , and , and our goal is to find the corre,
sponding monic third degree polynomial
Matching further like powers involves coefficients
and
higher, which are unknown. Note that the last three equations,
,
, and
, can be written as the
corresponding to
following linear system of equations:
This system is almost Toeplitz—only the signs of the odd
columns must be negated. The true Toeplitz system
where
,
,
gives the solution for
. Note that
are the coefficients of
defined in (15).
As this system is Toeplitz (the diagonals of the system matrix
are constant), the Levinson algorithm can be used to solve it
instead of
] [44].
efficiently [
The same procedure for general gives rise to a Toeplitz
system and comprises a generalization of Newton’s identities
to the case where only the first odd values are known. The
procedure for general follows likewise.
Algorithm ODDSOL: Given the sums of odd powers
for
, the polynomial
can be obtained as follows.
for
. Set
1) Set
for
.
using Equations (17) and
2) Compute for
(18).
3) Solve the Toeplitz system for
..
.
..
.
..
.
..
.
..
.
(19)
.
and set
Algorithm HESOL: Given and , the steps to solve the
harmonic elimination problem (2) and (3), are summarized as
follows.
CZARKOWSKI et al.: SOLVING THE OPTIMAL PWM PROBLEM FOR SINGLE-PHASE INVERTERS
469
TABLE I
PARAMETER VALUES COMPUTED IN ALGORITHMS ODDSOL AND HESOL FOR THE PWM HARMONIC-ELIMINATION PROBLEM WITH n = 4 AND A = 0:6
Fig. 2.
The PWM harmonic-elimination waveform obtained, with A = 0:6 and n = 15.
1) Set
for
.
,
, obtain the degree polyno2) From
for which the roots satisfy the equations (7)
mial
and (8). (Use algorithm ODDSOL.) Find the roots of
.
,
, with
.
3) Set
, set
. For
, set
For
. Sort the angles .
can be
For the general optimal PWM problem, the values
obtained by solving the linear system (6).
,
.
Example 1 (Harmonic Elimination): Let
Then
and
orthogonal. Numerically stable algorithms using properties of
orthogonal polynomials can therefore be developed and will be
presented in future publications.
IV. A RECURRENCE FOR
The fast recursive algorithms for solving Toeplitz systems
compute a solution using the solutions for smaller . The nowill be used to emphasize the dependence of
tation
on . Specifically,
denotes the monic degree- polynomial associated with the harmonic elimination problem, with
coefficients
With this notation, a recurrence relation
(20)
Then, the parameters computed in algorithms ODDSOL and
and ,
HESOL are shown in Table I. As ,
the shown in the table are obtained by letting
,
,
,
, and then
reordering . It can be verified numerically that the harmonic
elimination equations (2) and (3) are satisfied.
Example 2: Fig. 2 illustrates the PWM signal obtained with
and
.
It should be noted that for the optimal PWM problem, the
grows
Toeplitz matrix in (19) becomes ill-conditioned as
larger. By using extended precision arithmetic, the range of
for which this basic algorithm is useful can be extended.
However, in the following sections it will be shown that the
solution can also be expressed as the solution to a Padé approxare
imation problem, and consequently the polynomials
. For the harmonic elimination
can be used to compute
problem, the initial conditions can be taken to be
and
. The coefficients
in the recursion can
be computed using the following formula:
(21)
This recurrence exists because of the Toeplitz structure, see [45]
are then determined
for further details. The coefficients
recursively as [this implements (20)]
(22)
.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
In order for this algorithm to work, one still needs the coeffi. While can be computed
cients in the expansion of
using (17) and (18), in Section V, a more direct recursion will
be presented for computing the values themselves.
From (23) and (24) we have for the harmonic elimination
and
for
so that
problem, that
(25)
V. A RECURRENCE FOR
The algorithms presented above for the PWM harmonic elimination problem do not fully utilize the fact that the parameters
depend only on the single parameter . In this section we
will show a relationship between the original (transcendental)
]
formulation of the PWM problem [in terms of
).
and the converted (algebraic) formulation (in terms of
As a result, we obtain an expression for the function
that is important for developing a recursion for generating the
in the recurcoefficients that are needed for computing
.
sion
In this section, we show how the transcendental case can be
explicitly expressed using the introduced notations of
and
. The basic transformation is
and the new transforma-
Now, using
tion one gets
Therefore
with
one has
. With the previous transformation
. Solving for gives
Note that, as
Combining this with (25) and the identity (16) gives
from which we determine that
where
is given by
, one gets
so that the equations (4) and (5) can now be written as
(23)
..
.
The algorithm for computing is of complexity
only.
This algorithm follows the general power series algorithms described in [46], [47]. The key to this algorithm is to notice that
satisfies a second order linear differential equation
and an apparent singularity
(with singularities at
):
at
(24)
Therefore, if we define
If we look at the expansion of
then,
then we get the fourth-order recurrence on
the
product
has the roots
and as such the key identity
(16) can be applied directly to this problem—as it is of the
“sums of odd powers” form. One gets
The initial conditions are
at
for
and
where
From these values and the equation for
one derives the
factors in the three-term recurrence for orthogonal polynomials
. The first few
are
CZARKOWSKI et al.: SOLVING THE OPTIMAL PWM PROBLEM FOR SINGLE-PHASE INVERTERS
All
are rational functions in of rather special structure.
Since the case of the continued fraction expansion of
is not “explicitly solvable,”
is not a “known” function
and ; it belongs to the category of “higher Painleve”
of
transendencies (see examples of such functions in [48]). An
is the growth
important consequence of the “insolubility” of
as rational functions in
with integer
of coefficients of
coefficients. According to standard conjectures about explicit
and nonexplicit continued fraction expansions (cf., [49]), the
as a rational function in over grow as
coefficients of
for large . In fact, already for
, the coefficients
in
are large integers. This makes it impractical
of
for large . It is also
to precompute with full accuracy
explicitly,
unnecessary to analytically determine
only in the range of that is
since we need to know
significant for applications—this is the range where the weight
is positive. In this case a
of the orthogonal polynomials
much simpler best fit polynomial or rational approximation in
to
suffices.
can be summaThe recursive algorithm for computing
rized as follows.
Algorithm RECSOL: Given and , the polynomials
for be recursively computed as follows.
,
,
and for
to
1) Set
let
2) Set
let
and
and for
to
Find the roots of
.
3) Set
,
, with
.
, set
. For
, set
For
. Sort the angles .
Step 3 of algorithm RECSOL is the same as step 3 of algorithm HESOL. Using a computer algebra system, such as Maple
or Mathematica, this recursive algorithm allows one to obtain
as an explicit function of . A Maple program that implements algorithm RECSOL is given in the Appendix.
VI. COMPLEXITY OF THE PWM PROBLEM
What is the complexity (in the simplest definition, an operation count) of the problem of “sums of odd powers”—of the
optimal PWM problem discussed above? In the algebraic formulation one can first ask the same question about Newton’s
identities. If one uses Newton identities directly, the complexity
, but a significantly faster scheme can be found. The key
is
471
to this is the representation (14). Indeed, according to Brent’s
theorem,
terms of the power series expansion of
can
steps from the power series
be computed in only
(see [43, Sec. 4.7, Example 4]). This algoexpansion of
rithm requires only FFT-techniques for fast convolution [50].
Similar complexity considerations can be applied to the
that give
problem of fast computation of polynomials
the solution to the PWM problem of consecutive odd power
sums. A naive method of computation of Padé approximants
by solving dense systems of linear equations give a very high
. A classical Levinson algorithm
complexity bound of
of solving systems of Toeplitz linear equations, or algorithms
,
based on three-term recurrence relations satisfied by
give the complexity of computations of (all coefficients of)
at
. These algorithms are, perhaps, the best for
moderate because of their simple nature and the fact that
algorithm
they use almost no additional space. Also, the
but all
for
.
yields not only the single
However, for large these algorithms became impractical and
fast algorithms are needed.
A fast algorithm for computing (all coefficients of)
with total complexity of
works as follows. First,
terms of the
one applies Brent’s theorem to compute
power series expansion (at infinity) of
from the first
terms
with the complexity of only
. Then, one uses fast Padé approximation algorithms
(or equivalently, fast polynomial gcd algorithms). There is a
variety of such algorithms (for the earliest of Knuth, see [51]),
with the most popular belonging to Brent, Gustavson, and
complexity. Thus, we can compute
Yun [52] with
in at most
operations. Of course, this
method should be used only for a large (with additional precision of calculations) since it relies on a variety of extensions
of FFT methods that become advantageous only for very long
arrays.
We wish to note also that in [22] Sun and Grotstollen present
a Newton’s iteration directly on the transformed variables ,
and furthermore they note that the Jacobian in that case is a
algorithms are
Vandermonde matrix, and that efficient
algorithms are available, which
available. [In fact
become efficient for large ]. However, Newton’s iteration can
be very sensitive to initial values.
VII. CONCLUSIONS
This paper presents a contribution to the theory of optimal
pulse-width modulation and gives algorithms for efficient
on-line calculation of PWM switching patterns. Some specific
results for the harmonic elimination problem are also presented.
A number of new results regarding the PWM problem are
derived in this paper. It is shown that:
• by a transformation of variables the solution to the optimal
PWM problem is given by the roots of a polynomial
that are appropriately sorted;
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
• the polynomials
APPENDIX A
FIRST FEW
satisfy
Using the recursive algorithm HESOL, the polynomials
as explicit functions of can be obtained for the harmonic elimination problem using computer algebra software.
for
Here are the first few
where
and are the roots of -degree monic polynomial
;
can be found by solving a Toeplitz
• the polynomials
system of linear equations;
give also the solution to a Padé
• the polynomials
approximation problem and, therefore, constitute a set of
orthogonal polynomials;
are obtained through a simple re• the polynomials
currence
APPENDIX B
MAPLE PROGRAM
where an expression for
is given in (21);
• the complexity of a fast algorithm for the optimal PWM
.
problem is
An important consequence of the solution of the transcendental “sums of odd cosines” problem in PWM applications
lies in the ability to construct high-quality digital (step-function) approximations to arbitrary harmonic series. In this paper,
we have paid particular attention to the PWM harmonic eliminain (6) depend only on
tion problem in which coefficients
. In general, optimal PWM
the fundamental component
depend on the amplitudes of other harmonics
coefficients
which do not add to the complexity of the problem since
system (6) is linear. Since the proposed algorithms can be executed quite fast on any processor with high-performance DSP
capabilities, it opens a possibility of better on-the-fly construction of arbitrary (analog) waveforms using simple digital logic.
Areas of application include not only power electronic converters but also, for instance, control of microelectromechanical systems and digital audio amplification. It should be noted
that the design equations corresponding to optimal PWM with
bi-level waveforms [1] have the same structure as (2) and (3),
therefore the results presented in this paper are directly applicable.
While applying these basic algorithms, one should be aware
of the well-known inherent ill-conditioning of the corresponding Toeplitz matrix and Padé approximation calculations.
Techniques for the regularization of the basic algorithms
presented here will be dealt with in detail in future publications.
Future work will also focus on such practical applications as
three-phase inverters which eliminates harmonics with orders
divisible by three in (2) and (3).
A more detailed description of this work is given in the technical report [45].
The following Maple program implements algorithm
RECSOL for solving the PWM harmonic elimination problem.
Other programs are available from the authors.
A
2
2
2
2
n
i
i
CZARKOWSKI et al.: SOLVING THE OPTIMAL PWM PROBLEM FOR SINGLE-PHASE INVERTERS
473
Now, if we write the normalized (monic) polynomials
and
in terms of their roots
we get an identification of symmetric functions in
and
with the sequence of
in the definition of
. Namely, we get
m
APPENDIX C
RELATION TO THE THEORY OF PADÉ APPROXIMATION
The solution to the modified Newton problem, where one has
the first sums of odd powers, can also be obtained using the
theory of Padé approximation. Moreover, using the theory of
are orPadé approximation, we find that the polynomials
thogonal polynomials with respect to a specific weighting function that depends on only. Consequently, we can draw upon
methods developed for orthogonal polynomials to develop numerically stable fast algorithms for computing both the polynofor
and their roots.
mials
We start with the general relation between Padé approximations and the generalization of Newton relations between power
and elementary symmetric functions. Let us look at the Padé
to the series
at
.
approximation of the order
is defined via the generating function
Here the function
:
. The Padé apof the sequence
to
at (the neighborhood of)
proximation of the order
is a rational function
with
a polya polynomial of degree —such
nomial of degree and
matches the expansion of
that the expansion of
at
up to the maximal order. This means that
or
After taking the logarithmic derivative of this definition, we obtain the following representation of this definition:
for
(where
).
, one simply recovers Newton’s identities.
In the case
(the “diagonal” Padé approximations) is the
The case
case that solves the problem of “sums of odd powers”: Consider
for
and
the “anti-symmetric” case when
. In this case, in the notations above,
for
.
to
We, therefore, obtain the Padé approximation of order
the following function:
The Padé approximants
are related in this case by
. This relation between the numerator
must
and the denominator of the Padé approximants to
satisfies the simple functional identity:
hold because
. This gives the main result:
(
) to the problem
Theorem 1: The solution
of “sums of odd powers”
is given by the roots of the numerator
of the Padé approximation of order
to the function
Another way to verify this approximation without specialization from the case of general sequence , is to take the identity
(16).
Proof of Theorem 1: First of all, the Padé approxiof order
is
mation rational function
unique. [Indeed, if there would be two rational approximations
and
that approximate
up to
, then
, and since degrees of
are bounded by , then
.] Then,
is a Padé approximation of order
to
if
, we assume that this representation of the rational
and
are relatively
function is irreducible [i.e., that
is a Padé approximation of order
prime]. Then,
to
, and
is a Padé approxito
. Because of the functional
mation of order
, and the uniqueness of the Padé
equation
.
approximations, we get
. Moreover, since
This equation means that
at
starts at 1 [i.e.,
the expansion of
as
], we have
as
. This
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002
means that
and
into account the above-mentioned “main” identity
. Taking
we see that the right-hand side of this identity and the expanat
must agree up to (but not including)
sion of
. This means we have
for
.
, the value(s) of
Notice that in the definition of
for
have no impact on the definition of
because
only at
for
they enter the expansion of
. This is obvious because
.
This completes the proof of Theorem 1.
Since we identified the solution to the “sums of odd powers”
problem with numerator (or denominator) in the (diagonal)
Padé approximation problem, we infer from the standard theory
of continued fraction expansions that the rational functions
are partial fractions in the continued fraction
at
. This
expansion of the generating function
also means (see Szego for these and other facts of the theory
of continued fraction expansions and orthogonal polynomials
is the sequence
[53]) that the sequence of polynomials
of orthogonal polynomials [with respect to the weight that is
], and that the sequence of polyHilbert transform of
satisfies three-term linear recurrence relation.
nomials
Since the same recurrence is satisfied both by numerators and
denominators of the partial fractions, the recurrence is satisfied
and
. With the
by two sequences—
is 1, one obtains the particularly
leading coefficient of
simple three-term recurrence relation among
for
, which was introduced above where it was developed
using the matrix notation.
ACKNOWLEDGMENT
The authors wish to thank H. Huang for helpful comments on
the manuscript.
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Dariusz Czarkowski (M’97) received the M.S.
degrees in electronics and electrical engineering
from the University of Mining and Metallurgy,
Cracow, Poland, and from Wright State University,
Dayton, OH, in 1989 and 1993, respectively, and
the Ph.D. degree in electrical engineering from the
University of Florida, Gainesville, FL, in 1996.
He joined the Polytechnic University, Brooklyn,
NY, as an Assistant Professor of electrical engineering in 1996. His research interests are in the
areas of power electronics, electric drives, and power quality. He coauthored
a book entitled Resonant Power Converters (New York: Wiley Interscience,
1995).
475
David V. Chudnovsky (M’00) received the Ph.D. degree from the Institute of Mathematics of Ukranian
Academy of Science, Kiev, Ukraine, in 1972.
He was Senior Research Scientist in the Department of Mathematics at Columbia University,
New York, from 1978 to 1995, and a Research
Scientist with CNRS and Ecole Polytechnique Paris,
France, from 1977 to 1981. Since 1996, he is a
Distinguished Industry Professor of Mathematics,
Department of Mathematics, Polytechnic University,
Brooklyn, New York, and the Director of Institute for
Mathematics and Advanced Supercomputing (IMAS), Brooklyn, New York.
His research interests include: partial differential equations and Hamiltonian
systems, quantum systems, computer algebra and complexity, large scale
numerical mathematics, parallel computing and digital signal processing. Dr.
Chudnovsky is a member of ACM and APS.
Gregory V. Chudnovsky received the Ph.D. degree
from the Institute of Mathematics of Ukranian
Academy of Science, Kiev, Ukraine, in 1975.
He was Senior Research Scientist in the Department of Mathematics at Columbia University,
New York, from 1978 to 1995 and a Research
Scientist with CNRS at IHES, Paris, France, from
1977 to 1981. Since 1996, he is a Distinguished
Industry Professor of Mathematics, Department of
Mathematics, Polytechnic University and Director
of Institute for Mathematics and Advanced Supercomputing (IMAS), Brooklyn, NY. His research interests include: number
theory, mathematical physics, computer algebra and complexity, large scale
numerical mathematics, parallel computing and digital signal processing. Dr.
Chudnovsky is a member of APS.
Ivan W. Selesnick (S’91–M’95) received the B.S.,
M.E.E., and Ph.D. degrees in electrical engineering,
from Rice University, Houston, TX, in 1990, 1991,
and 1996, respectively.
In 1997, he was a visiting professor at the University of Erlangen, Nurnberg, Germany. He is currently an Aassistant Professor in the Department of
Electrical and Computer Engineering at Polytechnic
University, Brooklyn, NY. His current research interests are in the area of digital signal processing and
wavelet-based signal processing.
As a Ph.D. student Dr. Selesnick received a DARPA-NDSEG fellowship in
1991, and his Ph.D. dissertation received the Budd Award for Best Engineering
Thesis at Rice University in 1996 and an award from the Rice-TMC chapter of
Sigma Xi. He received an Alexander von Humboldt Award in 1997 and a National Science Foundation Career award in 1999. He is a member of Eta Kappa
Nu, Phi Beta Kappa, Tau Beta Phi.