CC BY
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Mathematical Structures and Modeling 2016. N. 1(37). PP. 25-29
UDC 519.216:517.518.45
how to explain the empirical success of generalized trigonometric functions in processing discontinuous signals
P. Barragan Olague
Student, e-mail: [email protected] V. Kreinovieh
Ph.D. (Phys.-Math.), Professor, e-mail: [email protected] University of Texas at El Paso, El Paso, Texas 79968, USA
Abstract. Trigonometric functions form the basis of Fourier analysis - one of the main signal processing tools. However, while they are very efficient in describing smooth signals, they do not work well for signals that contain discontinuities - such as signals describing phase transitions, earthquakes, etc. It turns out that empirically, one of the most efficient ways of describing and processing such signals is to use a certain generalization of trigonometric functions. In this paper, we provide a theoretical explanation of why this particular generalization is the most empirically efficient one.
Keywords: generalized trigonometric sums, signal processing, Gibbs effect,
Haar wavelets.
Fourier series and their limitations: a brief reminder. One of the discoveries of Isaac Newton was that if we place a prism in the path of (white) solar light, this light will decompose into lights of different colors. From the mathematical viewpoint, a monochromatic light is a sinusoid, i.e., the dependence x(t) of its intensity x on time t has the form x(t) = A ■ sin(w ■ t + ф) for some constants A, ш, and ф. The intensity of original white light is equal to the sum of these
n
components, i.e., to x(t) = A, ■ sin(w, ■ t + ф,).
i= 1
Newton showed that any light can be decomposed in this way, i.e., in effect, that any signal x(t) can be represented as a linear combination of sinusoids corresponding to different frequencies ш.
This idea was explored in the early 19 century by Jean-Baptiste Joseph Fourier, who showed that this representation helps in solving many physics-related differential equations. Computational methods based on such a representation are known as Fourier techniques. At present, these techniques are ubiquitous in science and engineering; see, e.g., [5].
However, the Fourier techniques have their limitations: while they work well for smooth signals, they do not work as well for discontinuous signals that describe abrupt transitions - such as phase transitions, earthquakes, etc. Specifically, if we
26 Р.В. Qlague, V. Kreinovich. How to Explain the Empirical Success...
represent a discontinuous signal as a sum of sinusoids, we get large oscillations near the discontinuity; this is known as the Gibbs phenomenon.
It is possible to avoid these oscillations if, instead of representing a signal as a linear combination of sinusoids, we represent it as a linear combination of discontinuous functions - e..g., Haar wavelets [4] - but the resulting representation is not very computationally efficient for smooth signals.
It is therefore desirable to come up with a representation which would be efficient both for smooth and for discontinuous signals.
Generalized trigonometric functions. A successful semi-heuristic approach to solving the above problem is the use of generalized trigonometric functions instead of the sinusoids. Specifically, a sinusoid can be defined as a function which is inverse to the integral
dt
y/T—T2
dt
(1 -12)1/2 ’
expanded by periodicity to the entire real line. A generalized trigonometric function can be defined as a periodic extension of an inverse function to a more general integral
Г dt
J (1 - tp)1/q
for general values p and q. The derivative of this generalized function is no longer everywhere continuous - and the farther p and q from the value 2, the larger this discontinuity.
Empirically, these functions - for appropriate p and q - are good approximations both for smooth and for discontinuous signals; see, e.g., [2, 3].
Challenge. The empirical success is here, but so far, there has been no convincing theoretical explanation for this success. In principle, we can think of many generalizations of trigonometric functions, and it is not clear why namely this generalization is empirically successful.
This absence of theoretical explanation prevents the wider use of this technique: the users are reluctant to use it, since they are not sure that the empirical success so far is not an artifact.
Our objective. In this paper, we provide a physics-motivated theoretical explanation for the empirical success of the generalized trigonometric functions.
Physical meaning of sinusoids: reminder. Sinusoidal signals are frequently observed in nature, because they correspond to simple oscillations. Namely, they
correspond to situations in which the potential energy Epot is equal to Epot = -■c-x2 for some constant c. In Newtonian mechanics, the kinetic energy is equal to Ekin = - ■ m ■ (X)2. Thus, the overall energy E = Epot + Ekin is equal to
E
2 - C - X +
1
2
■ m ■ (X)2.
Mathematical Structures and Modeling, 2016. N. 1(37)
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Sinusoidal oscillations correspond to the idealized case when we can ignore the friction and when, therefore, the energy is preserved:
1
2
c ■x +2
■ m ■ (x)2
Eo
const.
Thus, once we know the coordinate x, we can determine x as
(x)2
2E0 — c ■ x2 m
so ____________
dx V2E0 — c ■ x2
x = — =---------=------.
dt ym
This equation can be simplified if we separate the variables, i.e., if we move all the terms related to x to the left-hand side and all the terms related to t to the right-hand side. This can be done if we divide both sides of the above formula by the right-hand side and then multiply both sides by dt:
dx
m
\J2E0 — c ■ x2
dt.
In appropriately selected units of time and x, we have
dt
dx
Vl — x2 ’
thus, the dependence t(x) of t on x has the form
t
dx
Vl — x2
The desired dependence x(t) of x on t is the inverse function - which, as we have mentioned, is exactly the sinusoid.
Discussion. The formula for the potential energy Epot = 1 ■ c ■ x2 is scale-invariant in the sense that:
• if we change the measuring unit for x to a one which is A times smaller and thus, change all the numerical values from x to x' = A ■ x, •
• then, by appropriately re-scaling the unit for measuring energy, i.e., by taking E' = A2 ■ E, we will have the exact same dependence between E' and x' in
the new units: E' = - ■ c ■ (x')2.
Similarly, the dependence Ekin = - ■ c ■ (x)2 is also scale-invariant.
Our idea. Scale-invariance - i.e., the fact that the physical laws do not depend on the choice of measuring units - is an important physical principle. However,
28 Р.В. Qlague, V. Kreinovich. How to Explain the Empirical Success...
scale-invariance does not necessarily mean that the potential energy should be proportional to the square of x: e.g., the dependence Epot = x3 is also scale-invariant.
Let us therefore consider a general case in which both components Epot(x) and Ekin(X) of the overall energy E = Epot(x) + Ekin(X) are scale-invariant.
Our idea leads exactly to generalized trigonometric functions. Scale-invariance of the dependence Epot(x) means that for every parameter A describing re-scaling of the coordinate x, there exists an appropriate re-scaling p(A) of energy that preserves this dependence, i.e., for which E = Epot(x) implies that E' = = Epot(x'), where E' = p(A) ■ E and x' = A ■ x. Substituting the expressions for E' and x' into the above formula, we get p(A) ■ E = Epot(A ■ x). Since E = Epot(x), we thus get p(A) ■ Epot(x) = Epot(A ■ x).
It is known (see, e.g., [1]) that all continuous (or even integrable) solutions of this functional equation have the form Epot(x) = c ■ xp for some constants c and p. Similarly, scale-invariance of the expression for kinetic energy implies that Ekin(x) = m ■ (x)q for some constants m and q.
Thus, the overall energy E = Ekin + Epot takes the form E = c ■ xp + m ■ (x)q. In the no-friction approximation, energy is preserved, so the left-hand side is a constant. By selecting appropriate units for energy, we can make this constant equal to 1. Then, by selecting appropriate units for x and for time (hence for x), we can get a simplified expression 1 = xp + (x)q. In this case, (x)q = 1 — xp, hence
x = dx = (1 — xp)1/q,
so
dx
dt =
(1 — xp)1/q and
t(x) = /_____*_____
( ’ J (1 — xp)1/q'
The desired dependence x(t) is the inverse function to this integral t(x) - and is, thus, exactly the above-described trigonometric function.
Conclusion. We have shown that a seemingly arbitrary generalization of sinusoids can be naturally derived from a physically meaningful model - and the only functions obtained from this model are indeed the generalized trigonometric functions. This derivation provides a theoretical explanation of the empirical success of these functions - while there are many mathematically possible generalizations of sinusoids, these functions are the only ones which are consistent with the corresponding physical model.
Acknowledgments. This work was supported in part by the National Science Foundation grants HRD-0734825, HRD-1242122, and DUE-0926721.
The authors are thankful to Jan Lang for his inspiration and encouragement.
Mathematical Structures and Modeling, 2016. N. 1(37)
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References
1. Aezel J. Lectures on Functional Equations and Their Applications. New York : Dover, 2006.
2. Edmunds D.E., Gurka P., Lang, J. Properties of generalized trigonometric functions // Journal of Approximation Theory. 2012. V. 164. P. 47-56.
3. Lang J., Edmunds D. Eigenvalues, Embeddings, and Generalised Trigonometric Functions // Springer Lecture Notes in Mathematics. 2011. V. 2016.
4. Mallat S. A Wavelet Tour of Signal Processing: The Sparse Way. Burlington, Massachusetts, USA : Academic Press, 2008.
5. Stein E.M., Shakarchi R. Fourier Analysis: An Introduction. Princeton, New Jersey, USA : Princeton University Press, 2003.
как объяснить эмпирическим успех обобщенных тригонометрических функций в обработке разрывных
сигналов
П. Барраган Олаге
студент, e-mail: [email protected] В. Крейнович
к.ф.-м.н., профессор, e-mail: [email protected] Техасский университет в Эль Пасо, США
Аннотация. Тригонометрические функции лежат в основе анализа Фурье — одного из основных методов обработки сигналов. Тригонометрические функции очень эффективны, пока сигналы гладкие, но если сигнал содержит неожиданные скачки, например, если сигнал описывает фазовые переход или землетрясение, то применение Фурье-методов ведёт к известным трудностям (таким как эффект Гиббса). Численные эксперименты показывают, что для описания и обработки таких сигналов один из самых эффективных методов состоит в использовании подходящего обобщения тригонометрических функций. В этой статье мы проводим теоретическое объяснение этого эмпирического успеха.
Ключевые слова: обобщённые тригонометрические суммы, обработка сигналов, эффект Гиббса, вейвлеты Хаара.
