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How to Explain the Empirical Success of Generalized Trigonometric Functions in Processing Discontinuous Signals

Pedro Barragan Olague and Vladik Kreinovich

Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA pabarraganolague@miners.utep.edu, vladik@utep.edu

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1. Fourier Series and Their Limitations: A Brief Reminder

• Isaac Newton:

a prism decomposes each light into lights of different colors.

• Monochromatic light is a sinusoid

x(t) = A · sin(ω · t + ϕ).

• So, any signal can be represented as

x(t) =

n

• i=1

Ai · sin(ωi · t + ϕi).

• This Fourier representation helps in solving many

physics-related differential equations; however: – if we represent a discontinuous signal as a sum of sinusoids, – we get large oscillations near the discontinuity (Gibbs phenomenon).

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2. Fourier Series and Their Limitations (cont-d)

• We can avoid Gibbs phenomenon is we use a linear

combination of discontinuous functions.

• For example, we can use piecewise constant functions

such as Haar wavelets.

• However, the resulting representation is not very com-

putationally efficient for smooth signals.

• We need a representation which is efficient both for

smooth and for discontinuous signals.

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3. Generalized Trigonometric Functions: A Suc- cessful Semi-Heuristic Approach

• A sinusoid can be defined as a function which is inverse

to the integral

• dt

(1 − t2)1/2, expanded by periodicity.

• It is reasonable to consider periodic extension of an

inverse function to a more general integral

• dt

(1 − tp)1/q.

• The derivative of this generalized function is no longer

everywhere continuous.

• The farther p and q from the value 2, the larger this

discontinuity.

• Empirically, these functions are good approximations

both for smooth and for discontinuous signals.

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4. Generalized Trig Functions: Challenge

• However, so far, there have been no convincing theo-

retical explanation for this success.

• In principle, we can think of many generalizations of

trigonometric functions.

• It is not clear whey namely this generalization is em-

pirically 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.

• In this talk, we provide a physics-motivated theoretical

explanation for this empirical success.

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5. Physical Meaning of Sinusoids: Reminder

• Sinusoidal signals are frequently observed in nature,

because they correspond to simple oscillations.

• They correspond to situations in which the potential

energy Epot is equal to Epot = 1 2 · c · x2.

• In Newtonian mechanics, the kinetic energy is equal to

Ekin = 1 2 · m · ( ˙ x)2, so the overall energy is E = Epot + Ekin = 1 2 · c · x2 + 1 2 · m · ( ˙ x)2.

• Sinusoidal oscillations correspond to the idealized case

when we can ignore the friction: 1 2 · c · x2 + 1 2 · m · ( ˙ x)2 = E0 = const.

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6. Physical Meaning of Sinusoids (cont-d)

• Once we know the coordinate x, we can determine ˙

x as ˙ x = dx dt = √2E0 − c · x2 √m .

• This equation can be simplified if we separate the vari-

ables: √m · dx √2E0 − c · x2 = dt.

• In appropriately selected units of time and x, we have

dt = dx √ 1 − x2, thus t =

• dx

√ 1 − x2.

• The desired dependence x(t) of x on t is the inverse

function.

• As we have mentioned, this is exactly the sinusoid.

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7. Scale Invariance

• The formula for the potential energy Epot = 1

2 · c · x2 is scale-invariant in the following sense: – if we change the measuring unit for x to a one which is λ times smaller x′ = λ · x, – then, by appropriately re-scaling the unit for mea- suring energy, i.e., by taking E′ = λ2 · E, – we will have the exact same dependence between E′ and x′ in the new units: E′ = 1 2 · c · (x′)2.

• Similarly, the dependence Ekin = 1

2 · c · ( ˙ x)2 is also scale-invariant.

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8. Our Idea

• Physical laws should not depend on the choice of mea-

suring units.

• Thus, scale-invariance is an important physical princi-

ple.

• Scale-invariance

does not necessarily mean that Epot(x) = c · x2: e.g., Epot = x3 is also scale-invariant.

• Let us therefore consider a general case in which both

Epot(x) and Ekin( ˙ x) are scale-invariant.

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9. Our Idea Leads Exactly to Generalized Trigonometric Functions

• Scale-invariance of the dependence Epot(x) means that:

– for every parameter λ describing re-scaling of the coordinate x, – there exists an appropriate re-scaling µ(λ) of energy – that preserves this dependence, i.e., for which E = Epot(x) implies that E′ = Epot(x′), where E′ = µ(λ) · E and x′ = λ · x.

• Here, µ(λ) · E = Epot(λ · x), i.e.,

µ(λ) · Epot(x) = Epot(λ · x).

• It is known that all continuous solutions of this func-

tional equation have the form Epot(x) = c · xp.

• Similarly, scale-invariance of the expression for kinetic

energy implies that Ekin( ˙ x) = m · ( ˙ x)q.

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10. Our Idea (cont-d)

• Thus, E = Epot(x) + Ekin( ˙

x) = c · xp + m · ( ˙ x)q.

• In the no-friction approximation, energy is preserved:

E = const.

• By selecting appropriate units for E, x, and t (hence

for ˙ x), we get a simplified expression 1 = xp + ( ˙ x)q.

• In this case, ( ˙

x)q = 1−xp, hence ˙ x = dx dt = (1−xp)1/q, dt = dx (1 − xp)1/q, and t(x) =

• dx

(1 − xp)1/q.

• The desired dependence x(t) is the inverse function to

this integral t(x).

• This

is exactly the above-described generalized trigonometric function.

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11. Conclusion

• We have shown that:

– a seemingly arbitrary generalization of sinusoids – can be naturally derived from a physically mean- ingful 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 gen- eralizations of sinusoids, – these functions are the only one which are consis- tent with the corresponding physical model.

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12. Acknowledgments

• This work was supported in part by the National Sci-

ence Foundation grants: – HRD-0734825, – HRD-1242122, and – DUE-0926721.

• The authors are thankful to Jan Lang for his inspira-

tion and encouragement.