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Use of Daubechies Wavelets in the Representation of Analytical Functions

By Paulo César Linhares da Silva

Submitted: August 21st 2020Reviewed: September 4th 2020Published: October 1st 2020

DOI: 10.5772/intechopen.93885

Downloaded: 18

Abstract

This chapter aims to use Daubechies’ wavelets as basis functions to generate analytical functions, thus being able to rewrite the Taylor series using these wavelets. This makes it possible to analyze functions with a high degree of complexity, in problems that require a high degree of precision in their solution. Wavelet analysis can be applied to practical problems that require a high degree of precision, for example, in the study and analysis of electromagnetic propagation in optical fibers, solutions of differential equations involving engineering problems, in the transmission of WiFi signals, in the treatment and analysis of biomedical images, detection of oil sources through the study of seismic signals.

Keywords

  • wavelets
  • Daubechies
  • analytical functions
  • basis functions
  • Taylor series

1. Introduction

Wavelets [1] were born from the need to generate functions, especially those that present singularities, high gradients, discontinuities both in the time domain and in the frequency domain. Wavelets enable the high-resolution analysis of functions with these characteristics. An example of a problem that occurs when generating functions with a Fourier base is the Gibbs phenomenon. Such a phenomenon occurs because there is no way to represent functions that present discontinuities, even adding more elements in the base that will generate the function. A characteristic of wavelets is that they do not produce such an effect.

Wavelets are widely used in the solution of numerical problems in several areas of knowledge such as image compression, Numerical Harmonic Analysis [2], financial analysis, oil detection, differential Equations [3, 4], biomedical signals, analysis of electromagnetic integral Equations [5], optical fibers [6], among others. Many of these applications use the specific properties of wavelets, such as coefficients that are determined numerically, multi-resolution analysis to decompose a signal, integrals, and derivatives obtained numerically, energy concentrated in its compact and base with orthogonal elements.

2. Short introduction to wavelet theory

For the development of topics presented in this chapter, the reader must have as a prerequisite knowledge of functional analysis, linear algebra, measure theory and integration, differential and integral calculus. It is important to note that the wavelet basis is for the wavelet transform as well as the trigonometric basis is for the Fourier transform. Generally, the term wavelet is also used as a wavelet transform. The following subsections present these initial prerequisites to the reader.

2.1 Preliminaries on Hilbert spaces

In this subsection, some mathematical concepts necessary for a better formal understanding of the wavelet tool are defined. The definitions, contained in this section, are due to the author [2].

Definition 2.1 The space His said to be a Hilbert space, if an inner product <,>, associated with a standard =<,>has been defined in it. And a set of vectors vi, for iNan orthonormal system is said if the internal product <vn,vm>=δmn, for m,nN.

Definition 2.2 A set of vectors vnis orthonormal, if and only if, for every finite set of complex numbers xn, there is nanxn2=an2, for nN.

Definition 2.3 In Hilbert’s Hspace, a set of vectors vnis said to be a Riez system, if there are constants 0cC<such that for any finite set of complex numbers xnif you have:

can2nanxx2Cnan2E1

Definition 2.4 The space L2Ris said to be an integrable square function space, that is,

L2R=f:RC:Rfx2dx<E2

For f,gL2R, define the inner product <f,g>=Rfxgx¯dx. On what, gx¯is the complex conjugate of the function gx.

In particular f=f2=Rfx2dx12, and fis said to be an integrable square.

Definition 2.5 Let f:RCbe a function. The support of f, denoted by suppf, is the closing of the set xR:fx0. A function fis said to have compact support if the suppfset is compact.1

Definition 2.6 We say that a function fis generated by the basis functions f1fn, if coefficients exist c1cnsuch that:

f=i=1nficiE3

The concepts presented here about orthogonality and support of a ffunction, are fundamental to formalize the definition of wavelet. The following subsection presents the formal mathematical concept of wavelet.

2.2 Definition of wavelet

This subsection aims to define wavelet [2], the main mathematical tool used in the development of this chapter. However, it is necessary to define the expansion and translation of mathematical operations beforehand.

Definition 2.7 Given a>0, the expansion operator, Da, defined over a fxfunction in L1or L2over R, is given by, Da fx=a12fx.

Definition 2.8 Given bR, the translation operator, Tb, defined over a function fx, in L1or L2over R, is given by, Tb fx=fxb.

Thus, using the expansion and translation operations defined above, a family of functions ψj,kxwas built: L2R, base orthogonal to L2R.

ψj,kxj,kZ=2j2ψ2jxkj,kZ=D2jTkψxj,kZE4

The Definition 2.9, uses the family of functions ψj,kxj,kZ, to define the term mathematically wavelet.

Definition 2.9 A function ψxis called wavelet if the collection ψj,kxj,kZis an orthogonal basis on L2R. Where jand kare the resolution and translation of wavelet respectively.

By varying the values of jand/or k, it is possible to analyze with greater precision, for example, the behavior of functions that present abrupt changes in values and discontinuity. This type of analysis makes the wavelet a tool as or more efficient than the basic Fourier functions.

The definition 2.10 is another way used to define a wavelet.

Definition 2.10 A wavelet2 is a short duration wave, which has an average value equal to zero.

Due to the definition 2.10, wavelets resemble Fourier sine and cosine basis functions. Analogously to what is done in the Fourier transform, which has sine and cosine functions as base functions, in wavelet analysis, a function is decomposed into a base of wavelet functions.

The Fourier transform Fωexpression of a ftfunction is given by (5):

Fω=+fteiωtdtE5

The expression (5) means that the Fourier transform is the sum of every ftsign multiplied by a complex exponential, which can be separated into cosine and sinusoidal components in the real and complex parts, respectively.

Similarly, the expression of the wavelet transform Wj,kfof a function ft, is given by (6):

Wj,kf=+ftψj,ktdtE6

Similarly, the expression of the wavelet transform (6) is the internal product of the signal to be transformed by a wavelet function.

In the following subsection, among the most varied types of wavelets, the Daubechies wavelets are highlighted, which are the basis for the development of this chapter.

2.3 Daubechies wavelet properties

At 1988, a family of compact support wavelets [7] is built by Ingrid Daubechies. This family of wavelets has highly well-located elements. Each member wavelet is governed by a set of Ninteger coefficients and k=0.1N1coefficients through scale relations (7) and (8). The akand a1kcoefficients, which appear in the (7) and (8), are called filter coefficients and verify the following relations:

ϕx=k=0N1akϕ2xkE7
ψx=k=2N11ka1kϕ2xkE8

In the Figures 1 and 2 below, we have the graphical representation of the Daubechies wavelet functions ϕand ψof kind 4.

Figure 1.

Daubechies wavelets ϕ . Source: This figure was generated by the author using the python programming language.

Figure 2.

Daubechies wavelet ψ . Source: This figure was generated by the author using the python programming language.

The functions ϕin (7) and ψin (8) are called the scale function ϕand wavelet function ψ, respectively. The fundamental support of the scale function3 is the interval 0N1as the fundamental support of wavelet function ψxis the interval 1N2N2. In the case of N=4, we have the graphs of the Figures 1 and 2.

To determine the filter coefficients akand a1k, which appear in the (7) and (8), we use the relations (9)(12) below.

k=0N1ak=2E9
k=0N1akakm=δ0,mE10
k=0N11ka1kak2m=0E11
k=0N11kkmak=0,m=0.1,,N21,E12

where δk,mis the Kronecker Delta function.

3. Generating an analytical function of the type xkusing wavelets

Analytical functions are those that can be locally around a point x0expanded in a Taylor series, according to the following expression.

fx=n=0+fnx0n!xx0nE13

In general according to the author [8], any fxfunction can be represented in terms of a wavelet base, as follows:

fx=m=+ckϕxm=m=+cmϕmx.E14

The ckcoefficients are called moments of the scale functions. In particular, for fx=xk, we have the expression (15), below:

xk=m=+Mmk2jkϕ2jxm,E15

Since Mmkthe moment of the wavelet scales concerning the xkmonomial, where k is the degree of the polynomial, mand jare the translation and resolution of the ϕwavelet. The justification for the construction of the equation is found in the work of [8, 9, 10], in which the author concludes that the cmjcoefficients for approximating a monomial of the xkform, using a Daubechies wavelet base ϕ, looks like this:

cmj=Mmk2jkE16

The justification used in the approximation (15) of a polynomial function of type fx=xkderives from the number of null moments,

+xkψxdx=0,k=0.1,,N21E17

According to the Eq. (17), the NDaubechies Wavelet has N2vanish moments, being possible to represent a polynomial of degree at most N21, using the ϕxscale function. The polynomial approximation using the scale function is formalized in the following definition.

Definition 3.1 A wavelet has pvanish moments (18), if and only if, the wavelet scale function ϕcan generate polynomials of degree up to p1[Eq. (19)]. That is, the scale function alone can be used to represent these polynomials. The fact that it has more null moments means that the scale function can represent more complex functions.

+xmψxdx=0;m=0,1,,N21E18
fx=p1+p2x++pk1xk,kN21E19

In general, a Daubechies wavelet of kind N, properly translated and adjusted to the appropriate resolution level, generates a polynomial of degree k, with the relation between Nand kgiven by N=2k+2. For example, to generate a polynomial of degree 1a wavelet of Daubechies of kind 4is necessary.

To generate a polynomial with n+1terms, in the function of Daubechies wavelets of genres 4,6,8,,N1, we use the momentum equation and the polynomial expansion as a function of wavelets.

px=k=0nakxk,E20

where xk, takes the form

xk=m=+Mmk2jkϕ2jxmE21

Substituting the Eq. (21) in (20), we have:

px=k=0nak2jkm=+ϕ2jxmMmkE22

where kis the degree of the polynomial jand mare the resolution and translation of the wavelet respectively.

In the next subsection, the calculation of the moment generating function, which appears in the expression 21 as a coefficient of xk, is shown in detail.

4. Moment generating function

The calculation of the moment generating function according to the author [11] is of fundamental importance to approximate the functions by wavelets. The deduction of the moment-generating function now begins. For this, the mathematical expression is used

Mmk=+xkϕxmdxE23

which refers to the moment of the wavelet scale ϕin relation to the monomial xk.

For m=k=0, in (23), we have:

M00=+ϕxdx=1.E24

Substituting m=0in the Eq. (23), we have:

M0k=+xkϕxdx=+xks=0N1asϕ2xsdxE25
M0k=s=0N1as+xkϕ2xsdx.E26

Note that the variable s, in the Eq. (26), also represents a translation.

Making the substitution z=2x, dzdx=2, dx=dz2, we have:

M0k=s=0N1as+xkϕ2xsdxE27
M0k=12k+1s=0N1asMskE28

Using the substitution xm=t, dxdt=1, dx=dt, in (23), we have:

Mmk=+xkϕxmdx=r=0kkrmkrM0rE29

Now consider the equations:

M0k=12k+1s=0N1asMskE30
Msk=r=0kkrskrM0rE31

Substituting Mskin M0k, we have: (note that m=s)

M0k=12k+1s=0N1asr=0kkrskrM0rE32

Now separate the last term of the sum (32), r=k, to place the term on the left side of the equation:

M0k=12k+1s=0N1asr=0k1krskrM0r+12k+1s=0N1askrskkM0kE33

Using the fact that s=0N1as=2, we have:

M0k=122k1r=0k1krM0rs=0N1asskrE34

Thus, the equations are obtained:

Mmk=r=0kkrmkrM0rE35
M0k=122k1r=0k1krs=0N1asskrM0rE36

From (35), (34), and (24), we get the moment generating function Mmk:WR, where Wis wavelet space, mis the translation of the scale function and kis the degree of the polynomial to be approximated.

Mmk=122k1r=0k1krs=0N1asskrM0r,sem=0;k0r=0kkrmkrM0r,sem0;k01,sem=k=0,E37

The analytical expression for Mmkwas developed during the author’s research [11] and to validate the results found, a comparative study was made with other numerical results [12, 13] of the scientific literature.

Similar to what was done with the calculation of the moments for the function ϕ, there is also the calculation of the moments for the function ψ. This is given by integral (38)

+xmψxdx=0;m=0,1,,N21.E38

The following is an example of the calculation of the moments for the case of Daubechies wavelets of a kind N=4.

Example 4.1 In this example, the Daubechies wavelet of kind 4is used to generate the analytical polynomial function fx=x. According to the definition 3.1, the scale function of Daubechies of genus N=4, generates a line (polynomial of degree 1). To represent a 1monomial with a 4Daubechies wavelet in the 01range, the translations ϕx,ϕx+1,ϕx+2, whose supports are 0.3,1.2,2.1, that is:

x=m=20Mm12jkϕ2jxm=M212jϕx+2+M112jϕx+1+M012jϕxE39

The support of the linear combination (39), represented in Figure 3, is obtained by the intersection of the supports of the translations of the function ϕx. This intersection results in the interval I=0.1. This fact defines well the function to be integrated in the Irange. In Figure 3, the number of translations of the function ϕxto generate fx=xis illustrated.

Figure 3.

Translations required to represent the analytical function f x = x using Daubechies wavelets of kind 4 . Source: Own authorship.

Figure 4 shows the graph of translated functions ϕx,ϕx+1and ϕx+2respectively, that form a base to generate the function fx=x.

Figure 4.

Translations required to represent the analytical function f x = x using Daubechies wavelets of kind 4 . Source: This figure was generated by the author using the python programming language.

The calculation using the moment generating function depends on the Daubechies wavelet coefficients of kind 4. These coefficients are obtained by the Eqs. (9)(12), which gives rise to the following non-linear system.

a0+a1+a2+a3=2a02+a12+a22+a32=2a0a2+a1a3=0a1+2a23a3=0E40

The solution of this system is the irrational numbers a0,a1,a2,a3, given by:

a0=0,683012701892219a1=1.183012701892219a2=0,316987298107781a3=0,183012701892219E41

Using the moment generating function for the case where m=0, we have:

M01=12r=001rM0rs=03ass1r=12M00a1+2a2+3a3=a1+2a2+3a32=0,633974600E42

Proceeding with the calculations, we obtain:

Mmk=r=011rm1rM0r=m+M01=m+a1+2a2+3a32=m+0,633974600E43

Replacing the value of mby m=1, m=2and k=1, we obtain:

M11=0,366025400E44
M21=1,366025400E45

So, the representation for the xpolynomial (for a resolution j=0) is:

x=0,634ϕx0,366ϕx+11.366ϕx+2E46

In Figure 5, we have the graphical representation of the function obtained of the expression (46). Here the function fx=xis generated by linear combination of wavelets ϕx,ϕx+1and ϕx+2.

Figure 5.

Function f x = x using Daubechies wavelets of kind 4 . Source: This figure was generated by the author using the python programming language.

The representation for the expression (46) using the summation is given by,

x=m1=20Mm112jkϕ2jxm1E47

The expression for writing polynomials of degrees k=2and k=3in terms of Daubechies wavelets is given by

x2=m2=42Mm222jkϕ2jxm2E48
x3=m3=60Mm332jkϕ2jxm3E49

See that to generate the polynomials (48), (49) is necessary to use Daubechies wavelets of kind 6and 8, according with the definition 3.1.

4.1 Taylor polynomial using Daubechies wavelets

The Taylor polynomial or Taylor series is an expression that allows the calculation of the local value of a function fusing your derivatives. For this, the function fmust be of class Cinfinite (represented by C) which implies that the f is infinitely derivable in an interval containing a point x0. The expression for the Taylor polynomial for the function fis as follows,

fx=k=0+fkx0k!xx0kE50

The expression (50) developed around x0=0is:

fx=k=0+fk0k!xkE51

Making use of the expression (21), we have:

fx=k=0+m=+fk0k!Mmk2jkϕ2jxmE52

The expression (52) is another way of writing Taylor’s polynomial using Daubechies Wavelets.

Example 4.2 Consider the analytical function fx=ex, using Daubechies wavelet of kind a N=4is possible to write this function fin terms of this wavelet. For this, Taylor’s series development around the point x0=0of this function is given by:

ex=n=0+xnn!E53

Using only two summation terms in the expression (53), we have:

exn=01xnn!=1+xE54

Using the expression (46), we have:

ex1+x=1+0,634ϕx0,366ϕx+11.366ϕx+2E55

The expression (55) allows us to approximate the exponential function using a base of Daubechies wavelets. This type of approximation, although simple for this case, is very useful in the case of representation for functions other types.

In the following example, the expression (46) is used to approximate Taylor’s series developments for the functions sx=ex,fx=coshx,gx=sinhxand hx=ln1+x.

Example 4.3 For the functions fx=coshx,gx=cosxand hx=secx. Taylor’s series development of these functions around the point x0=0is:

coshx=n=0+x2n2n!E56
sinhx=n=0+x2n+12n+1!E57
ln1+x=n=1+1n+1xnnE58

In order to verify the potentiality of the application of Daubechies wavelets we will calculate the value of the functions in (53), (56), (57) and (58) evaluated at point x = 1. Considering only 7terms in each summation. For obtain the results using Daubechies wavelets we apply the expression (55) in each summation (53), (56), (57) and (58). In the Table 1 we have a comparison between the calculation of the values of the functions sx=ex,fx=coshx,gx=sinhxand hx=ln1+xevaluated at point x=1, using the Taylor series and the Daubechies wavelets of kind 4.

FunctionValue in x=1, Taylor Series4Value in x=1, Daubechies Wavelets5Error %
sx=ex2.7166666672.7167354694433290.0025%
fx=coshx1.5430881617917531.5430583112874780.0019%
gx=sinhx1.17501998401278971.17505915211088220.00376%
hx=ln1+x0.64563492031220080.64563491902143071,9.107%

Table 1.

Comparison of the values obtained by the Taylor series and by Daubechies wavelets.

Calculation using Taylor Series.


Calculation using Daubechies wavelets of kind 4.


Table 1 appears here only as a way of showing the quality of the approximations using the Daubechies wavelets of kind 4. Obviously if we want more precise values, we must use Daubechies wavelets of the kind greater than 4. This will cause changes in the resolution and translation of each wavelet, but the result will be even better.

5. Conclusions

Daubechies wavelets are quite versatile mathematical tools. They can be used to analyze, generate, decompose a function, or even a signal that is represented by an analytical function. This type of application is widely used, for example, in electrical engineering in studies of magnetic fields and electric fields. The theory exposed in this chapter provides tools to carry out these studies. The use of the Taylor series as a way of approximating analytical functions is a very used technique in applied mathematics. Making use of the Taylor series with wavelets is another option to perform an approximation of analytical functions. In future work, we are researching other wavelets, for example Deslauries-Dubuc interpolets, that have an even better approach quality. As Deslauriers-Dubuc interpolets and others in research.

Acknowledgments

The author would like to thank UFERSA for support during my doctoral studies.

Notes

  • A set is said to be compact if it is limited and closed
  • Anglophone term to designate a small wave, in the sense of having a fast duration.
  • We emphasize that the scale function has energy concentrated in its support that is determined by the genus of the wavelet, that is, supp ϕ = 0 N − 1 , and that the total energy of the scale function is unitary, that is, ∫ − ∞ + ∞ ϕdx = 1 .

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Paulo César Linhares da Silva (October 1st 2020). Use of Daubechies Wavelets in the Representation of Analytical Functions [Online First], IntechOpen, DOI: 10.5772/intechopen.93885. Available from:

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