An Introduction to Wavelets · An Introduction to Wavelets Dr. Sunil Kumar Singh Associate Professor Department of Mathematics Mahatma Gandhi Central University Motihari-845401, Bihar,

An Introduction to Wavelets

Dr. Sunil Kumar Singh

Associate ProfessorDepartment of Mathematics

Mahatma Gandhi Central UniversityMotihari-845401, Bihar, India

E-mail: sks [email protected]

April 4, 2020

Dr. Sunil Kr. Singh (M.G.C.U.B.) An Introduction to Wavelets April 4, 2020 1 / 21

Wavelet analysis is a rapidly developing area in Mathematical Sciences. Mor-let and Grossman were the first introduced the wavelets in the beginning of1980. Based on the idea of wavelets as a family of functions constructedfrom translation and dilation of a single function ψ, called the ‘motherwavelet’, we define mother wavelet as [4]

ψa,b(t) =1√|a|

ψ

(t − b

a

); a, b ∈ R, a 6= 0,

where a is called a scaling parameter, and b is a translation parameter whichdetermines the location of the wavelet.


Definition (Wavelets)

A function ψ is a wavelet if it satisfies the following conditions

(i) ψ(t)→ 0 as |t| → ∞(ii)

∫∞−∞ ψ(t) dt = 0.

Definition

Wavelets in L2(R) : A function ψ ∈ L2(R) is a wavelet if it satisfies the‘admissibility condition’

Cψ :=

∫ ∞−∞

|ψ(ξ)|2

|ξ|dξ <∞, (1)

where ψ(ξ) is the Fourier transform of ψ(t).


Example (The Haar Wavelet)

The Haar wavelet is one of the classic examples. It is defined by

ψ(t) =

1, if 0 ≤ t < 1

2

−1, if 12 ≤ t < 1

0, otherwise.

The Haar wavelet has compact support. It is obvious that∫ ∞−∞

ψ(t)dt = 0, and

∫ ∞−∞|ψ(t)|2 dt = 1.

This wavelet is well-localized in the time domain but it is not continuous.


-1 1/2 1

−1

1

t

ψ(t)

Figure: Haar Wavelet


Example

The function ψ(t) = t e−t2

satisfies the admissibility condition and hencewavelet.Note that: ∫

R|ψ(t)|2 dt =

√π

4√

2<∞,

ψ(ξ) = − i√π ξ

2e−ξ

2/4,

and

Cψ =

∫R

|ψ(ξ)|2

|ξ|dξ =

π

2<∞.


-3 -2 -1 1 2 3

−1

1

t

ψ(t)

Figure: ψ(t) = t exp(−t2)


Example (The Mexican Hat wavelet)

The Mexican hat wavelet is defined by

ψ(t) = (1− t2) exp(− t2

2 ),

ψ(ξ) =√

2πξ2 exp(− ξ2

2 ).


-3 -2 -1 1 2 3

−1

1

t

ψ(t)

Figure: ψ(t) = (1− t2) exp(− t2

2 )


Example (The Shannon wavelet)

The Shannon function ψ whose Fourier transform satisfies

ψ(ξ) = χI (ξ),

where I = [−2π,−π]∪ [π, 2π], is called the Shannon wavelet. The waveletψ(t) is obtained from the inverse Fourier transform of ψ(ξ), so that

ψ(t) =sin(πt2 )

(πt2 )cos(3πt2 ).

Example (The Morlet wavelet)

The Morlet wavelet is defined by

ψ(t) = exp(iξ0t − t2

2 ),

ψ(ξ) =√

2π exp(− (ξ−ξ0)22 ).


Theorem

If ψ ∈ L2(R) is a wavelet and φ is an absolutely integrable function, thenthe convolution function (ψ ∗ φ) is a wavelet.


Proof.

∫ ∞−∞|(ψ ∗ φ)(x)|2 dx =

∫ ∞−∞

∣∣∣∣∫ ∞−∞

ψ(x − u)φ(u)du

∣∣∣∣2 dx≤∫ ∞−∞

(∫ ∞−∞|ψ(x − u)| |φ(u)| du

)2

dx

=

∫ ∞−∞

(∫ ∞−∞|ψ(x − u)| |φ(u)|1/2 |φ(u)|1/2 du

)2

dx

≤∫ ∞−∞

(∫ ∞−∞|ψ(x − u)|2 |φ(u)| du

∫ ∞−∞|φ(u)| du

)dx

=

∫ ∞−∞|φ(u)| du

∫ ∞−∞

∫ ∞−∞|ψ(x − u)|2 |φ(u)| dudx

=

∫ ∞−∞|φ(u)| du

∫ ∞−∞|φ(u)|

∫ ∞−∞|ψ(x − u)|2 dxdu

=

(∫ ∞−∞|φ(u)| du

)2

‖ψ‖22 <∞.


Thus ψ ∗ φ ∈ L2(R). Moreover

∫ ∞−∞

|F (ψ ∗ φ)|2

|ξ|dξ =

∫ ∞−∞

∣∣∣ψ(ξ)φ(ξ)∣∣∣2

|ξ|dξ

=

∫ ∞−∞

∣∣∣ψ(ξ)∣∣∣2 ∣∣∣φ(ξ)

∣∣∣2|ξ|

dξ

≤(

sup∣∣∣φ(ξ)

∣∣∣2)∫ ∞−∞

∣∣∣ψ(ξ)∣∣∣2

|ξ|dξ <∞.

Thus the convolution function ψ ∗ φ is a wavelet.


Example

If we take the Haar wavelet

ψ(t) =

1, if 0 ≤ t < 1

2

−1, if 12 ≤ t < 1

0, otherwise,

and convolute it with the following function

φ(t) =

0, if t < 0

1, if 0 ≤ t < 1

0, t ≥ 1


we obtain a simple wavelet

(ψ ∗ φ)(t) =

∫ ∞−∞

ψ(t − u)φ(u)du

=

∫ 0

−∞ψ(t − u)φ(u)du +

∫ 1

0ψ(t − u)φ(u)du

+

∫ ∞1

ψ(t − u)φ(u)du

=

∫ 1

0ψ(t − u)du

=

∫ t

t−1ψ(x)dx .


-1 1/2 1 3/2 2

-1/2

1/2

t

(ψ ∗ φ)(t)

Figure: The wavelet (ψ ∗ φ)(t)


Example

The convolution of the Haar wavelet with φ(t) = exp(−t2) generates asmooth wavelet.


Figure: The wavelet (ψ ∗ φ)(t)


References I

[1] A. Boggess, and F. J. Narcowich, A First Course in Wavelets withFourier Analysis, Wiley; 2nd ed., 2009.

[2] C. K. Chui : An Introduction to Wavelets, Academic Press, New York,1992.

[3] I. Daubechies : Ten Lectures on Wavelets, CBMSNSF Regional Con-ference Series in Applied Mathematics (SIAM,1992).

[4] Lokenath Debnath and Firdous Ahmad Shah : Wavelet Transforms andTheir Applications, 2nd ed., Brkhauser, Boston, 2015.

[5] E. Hernaddez and G. Weiss : A First Course on Wavelets, CRC Press,Boca Raton, Florida, 1996.

[6] T. H. Koornwinder : Wavelets, World Scientific PublishingCo.Pvt.Ltd., Singapore , 1993.

[7] R. S. Pathak : The Wavelet transform, Atlantis Press/ world Scientific,France, 2009.


References II

[8] P. Wojtaszczyk : A Mathematical Introduction to Wavelets, CambridgeUniversity Press, 1997.

[9] D. F Walnut : An Introduction to Wavelet Analysis, Brkhauser, 2004.

[10] A. H. Zemanian : Generalized Integral Transformations, IntersciencePublishers, New York, 1996.


THANK YOU


Documents

An Introduction to Wavelets · An Introduction to Wavelets Dr. Sunil Kumar Singh Associate Professor Department of Mathematics Mahatma Gandhi Central University Motihari-845401, Bihar,