This article consists of basics of wavelet analysis required for understanding of and use of wavelet theory. In this article we briefly discuss about HAAR wavelet transform their space and structures.

1. International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 4 Issue 2 ǁ February. 2016 ǁ PP.31-35
www.ijres.org 31 | Page
A Short Report on Different Wavelets and Their Structures.
Sumana R. Shesha* and Achala L. Nargund*
*Post Graduate Department of Mathematics and Research Centre in Applied Mathematics MES College,
Malleswaram 15th
cross, Bangalore-03
Abstract: This article consists of basics of wavelet analysis required for understanding of and use of wavelet
theory. In this article we briefly discuss about HAAR wavelet transform their space and structures.
Keywords: HAAR wavelet transform, Multiresolution analysis (MRA), Daubechies wavelet transforms,
Biorthogonal wavelets.
Mathematical Classification: 42C40
I. Introduction to wavelets [1,2,6,7]
Wavelet analysis is used to decomposes sounds and images into component waves of varying durations,
called wavelets which are localised vibrations of a sound signal or localized detail in an image. This analysis can
be used in signal processing for removing noise. Jean Baptist Joseph Fourier (1807) has introduced frequency
analysis leading Fourier analysis. He also explained the fact that functions can be represented as the sum of sine
and cosine which led to Fourier Transform.
Alfred Haar in the year 1909 introduced wavelets. Haar’s contribution to wavelets is very evident so the
entire wavelet family named after him. The Haar wavelets are the simplest of the wavelet families.
II. Haar wavelets [2,3,4]
A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a
mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other
wavelet transforms.
III. Haar spaces
Let  t be the box function defined by,
 





 

otherwise
t
t
0
101
 . (1)
The equation,
     122  ttt  (2)
is called the dilation equation. The Haar function  t is typically called a scaling function. The space
    RLktspanV Zk
2
0   is called the Haar space 0V generated by the Haar function  t . The set
   Zkkt  forms an orthonormal basis for 0V . The set    Zkkt  also forms a Riesz basis for 0V .
[Riesz basis: Suppose that    RLt 2
 and BA 0 are constants such that
   

Zn
n
Zn
nn
Zn
n cBtccA 22
 (3)
for any square-summable sequence c

. Then we say that the translates    Znn t 
 form a Riesz basis of
   Znn tspanV 
 0 .] The projection of the functions    RLtf 2
 into 0V is given by,
        ktkttftfP
Zk
 
,
(4)
The vector space     RLktspanV Zk
j
j
2
2   is called the Haar space jV .  RLVj
2
 is a vector
space of piecewise constant functions with possible breakpoints at Zj
2 . For each Zk  , define the function








 kt
jj
kj
22
, 22  . (5)
The set    Zkkj t , is an orthonormal basis for jV . The compact support for the functions  tkj, is given
by,

2. A short report on different Wavelets and their structures.
www.ijres.org 32 | Page
  


 
 jj
kk
2
1
,
2
supp kj, . (6)
The dyadic interval ZkjI kj ,,,
, dilation equation and projection function are defined by





 




 
 jjjjkj
k
t
k
Rt
kk
I
2
1
22
1
,
2
,
. (7)
     ttt kjkjkj 12,12,1,
2
2
2
2
   . (8)
        ttfttfP kj
Zk
kjj ,, , 
 , (9)
where jP is called the approximation operator and the space jV is also known as approximation space. The
Haar spaces   ZjjV 
satisfy
   2101 VVVV
 The function   jVtf  if and only if   12  jVtf .

 02101 

  VVVVV
Zj
j
. (10)
  RLVVVVV
Zj
j
2
2101  


IV. Haar wavelet spaces
The wavelet function  t is given by,
     





















otherwise
t
t
ttt
0
1
2
1
1
2
1
01
122 
. (11)
The equation,
     122  ttt  ,
is called the dilation equation. The space     RLktspanW Zk
2
0   is
called the Haar wavelet space 0W generated by the Haar wavelet function  t . The set    Zkkt  forms
an orthonormal basis for 0W . The set    Zkkt  also forms a Riesz basis for 0W . The Haar wavelet space
0W satisfies the following properties.
 If   0Vtf  and   0Wtg  , then     0, tgtf .
 0V and 0W are perpendicular to each other. (12)
 001 WVV  .
The vector space     RLktspanW Zk
j
j
2
2   is called the Haar wavelet space jW .
For each Zk  , define the function  /22
, 2 2
j
j
j k t k   . The set    Zkkj t , is an orthonormal
basis for jW . The compact support for the functions  tkj, is given by,
  


 
 jj
kk
2
1
,
2
supp kj, . (13)
For Zkj , , the dilation equation for  tkj, is given by,

3. A short report on different Wavelets and their structures.
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     ttt kjkjkj 12,12,1,
2
2
2
2
   . (14)
For each Zj  , the detail operator jQ on functions    RLtf 2
 is defined by,
        tfPtfPtfQ jjj  1
. (15)
The Haar wavelet spaces   ZjjW 
satisfy the following properties.
If j and l are integers with jl  , then jV and jW are perpendicular to each other.
If j and l are integers with jl  , then jW and lW are perpendicular to each other.
jjj WVV 1
.
Let   11   jj Vtf be defined as    
 
Zm
mjmj tatf ,1  and suppose that h

is the vector given by
 
T
T
hhh 






2
2
,
2
2
, 10

, then the projection of  tf j 1
into jV can be written as
           

Zk
kj
Zk
kjjkjkj tttftbtf ,,1, ,  (16)
where   k
kkk ahaab

 122
2
2 for Zk  and  T
kk
k
aaa 122 , 

.
Let   11   jj Vtf be defined as
   
 
Zm
mjmj tatf ,11  . (17)
Suppose that  tf j
is the projection of  tf j 1
into jV and g

is the vector given by
 
T
T
ggg 






2
2
,
2
2
, 10
 . (18)
If      tftftg jjj  1
is the residual function in 1jV , then   jj Wtg  and is thus given by
   

Zk
kjkj tctg , (19)
where   k
kkk agaac

 122
2
2 for Zk  and  T
kk
k
aaa 122 , 

.
Thus, a piecewise constant function   0, jtf j
with possible breakpoints at points in Zj
2 can be
decomposed into a piecewise constant approximation function  tf j 1
with possible breakpoints on the coarser
grid Zj 1
2 
and a piecewise constant detail function  tg j 1
with possible breakpoints in Zj
2 . This process
can be iterated and finally  tf j
can be written as the sum of an approximation function  tf0
with possible
breakpoints at the integers and detail functions      tgtgtg j 110 ,,,  with possible breakpoints at
ZZZ j
2,,
4
1
,
2
1

respectively. The next step is to model the decomposition in terms of linear
transformations (matrices). The technique to process discrete data via a discrete version of the decomposition
process is called the discrete Haar wavelet transformation.
V. Discrete Haar wavelet transformation

4. A short report on different Wavelets and their structures.
www.ijres.org 34 | Page
Suppose that N is an even positive integer. The discrete Haar wavelet transformation is defined as,






















































2
2
2
2
0000
00
2
2
2
2
00
0000
2
2
2
2
2
2
2
2
0000
00
2
2
2
2
00
0000
2
2
2
2
2
2
N
N
N
G
H
W
(20)
The N
N

2
block 2NH is called the averages block and the N
N

2
block 2NG is called the details block.
We have, baHN

2
and caGN

2
. The inverse discrete Haar wavelet transformation is given by,







c
b
Wa T
N 

 . (21)
The two-dimensional discrete Haar wavelet transformation B of a matrix A is defines as,
T
NM WAWB  .
The inverse of two-dimensional discrete Haar wavelet transformation is given by,
N
T
M WBWA  .
VI. Advantages of Haar Wavelet Transform
a) It is very simple.
b) It is fast.
c) It can be calculated without any need of temporary array so it needs less memory.
d) It can be reversed exactly without the edge effects.
VII. GRAPHS [5]
Original function xf sin

5. A short report on different Wavelets and their structures.
www.ijres.org 35 | Page
Haar Wavelet
Haar Wavelet for xf sin
VIII. Acknowledgement
We are very much thankful to Prof. N. M. Bujurke, UGC Emeritus professor and INSA Senior Scientist for
introducing us to such a deep subject and helping us in preparing this report. The first author would like to thank
the three academies, namely, the Indian Academy of Sciences, Bangalore, the Indian National Science
Academy, New Delhi and the National Academy of Sciences India, Allahabad for providing an opportunity to
study this subject under Prof. N. M. Bujurke through Summer Research Fellowship Programme.
Refernces
[1.] A brief history of Wavelets – Brendon Keift - Internet
[2.] A Primer on WAVELETS and their Scientific Applications – James S. Walker –Chapman &
Hall/CRC.
[3.] WAVELET THEORY: An Elementary Approach with Applications – David K. Ruch & Patrick J. Van
Fleet – Wiley.
[4.] An Introduction to Wavelet Analysis – David F. Walnut – Birkhäuser.
[5.] Wavelets and Multiwavelets – Fritz Keinert – Chapman & Hall/CRC.
[6.] Wavelets: Tools for Science and Technology – Stephane Jaffard, Yves Meyer, Robert D. Ryan – Siam.
[7.] THE WORLD According to WAVELETS: The Story of a Mathematical Technique in the Making –
Barbara Burke Hubbard – Universities Press.