Applied Mathematics and Computation 210 (2009) 473–478

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

New construction of wavelets base on floor function

Zulkifly Abbas a, S. Vahdati a,*, M. Tavassoli Kajani b, K.A. Atan c

a Laboratory of Computational Science and Informatics, Institute for Mathematical Research, University Putra Malaysia, Serdang 43400, Selangor, Malaysiab Department of Mathematics, Islamic Azad University, P.O. Box 81595-158, Khorasgan, Isfahan, Iranc Laboratory of Theoretical Studies, Institute for Mathematical Research, University Putra Malaysia, Serdang 43400, Selangor, Malaysia

a r t i c l e i n f o

Keywords:Floor functionStepwise functionHaar waveletSine–Cosine waveletBlock-Pulse functionsHybrid Fourier Block-Pulse functions

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.01.065

* Corresponding author.E-mail address: sdvahdati@gmail.com (S. Vahdat

a b s t r a c t

In this paper, the properties of the floor function has been used to find a function which isone on the interval [0,1) and is zero elsewhere. The suitable dilation and translationparameters lead us to get similar function corresponding to the interval ½a; bÞ. These func-tions and their combinations enable us to represent the stepwise functions as a function offloor function. We have applied this method on Haar wavelet, Sine–Cosine wavelet, Block-Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations ofthese functions.

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1. Introduction

In the recent years, there has been an increase usage among scientists and engineers to apply wavelet technique based onspecial kind of stepwise functions as well as a numerical solution to solve both linear and nonlinear problems. The mainadvantage of the wavelet technique is its ability to transform complex problems into a system of algebric equations. As over-view of this method can be found in [1,2,4–7]. The accuracy of the wavelet technique increase with the order of the system aswell as function’s coefficients values. Calculations of these coefficients usually are done by computer programming. The floorfunction f ðxÞ ¼ ½x� is a known function for the most of mathematical and engineering programming languages so the pro-gramming for the problems which are involved with stepwise functions will become easier if we can change the constructionof the stepwise functions to a function which does not have any limitation on the domain.The present paper introduces thefunction � ðtÞ and � a;bðtÞ which are described by floor function. The first function is one on the interval [0,1) and is zero else-where and the second one has similar behavior on the interval ½a; bÞ. We can also define the similar functions on the integernumbers Z which the values of the function are one on the set A ¼ fn;nþ 1; . . . ;mgðn;m 2 Z; n 6 mÞ and are zero on the setZ� A. Using the combinations of these functions with the suitable dilation and translation parameters have applied to getthe new construction for wavelets.

2. Floor function

The function � ðtÞ is defined as

� ðtÞ ¼ 1

½t�2 þ 1

" #; ð2:1Þ

which ½t� is the highest integer number less than or equal to t. As to the formula (2.1) then it should be said that this functionis zero outside of the interval [0, 1). The graph of � ðtÞ is given in Fig. 1.

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i).

Fig. 1. Graph of � ðtÞ and /ðtÞ.

474 Z. Abbas et al. / Applied Mathematics and Computation 210 (2009) 473–478

Using the suitable dilation and translation parameters enable us to get the function � a;bðtÞ which has similar behavior onthe interval ½a; bÞ.

� a;bðtÞ ¼1

t�ab�a

� �2 þ 1

" #: ð2:2Þ

Now, consider the below stepwise function which is similar to the construction of many kinds of wavelets.

f ðtÞ ¼

f1ðtÞ a1 6 t < b1

f2ðtÞ a2 6 t < b2

..

. ...

fnðtÞ an 6 t < bn

0 otherwise:

8>>>>>>><>>>>>>>:

ð2:3Þ

Eq. (2.4) shows its floor construction:

f ðtÞ ¼Xn

i¼1

� ai ;biðtÞfiðtÞ ¼

Xn

i¼1

1t�aibi�ai

h i2þ 1

264

375fiðtÞ: ð2:4Þ

Let n and m are two integer numbers in which n 6 m, the aim is to find the function as its values on the setA ¼ fn;nþ 1; . . . ;mg are one and zero on the set Z� A. With consideration the properties of the function � a;bðtÞ with e (asany positive real number, for convenience we consider e ¼ 1) if we put a ¼ n, b ¼ mþ e and t 2 Z then we will have the desir-able function:

� n;m : Z! f0;1g

k # 1k�n

m�nþ1½ �2þ1

" # : ð2:5Þ

The graph of the function � n;mðkÞ is given in Fig. 2.

Fig. 2. Graph of � n;mðkÞ.

Z. Abbas et al. / Applied Mathematics and Computation 210 (2009) 473–478 475

For the cases that we need the similar functions at some point t ¼ a we can put � aðtÞ ¼ 1ðt�aÞ2þ1

h i, for the case t P a we

define � a;1ðtÞ ¼ 1

t�ajt�ajþ1

h i2

þ1

264

375 and also for the case t > a, it may define:

2 3

� a;1ðtÞ � � aðtÞ ¼

1

t�ajt�ajþ1

h i2þ 1

64 75� 1

ðt � aÞ2 þ 1

" #: ð2:6Þ

With the similar method we can get the suitable floor construction for other kinds of intervals.

3. Applications

In this section, we have applied the function � a;bðtÞ on Haar wavelets, Sine–cosine wavelet, Block-Pulse functions and Hy-brid Fourier Block-Pulse functions to get the new construction.

3.1. Haar wavelets

The Haar scaling function is defined as

/ðxÞ ¼1 0 6 x < 1;0 otherwise;

�ð3:1Þ

and the Haar wavelet is the function

wðxÞ ¼ /ð2xÞ � /ð2x� 1Þ: ð3:2Þ

The function / is sometimes called the Father wavelet and w, the Mother wavelet[3]. Their graphs are given in Figs. 1 and 3,respectively. The below equation is floor representation of Haar wavelet:

wðxÞ ¼ 1

½2x�2 þ 1

" #� 1

½2x� 1�2 þ 1

" #: ð3:3Þ

3.2. Sine–Cosine wavelets

Sine–Cosine wavelets (SCW) wn;mðtÞ ¼ wðn; k;m; tÞ have four arguments; n ¼ 0;1;2; . . . ;2k � 1; k ¼ 0;1;2; . . ., the values ofm are given in Eq. (3.5) and t is the normalized time. They are defined on the interval [0,1) as

wn;mðtÞ ¼2

kþ12 f ð2kt � nÞ; n

2k 6 t < nþ12k

0; otherwise

(ð3:4Þ

with

fmðtÞ ¼

1ffiffi2p ; m ¼ 0

cosð2mptÞ; m ¼ 1;2; . . . ; L

sinð2ðm� LÞptÞ; m ¼ Lþ 1; Lþ 2; . . . ;2L;

8><>: ð3:5Þ

Fig. 3. Graph of wðtÞ.

476 Z. Abbas et al. / Applied Mathematics and Computation 210 (2009) 473–478

where L is any positive integer. The set of SCW are an orthonormal set. In SCW we have a limitation for t which has to be inthe interval n

2k ;nþ12k

h �. For the integer number m we have three conditions, the first m ¼ 0, the second m ¼ 1;2; . . . ; L and the

hast one m ¼ Lþ 1; Lþ 2; . . . ;2L. Corresponding to these conditions we define four floor construction functions:

� n2k ;

nþ12kðtÞ ¼ 1

½2kt � n�2 þ 1

" #; � 0ðmÞ ¼

1m2 þ 1

� �;

� 1;LðmÞ ¼1

m�1L

� �2 þ 1

" #; � Lþ1;2LðmÞ ¼

1m�L�1

L

� �2 þ 1

" #:

ð3:6Þ

The following equation shows the floor construction of SCW:

wn;mðtÞ ¼ 2kþ1

2 � n2k ;

nþ12kðtÞ � 0ðmÞ

1ffiffiffi2p þ � 1;LðmÞ cosð2kþ1mptÞ þ � Lþ1;2LðmÞ sinð2kþ1ðm� LÞptÞ

: ð3:7Þ

For example, consider k ¼ 1 and L ¼ 2, we have

wn;mðtÞ ¼ 21

½2t � n�2 þ 1

" #1

m2 þ 1

� �1ffiffiffi2p þ 1

m�12

� �2 þ 1

" #cosð4mptÞ þ 1

m�32

� �2 þ 1

" #sinð4ðm� 2ÞptÞ

!

so,

wn;mðtÞ

n ¼ 0 n ¼ 1

m ¼ 0

2ffiffi2p 1

½2t�2þ1

h ih i

2ffiffi

2p 1

½2t�1�2þ1

h ih i

m ¼ 1

2 1½2t�2þ1

cosð4ptÞ

2 1½2t�1�2þ1

cosð4ptÞ

m ¼ 2

2 1½2t�2þ1

h icosð8ptÞ

2 1

½2t�1�2þ1

h icosð8ptÞ

m ¼ 3

2 1½2t�2þ1

h isinð4ptÞ

2 1

½2t�1�2þ1

h isinð4ptÞ

m ¼ 4

2 1½2t�2þ1

h isinð8ptÞ

2 1

½2t�1�2þ1

h isinð8ptÞ

3.3. Block-Pulse functions

A set of Block-Pulse functions biðkÞ, i ¼ 1;2; . . . ;m on the interval [0,1) are defined as follows [4]:

biðkÞ ¼1 i�1

m 6 k < im ;

0 otherwise:

(ð3:8Þ

The Block-Pulse functions on [0,1) are disjoint, that is, for i ¼ 1;2; . . . ;m, j ¼ 1;2; . . . ;m we have: biðtÞbjðtÞ ¼ dijbiðtÞ, also thesefunctions have the property of orthogonality on [0,1).

For these functions is enough to put a ¼ i�1m and b ¼ i

m for i ¼ 1;2; . . . ;m to get the below floor construction of Block-Pulsefunctions:

biðkÞ ¼ � i�1m ; i

mðkÞ ¼ 1

½mk� iþ 1�2 þ 1

" #; ði ¼ 1;2; . . . ;mÞ ð3:9Þ

for example, let m ¼ 4 we have

biðkÞ ¼ � i�14 ; i4ðkÞ ¼ 1

½5k� iþ 1�2 þ 1

" #; ði ¼ 1;2; . . . ;5Þ

then

i ¼ 1

i ¼ 2 i ¼ 3 i ¼ 4 i ¼ 5

biðkÞ

1½5k�2þ1

h i

1

½5k�1�2þ1

h i

1

½5k�2�2þ1

h i

1

½5k�3�2þ1

h i

1

½5k�4�2þ1

h i

Z. Abbas et al. / Applied Mathematics and Computation 210 (2009) 473–478 477

3.4. Hybrid Fourier Block-Pulse functions

Fourier functions on the interval ½0;2p�:

/0ðtÞ ¼ 1;/mðtÞ ¼ cos mt; m ¼ 1;2;3; . . . ;

/�mðtÞ ¼ sin mt; m ¼ 1;2;3; . . .

ð3:10Þ

These functions are orthogonal in Hilbert space L2½0;2p� [8].For m ¼ 0;1;2; . . . ;2r and n ¼ 1;2; . . . ;N the Hybrid Fourier Block-Pulse functions are defined as:

bðn;m; tÞ ¼

1 m ¼ 0/mð2pNtÞ m ¼ 1;2; . . . ; r

/�m�rð2pNtÞm ¼ r þ 1; r þ 2; . . . ;2r

9>=>; n�1

N 6 t < nN

0 otherwise:

8>>><>>>:

ð3:11Þ

In the Hybrid Fourier Block-Pulse functions we have a limitation for t which has to be in the interval n�1N ; n

N

� �. For the integer

number m we have three conditions, the first m ¼ 0, the second m ¼ 1;2; . . . ; r and the hast one m ¼ r þ 1; Lþ 2; . . . ;2r. Cor-responding to these conditions we define four floor construction functions:

� n�1N ;nNðtÞ ¼ 1

Nt � nþ 1½ �2 þ 1

" #; � 0ðmÞ ¼

1m2 þ 1

� �;

� 1;rðmÞ ¼1

m�1r

� �2 þ 1

" #; � rþ1;2rðmÞ ¼

1m�r�1

r

� �2 þ 1

" #:

ð3:12Þ

The following equation shows the floor construction of Hybrid Fourier Block-Pulse functions:

bðn;m; tÞ ¼ � n�1N ;nNðtÞð� 0ðmÞ þ � 1;rðmÞ cosð2mpNtÞ þ � rþ1;2rðmÞ sinð2ðm� rÞpNtÞÞ: ð3:13Þ

As an example in Hybrid Fourier Block-Pulse functions consider r ¼ 2 and N ¼ 2 we have

bðn;m; tÞ ¼ 1

2t � nþ 1½ �2 þ 1

" #1

m2 þ 1

� �þ 1

m�12

� �2 þ 1

" #cosð4mptÞ þ 1

m�32

� �2 þ 1

" #sinð4ðm� rÞptÞ

!

so,

bðn;m; tÞ

n ¼ 1 n ¼ 2

m ¼ 0

1½2t�2þ1

h i

1

½2t�1�2þ1

h i

m ¼ 1 1

½2t�2þ1

h icosð4ptÞ

1

½2t�1�2þ1

h icosð4ptÞ

m ¼ 2

1½2t�2þ1

h icosð8ptÞ

1

½2t�1�2þ1

h icosð8ptÞ

m ¼ 3

1½2t�2þ1

h isinð4ptÞ

1

½2t�1�2þ1

h isinð4ptÞ

m ¼ 4

1½2t�2þ1

h isinð8ptÞ

1

½2t�1�2þ1

h isinð8ptÞ

4. Conclusion

In this work, we introduce the function � ðtÞ which is equal to Haar scaling function. The construction of this function isbased on floor function which is a known function for most of mathematical and engineering programming languages. Wehave used the suitable dilation and translation parameter to find the similar function on the interval ½a; bÞ. The extension ofthis function for other kinds of intervals, a single point and also for a subset of integer numbers has provided. These functionsenable us to change the stepwise function to a function with floor construction. We have changed the construction of somewavelets and functions to get the new formation which is more convenient for computer programming.

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