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: [email protected] (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 ¼1
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: ð3:2ÞThe 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 1m ¼ 0
2ffiffi2p 1½2t�2þ1
h ih i
2ffiffi2p 1
½2t�1�2þ1
h ih i
m ¼ 1
2 1½2t�2þ1cosð4ptÞ
2 1½2t�1�2þ1cosð4ptÞ
m ¼ 2
2 1½2t�2þ1h icosð8ptÞ
2 1½2t�1�2þ1
h icosð8ptÞ
m ¼ 3
2 1½2t�2þ1h isinð4ptÞ
2 1½2t�1�2þ1
h isinð4ptÞ
m ¼ 4
2 1½2t�2þ1h isinð8ptÞ
2 1½2t�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 5biðkÞ
1½5k�2þ1h i
1½5k�1�2þ1
h i
1½5k�2�2þ1
h i
1½5k�3�2þ1
h i
1½5k�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 2m ¼ 0
1½2t�2þ1h i
1½2t�1�2þ1
h i
m ¼ 1 1½2t�2þ1
h icosð4ptÞ
1½2t�1�2þ1
h icosð4ptÞ
m ¼ 2
1½2t�2þ1h icosð8ptÞ
1½2t�1�2þ1
h icosð8ptÞ
m ¼ 3
1½2t�2þ1h isinð4ptÞ
1½2t�1�2þ1
h isinð4ptÞ
m ¼ 4
1½2t�2þ1h isinð8ptÞ
1½2t�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.
References
[1] M. Razzaghi, S. Yousefi, Sine–Cosine wavelets operational matrix of integration and its applications in the calculus of variations, International Journal ofSystems Science 33 (10) (2002) 805–810.
478 Z. Abbas et al. / Applied Mathematics and Computation 210 (2009) 473–478
[2] M. Tavassoli Kajani, M. Ghasemi, E. Babolian, Numerical solution of linear integro-differential equation by using sine–cosine wavelets, AppliedMathematics and Computation 180 (2006) 269–574.
[3] A. Boggess, F.J. Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice-Hall, 2001.[4] K. Maleknejad, M. Tavassoli Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legendre and block-pulse functions,
Applied Mathematics and Computation 145 (2003) 623–629.[5] Ü. Lepik, Haar wavelet method for nonlinear integro-differential equations, Applied Mathematics and Computation 176 (2006) 324–333.[6] C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEEE Proceedings of Control Theory and Applications
144 (1997) 87–93.[7] M. Razzaghi, H.R. Marzban, Direct method for variational problems via hybrid of block-pulse and Chebyshev functions, Mathematical Problems in
Engineering 6 (2000) 85–97.[8] B. Asady, M. Tavassoli Kajani, A. Hadi Vencheh, A. Heydari, Direct method for solving integro differential equations using hybrid Fourier and block-pulse
functions, International Journal of Computer Mathematics 82 (7) (2005) 889–895.
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