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Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 29 000–000 (2012) 2950 – 2954 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE)
A Novel Approach to Processing Fractal Signals Using the Yang-Fourier Transforms Xiao-Jun Yang a*, Meng-Ke Liao b, Jiang-Wen Chen b a
Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, P. R. C b College of Water conservancy, Shihezhi University, Shihezhi, Xinjiang, 832003, P. R. C
Abstract In the present paper, local fractional continuous non-differentiable functions in fractal space are studied, and the signals in fractal-time space are reflectively investigated using the Yang-Fourier transforms based on the local fractional calculus. Two illustrative examples are given to elaborate the signal process and reliable results.
© 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Keywords: Non-differentiable function, fractal signals, Yang-Fourier transform, fractal-time space, local fractional calculus;
1. Introduction Generally speaking, signal is that anything carries information. Fourier analysis is one of the most frequently used tools is signal processing [1]. A signal is defined as a function of time. However, some signal functions are fractal curves, which are everywhere continuous but nowhere differentiable [2-9]. As a result, we cannot employ the classical Fourier analysis, which requires that the defined functions should be differentiable, to describe the signals in fractal time–space. Recently, local fractional calculus (fractal calculus), which is dealing with fractal functions, has been proposed and developed. For these merits, local fractional calculus was successfully applied in the fractal elasticity [4], the fractal wave equation [9], the Yang-Laplace transforms[6,7,9], the Yang-Fourier transforms[6-8], the local fractional short time transforms[6,7] and the local fractional wavelet transforms[6,7]. In this paper, we apply the Yang-Fourier transform to deal with the fractal signals. _______________ Email address:
[email protected] Supported by the National Natural Science Foundation of China (Grant No. 50904045) .
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.420
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2. Fractal-time signals and local fractional calculus In this section, we mainly study the fractal-time signals and local fractional calculus. As is known, the signals are functions of time and the signals and functions are interchangeable. Here, both local fractional continuity of the functions and local fractional calculus are investigated. 2.1. Notations Definition 1 If there exists the relation [6,7]
f ( x ) − f ( x0 ) < ε α
(1)
with x − x0 < δ ,for ε , δ > 0 and ε , δ ∈ . Now f ( x ) is called local fractional continuous at x = x0 , denote by lim f ( x ) = f ( x0 ) .Then f ( x ) is called local fractional continuous on the interval ( a, b ) , x → x0
denoted by
f ( x ) ∈ Cα ( a, b ) .
(2)
Definition 2 A function f ( x ) is called a non-differentiable function of exponent α , 0 < α
≤ 1 , which
satisfy Hölder function of exponent α , then for x, y ∈ X such that [6,7]
f ( x) − f ( y) ≤ C x − y
α
.
(3)
Definition 3 A function f ( x ) is called to be continuous of order α , 0 < α ≤ 1 , or shortly α continuous, when we have the following relation [6,7]
(
f ( x ) − f ( x0 ) =o ( x − x0 )
α
).
(4) Remark 1. Compared with (2.4), (2.1) is standard definition of local fractional continuity. Here (2.3) is unified local fractional continuity.
Definition 4 Setting f ( x ) ∈ Cα ( a, b ) , local fractional derivative of f ( x ) of order α at x = x0 is defined [6-9]
= f ( ) ( x0 ) α
α
where Δ
Δ d α f ( x) = lim x = x0 α x→x0 dx
α
( f ( x) − f ( x ) ) ≅ Γ(1+α) Δ( f ( x) − f ( x )) . 0
( f ( x) − f ( x ) ) , 0
( x − x0 )
α
(5)
0
Definition 5Setting f ( x ) ∈ Cα ( a, b ) , local fractional integral of f ( x ) of order α in the
[
]
interval a, b is defined [6,9] ( ) = f ( x) a Ib α
=j N −1 b α 1 1 α lim = f t dt f ( t j )( Δt j ) , ( )( ) ∑ ∫ a Δ → 0 t Γ (1 + α ) Γ (1 + α ) j =0
(6)
where Δt j = t j +1 − t j ,= Δt max {Δt1 , Δt2 , Δt j ,...} and ⎡⎣t j , t j +1 ⎤⎦ , j = 0,..., N − 1 , t0 = a, t N = b , is a partition
[
]
of the interval a, b . Here, it follows that
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I
(α )
a a
and
f ( x ) = 0 if a = b
(7)
I ( ) f ( x) = −b Ia( ) f ( x) if a < b . α
3
α
(8)
a b
2.2. Recent results Suppose that f ( x ) , g ( x ) ∈ Dα ( a, b ) , the following differentiation rules are valid [6-9]:
d α ( f ( x) ± g ( x)) d α f ( x) d α g ( x) = ± ; dxα dxα dxα d α ( f ( x ) g ( x )) dα f ( x) dα g ( x) g x f x = + ( ) α ( ) α ; dxα dx dx Theorem 1 [6-9] Suppose that f ( x ) , g ( x ) ∈ Cα [ a, b ] , then I
(α )
a b
⎡⎣ f ( x ) ± g ( x= ) ⎤⎦
f ( x) Theorem 2 [6-9] Suppose that=
I
(α )
a b
f ( x ) ± a I b (α ) g ( x ) .
( x ) ∈ Cα [ a, b] , then we have (α ) f= ( x ) g (b ) − g ( a ) . a Ib g(
(9) (10)
(11)
α)
(12)
3. Yang-Fourier transforms In this section, we start with the Yang-Fourier transforms and some results.
Definition 6 Suppose that f ( x ) ∈ Cα ( −∞, ∞ ) , the Yang-Laplace transform, dented
{
}
by Fα f ( x) ≡ fω
F,α
(ω) , is given in the form [6-8] F,α Fα = (ω) : { f ( x)} fω=
∞ 1 α Eα ( −iαωα xα ) f ( x)( dx) , ∫ −∞ Γ(1+α)
(12)
where the latter converges. And of course, a sufficient condition for convergence is ∞ ∞ 1 1 α α f ( x ) Eα ( −iαωα xα ) ( dx) ≤ f ( x ) ( dx) < K < ∞. ∫ ∫ −∞ −∞ Γ (1+ α ) Γ (1+ α )
{
}
Definition 7 Suppose that Fα f ( x) ≡ fω −1 = f ( x) Fα= ( fωF,α (ω)) :
F ,α
1
(14)
(ω) , its inverse is given by the expression
E ( i ω x ) fω (ω)( dω) , x > 0 . (15) ( 2π ) ∫ α α Suppose that Fα { f ( x)} ≡ fω (ω) and Fα { g ( x)} ≡ gω (ω) , the following formulas are valid [6-8]: (16) Fα {af ( x ) + bg ( x )}= afω α (ω ) + bgω α (ω ) ; ∞
α −∞ α
F,
α α α
F,α
F,
F,
F,
α
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If lim
x →∞
{
}
f ( x ) = 0 , Fα f (α ) ( x ) = iα ω α fωF ,α (ω ) ; Fα { f ( ax )} =
1 F ,α ⎛ ω ⎞ fω ⎜ ⎟ , a > 0 ; aα ⎝a⎠
Fα { f ( x − c )} = Eα ( −iα cα ω α ) Fα { f ( x )} ;
{
}
Fα f ( x) Eα ( −iα xαω0α ) =fωF ,α (ω −ω0 ) .
(17) (18) (19) (20)
4. Two illustrative examples In this section, we give some applications of the local Yang-Fourier transforms to the fractal signals.
Example 1. Let a non-periodic signal X ( t ) be defined by the relation
⎧ A, −t0 ≤ t < t0 ; X (t ) = ⎨ ⎩0, else.
(21)
Taking the Yang-Fourier transforms, we have
X ωF ,α (ω )
∞ 1 α X ( t ) Eα ( −iα ω α xα ) ( dx ) ∫ −∞ Γ (1 + α )
=
t 1 α AEα ( −iα ω α xα ) ( dx ) ∫ − t Γ (1 + α )
= = Taking into account [6, 7] Eα
( −i
α
AEα ( −iα ω α xα ) t0 −iα ω α
α
t 0α =
.
xα )= cosα xα − iα sinα xα , we have
= X ωF ,α (ω ) where sin α C ω
−t0
(22)
2 A sinα ω α t0α = 2 At α sinα C ω α t0α . α
ω
(23)
sinα ω α t0α . ω α t 0α
Hence we obtain the pairs
⎧ A, −t0 ≤ t < t0 ; YFT 2 At0α sinα C ω α t0α . X (t ) = ⎨ ⇔ ⎩0, else. Example 2. Let a non-periodic signal X ( t ) be defined by the relation
X (t ) =
2sinα ω0α t α . ω0α t α
From (4.4), we directly obtain the relation
(24)
(25)
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Xiao-Jun Yang et al. / Procedia Engineering 29 (2012)00 2950 – 2954 Xiao-Jun Yang, Meng-Ke Liao, Jiang-Wen Chen / Procedia Engineering (2011) 000–000
⎧1, −ω0 ≤ ω < ω0 ; X ωF ,α (ω ) = ⎨ ⎩0, else.
5
(26)
Therefore we get the pairs
= X (t )
⎧1, −ω0 ≤ ω < ω0 ; 2sin α ω0α t α YFT F ,α = X ω (ω ) ⎨ ⇔ α α ω0 t ⎩0, else.
(27)
5. Conclusions In this paper, we point out a novel method for processing the signals in fractal-time space are investigated using the Yang-Fourier transforms based on the local fractional calculus. Some typical examples are given to elaborate the signal process and reliable results. Acknowledgements This work is grateful for the finance supports of the National Natural Science Foundation of China (Grant No. 50904045). References [1] Vretblad A. Fourier Analysis and Its Applications. New York: Springer-Verlag, 2003. [2] Kolwankar KM, Gangal AD. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos1996; 6 (4):505–513. [3] Adda F B, Cresson J. About non-differentiable functions. J. Math. Anal. Appl. 2001, 263, , 721–737. [4] Carpinteri A, Chiaia B, Cornetti P. Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Engrg. 2001; 191, 3-19. [5] Chen Y, Yan Y, Zhang K. On the local fractional derivative. J. Math. Anal. Appl. 2010; 362, 17–33. [6] Yang X. Local Fractional Integral Transforms. Prog. in Nolin. Sci. 2011; 4, 1-225. [7] Yang X. Local Fractional Functional Analysis and Its Applications. Hong Kong: Asian Academic publisher Limited, 2011. [8] Zhong W P, Gao F, Shen X M. Applications of Yang-Fourier Transform to Local Fractional Equations with Local Fractional Derivative and Local Fractional Integral. In: The 2011 International Conference on Computer Science and Education, Springer, 2011, in press. [9] Zhong W P, Gao F. Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with fractional derivative. In: The 2011 3rd International Conference on Computer Technology and Development, ASME, 2011, in press.
