Local Fractional Discrete Wavelet Transform for Solving Signals on Cantor Sets

Abstract

The discrete wavelet transform via local fractional operators is structured and applied to process the signals on Cantor sets. An illustrative example of the local fractional discrete wavelet transform is given.

1. Introduction

In recent years, the classical wavelet theory [1–7] has played an important role in many scientific fields such as signal processing [8], electrical systems [9], image processing [10], and differential equations [11]. The continuous wavelet transform is applied to handle the analyzing nonstationary signals, which have some characteristics of instantaneous peaks or discontinuities, where the mother wavelet met scaling and translation operations [3]. Two major categories of wavelet transforms are continuous and discrete [5]. When the mother wavelet functions are orthonormal, the discrete wavelet transform [12] gives multiresolution algorithm decomposing signals into scales with different time and frequency resolution, which leads to finite number of wavelet comparisons of signals, and improves the computational speeds because of the functions that are stretched or compressed and placed at many positions along the signals [13].

Based on the fractional Fourier transform [14–17], the fractional wavelet transform, which was a good tool for processing transient signals and compressing images, was structured in [18, 19]. The fractional wavelet transform has some applications in various branches of science and engineering [20–23]. For example, the simultaneous spectral analysis of a binary mixture system was presented in [20] by using the fractional wavelet transform. Application of the fractional wavelet transform to the simultaneous determination of ampicillin sodium and sulbactam sodium in a binary mixture was considered in [21]. The fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two-component mixture was suggested in [22]. The optical image encryption based on fractional wavelet transform was given in [23]. By discretizing continuous fractional wavelet transform, the discrete fractional wavelet transform was reported and its application to multiple encryptions was considered in [24].

The wavelet method and its fractional counterpart have many applications in various branches of science and engineering. However, they are invalid for solving the signals defined on Cantor sets. The local fractional calculus theory [25–34] was applied to handle the functions defined on Cantor sets, which are local fractional continuous. A natural question is to generalize signals concepts on the Cantor set, which are the nondifferentiable functions defined on Cantor sets [24, 26] and the Cantor function [35]. The mathematical theory of the local fractional wavelet transform of the local fractional continuous signal was structured in [25, 36] based on the basic idea.

One of the open problems in this area is how to improve the computational speeds of the local fractional wavelet theory as in the classical one. The aim of this paper is to structure the discrete version of the local fractional wavelet transform based on the generalized inner production space. The paper has been organized as follows. In Section 2, we introduce some basic notations and theorems of the generalized inner product space. In Section 3, we propose the local fractional discrete wavelet transform. In Section 4, one example is presented. Finally, Section 5 is conclusions.

2. Preliminaries

In this section, we give some basic notations and theorems of the generalized inner product space.

Let [25] 𝐿_2,𝛼[𝑅]=𝑓(𝑥)∈𝐶_𝛼[𝑅]∶1Γ(1+𝛼)^∞_−∞𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝<∞,1≤𝑝<∞. (1) Here, the local fractional integral operator 𝑓(𝑥) in the interval [𝑎,𝑏] was defined in [25–30] as _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^{𝑗=𝑁−1}_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼, (2) where a partition of the interval [𝑎,𝑏] is denoted as Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗, Δ𝑡=max{Δ𝑡₀,Δ𝑡₁,Δ𝑡_𝑗,…} and 𝑗=0,…,𝑁−1, 𝑡₀=𝑎, 𝑡_𝑁=𝑏. Local fractional operators were applied to model some nondifferentiable problems [25–32].

From (1) the generalized inner product space of 𝐿_2,𝛼[𝑅] is defined as follows [25]: 𝑓,𝑔_𝛼=1Γ(1+𝛼)^∞_−∞𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼. (3) The two useful theorems are presented as follows.

Theorem 1 (see [25]). Let 𝑋 be an inner product space. If??{𝑒^𝛼_𝑛} is an orthonormal system in 𝑋, then one has that𝑓²_𝛼=^∞_𝑖=1𝑓,𝑒^𝛼_𝑖𝛼²,𝑓=^∞_𝑖=1𝑓,𝑒^𝛼_𝑖𝛼𝑒^𝛼_𝑖 (4)(5) are equivalent, where ‖𝑓‖²_𝛼 is a norm of the function 𝑓 and {𝑒^𝛼_𝑛} has the following properties: 𝑒^𝛼_𝑛𝛼𝑒=1,_𝛼𝑖,𝑒^𝛼_𝑗=0,𝑖≠𝑗,1,𝑖=𝑗. (6)

Proof. See [25].

Theorem 2 (see [25]). Let 𝑋 be an inner product space and {𝑒^𝛼_𝑛} be an orthonormal system in 𝑋. If 𝑥^𝛼∈span{𝑒^𝛼₁,…,𝑒^𝛼_𝑛}, then for all 𝑥^𝛼∈𝑋 one has 𝑥^𝛼=^𝑛_𝑖=1𝑥^𝛼,𝑒^𝛼_𝑖𝛼𝑒^𝛼_𝑖, (7) where span{𝑥^𝛼₁,…,𝑥^𝛼_𝑛} is the linear subspace of 𝑋 of the linear span of the local fractional vectors [25], namely, span𝑥^𝛼₁,…,𝑥^𝛼_𝑛=𝑥^𝛼=^𝑛_𝑖=1𝑎_𝑖𝑥^𝛼_𝑖∶𝑎_𝑖∈𝐸. (8)

Proof. See [25].

3. Local Fractional Discrete Wavelet Transform for Signals on Cantor Sets

3.1. Local Fractional Continuous Wavelet Transformation for Signals on Cantor Sets

The local fractional continuous wavelet transform of the local fractional continuous signal 𝑓(𝑡) was presented in [25, 26, 36] as 𝑊_𝜑,𝛼𝑓()=𝑎𝑎,𝑏^−(𝛼/2)Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼,0<𝛼≤1, (9) where the local fractional daughter’s wavelets were suggested in [25, 26, 36] by 𝜑_{𝑎,𝑏,𝛼}(𝑡)=1𝑎^𝛼/2𝜑𝑡−𝑏𝑎, (10) where 𝑎 is the dyadic dilation, 𝑏 is the dyadic position, and 𝑎^−(𝛼/2) is the normalization Cantor factor. The inverse formula of local fractional wavelet transform was given in [25, 36] by 𝑓(𝑥)=𝐶_𝜑,𝛼Γ²()×1+𝛼_{∞−∞∞−∞}𝑎^−2𝛼𝑊_𝜑,𝛼𝑓(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑎)^𝛼(𝑑𝑏)^𝛼,0<𝛼≤1, (11) where the parameter is [25, 36] 𝐶_𝜑,𝛼=1Γ(1+𝛼)^∞_−∞𝑓(𝑥)²|𝑥|^𝛼(𝑑𝑥)^𝛼,0<𝛼≤1. (12) We notice that the classical continuous wavelet transform is the local fractional one in case of fractal dimension 𝛼=1.

3.2. Local Fractional Discrete Wavelet Transform for Signals on Cantor Sets

Let us structure the local fractional daughter wavelet in the form 𝜑_{𝑎,𝑏,𝛼}(𝑡)=1𝑎^𝛼/2𝜑𝑡−𝑏𝑎, (13) where 𝜑∈𝐿_2,𝛼[𝑅].

When 𝑎=2^−𝑗 and 𝑏=𝑘2^−𝑗, we get 𝜑_{𝑎,𝑏,𝛼}(𝑡)=𝜑_{𝑗,𝑘,𝛼}(𝑡)=𝜑_{2−𝑗,𝑘2−𝑗,𝛼}(𝑡)=2^𝑗𝛼/2𝜑2^𝑗𝑡−𝑘 (14) for integers 𝑗,𝑘∈Ζ.

Let 𝜑_{𝑗,𝑘,𝛼}(𝑡)=2^𝑗𝛼/2𝜑(2^𝑗𝑡−𝑘) be orthogonal set of local fractional wavelets. Then we can obtain 𝜑_{𝑗,𝑘,𝛼},𝜑_{𝑚,𝑛,𝛼𝛼}=𝛿^𝛼_𝑗,𝑚𝛿^𝛼_𝑘,𝑛,𝑗,𝑘,𝑚,𝑛∈Ζ, (15) where 𝛿^𝛼_𝑗,𝑚 and 𝛿^𝛼_𝑘,𝑛 are local fractional Kronecker delta [27].

Making use of (7), for 𝑗,𝑘,𝑚∈Ζ we have 𝑓(𝑥)=^∞∞_{𝑗=−∞𝑚=−∞}𝑎_{𝑗,𝑘,𝛼}𝑒^𝛼_𝑗,𝑘, (16) where its coefficients are 𝑎_𝑗,𝑘=𝑓(𝑥),𝑒^𝛼_{𝑗,𝑘𝛼}=𝑊_𝜑,𝛼𝑓2^−𝑗,𝑘2^−𝑗. (17) Here, 𝑎_𝑗,𝑘 is called as the local fractional discrete wavelet transform of the signal 𝑓(𝑥).

4. An Illustrative Example

Local fractional mother wavelet is defined in [26] as 𝜑_𝐻(𝛼)(𝑡)(𝑡)=1=𝑀1,0≤𝑡<21−1,2≤𝑡<10,else (18) and local fractional integral of local fractional mother wavelet reads as 𝜙_𝐻(𝛼)(𝑡)(𝑡)=𝑡=𝑁^𝛼Γ()11+𝛼,0≤𝑡<2(1−𝑡)^𝛼Γ(),11+𝛼2≤𝑡<10,else. (19) Figure 1 shows the graph of the local fractional mother wavelet and Figure 2 shows the graph of the local fractional integral of local fractional mother wavelet.

Figure 1 

The graph of the local fractional mother wavelet.

Figure 2 

The graph of the local fractional integral of local fractional mother wavelet.

When fractal dimension 𝛼=1, we have 𝜑_𝐻(1)(𝑡)=𝑀(𝑡) (20) so that 𝜙_𝐻(1)(𝑡)(𝑡)=1=𝐿𝑡,0≤𝑡<211−𝑡,2≤𝑡<10,else. (21) Figure 3 shows the graph of the integral of mother wavelet 𝜑_𝐻(1)(𝑡).

Figure 3 

The graph of the integral of the mother wavelet.

For integers 𝑗,𝑘∈Ζ, we have [26] 𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)=2^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘, (22) where 𝜑_𝐻(𝛼)(𝑡)=11,0≤𝑡<2,1−1,2≤𝑡<1,0,else. (23) Hence, we have 𝜑_𝐻(𝛼)^𝑗,𝑘,𝜑_𝐻(𝛼)^𝑚,𝑛_𝛼=1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)𝜑_𝐻(𝛼)^𝑚,𝑛(𝑡)(𝑑𝑡)^𝛼=1Γ()×1+𝛼_∞−∞2^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘2^𝑚𝛼/2𝜑_𝐻(𝛼)×2^𝑚𝑡−𝑛(𝑑𝑡)^𝛼=2^{(𝑗+𝑚)𝛼/2}1Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘𝜑_𝐻(𝛼)2^𝑚𝑡−𝑛(𝑑𝑡)^𝛼=2^{(𝑚−𝑗)𝛼/2}1Γ(1+𝛼)×^∞_−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑚−𝑗(𝑠+𝑘)−𝑛(𝑑𝑠)^𝛼, (24) where 𝑠=2^𝑗𝑡−𝑘.

In view of (24), we obtain [15] 1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)²(𝑑𝑡)^𝛼1=1,Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)(𝑑𝑡)^𝛼=0, (25) where 𝑗=𝑚?and?𝑘=𝑛,??𝑗,𝑘∈Ζ.

When 𝑗=𝑚,??𝑗,𝑘,𝑚∈Ζ, from (24) we obtain 𝜑_𝐻(𝛼)^𝑗,𝑘,𝜑_𝐻(𝛼)^𝑗,𝑛_𝛼=1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)𝜑_𝐻(𝛼)^𝑚,𝑛(𝑡)(𝑑𝑡)^𝛼=2^{(𝑗+𝑚)𝛼/2}1Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘𝜑_𝐻(𝛼)2^𝑚𝑡−𝑛(𝑑𝑡)^𝛼=2^{(𝑚−𝑗)𝛼/2}1Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑚−𝑗(𝑠+𝑘)−𝑛(𝑑𝑠)^𝛼=1Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)(𝑠+𝑘−𝑛)(𝑑𝑠)^𝛼=𝛿^𝛼_{0,𝑘−𝑛}=𝛿^𝛼_𝑘,𝑛, (26) where 𝑠=2^𝑗𝑡−𝑘.

When 𝑔=𝑚−𝑗>0,??𝑗,𝑘,𝑚,𝑛∈Ζ, from (24) we have 𝜑_𝐻(𝛼)^𝑗,𝑘,𝜑_𝐻(𝛼)^𝑚,𝑛_𝛼=1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)𝜑_𝐻(𝛼)^𝑚,𝑛(𝑡)(𝑑𝑡)^𝛼=2^𝑔𝛼/21Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑔(𝑠+𝑘)−𝑛(𝑑𝑠)^𝛼=2^𝑔𝛼/21Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑔𝑠+𝜂(𝑑𝑠)^𝛼, (27) where 𝑠=2^𝑗𝑡−𝑘 and 𝜂=2^𝑔𝑘−𝑛. Consider 𝜑_𝐻(𝛼)^𝑗,𝑘,𝜑_𝐻(𝛼)^𝑚,𝑛_𝛼=2^𝑔𝛼/21Γ(1+𝛼)×^∞_−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑔𝑠+𝜂(𝑑𝑠)^𝛼=2^𝑔𝛼/21Γ()×1+𝛼_∞−∞𝜑_𝐻(𝛼)(𝑠)𝜑_𝐻(𝛼)2^𝑔𝑠+𝜂(𝑑𝑠)^𝛼=2^𝑔𝛼/21Γ(1+𝛼)₀^1/2𝜑_𝐻(𝛼)2^𝑔𝑠+𝜂(𝑑𝑠)^𝛼−1Γ(1+𝛼)¹_1/2𝜑_𝐻(𝛼)2^𝑔𝑠+𝜂(𝑑𝑠)^𝛼=2^{−𝑔𝛼/2}1Γ(1+𝛼)^2𝑔−1_𝜂^+𝜂𝜑_𝐻(𝛼)𝑞𝑑𝑞^𝛼−1Γ(1+𝛼)^2𝑔₂^{+𝜂𝑔−1}_+𝜂𝜑_𝐻(𝛼)𝑞𝑑𝑞^𝛼, (28) where 𝑞=2^𝑔1𝑠+𝜂,Γ(1+𝛼)^2𝑔−1_𝜂^+𝜂𝜑_𝐻(𝛼)𝑞𝑑𝑞^𝛼1=0,Γ(1+𝛼)^2𝑔₂^{+𝜂𝑔−1}_+𝜂𝜑_𝐻(𝛼)𝑞𝑑𝑞^𝛼=0, (29) with 𝜂>1,??2^𝑔−1+𝜂>1, and 2^𝑔+𝜂>1.

Hence, taking 𝑒^𝛼_𝑗,𝑘=𝜑^𝑗,𝑘_𝐻(𝛼) gives 𝑓(𝑥)=^∞∞_{𝑗=−∞𝑚=−∞}𝑎_{𝑗,𝑘,𝛼}𝜑_𝐻(𝛼)^𝑗,𝑘(𝑥), (30) where 𝑎_𝑗,𝑘=𝑓(𝑥),𝜑_𝐻(𝛼)^𝑗,𝑘(𝑥)_𝛼=𝑊_𝜑,𝛼𝑓2^−𝑗,𝑘2^−𝑗=2^𝑗𝛼/21Γ(1+𝛼)^∞_−∞𝑓(𝑥)𝜑_𝐻(𝛼)^𝑗,𝑘(𝑥)(𝑑𝑥)^𝛼. (31) Appling (4), we have 𝑓²(𝑥)=^∞_𝑖=1𝑎_𝑗,𝑘² (32) with 𝑎_𝑗,𝑘=2^𝑗𝛼/21Γ(1+𝛼)^∞_−∞𝑓(𝑥)𝜑_𝐻(𝛼)^𝑗,𝑘(𝑥)(𝑑𝑥)^𝛼. (33) Hence, from (32) we find that the energy is conserved.

5. Conclusions

In this work the local fractional discrete wavelet transform based on the local fractional calculus theory was proposed. By using the basic theorems of generalized inner product space, the local fractional discrete wavelet transform and its reconstruction formula were discussed. We find that the energy of the signal on Cantor sets is conserved. An illustrative example for the local fractional wavelet transform of the signal on Cantor sets was given. It is shown that the classical discrete wavelet transform is the local fractional one in case of fractal dimension 𝛼=1.

Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

Acknowledgment

The authors would like to thank the referees for their useful comments and remarks.