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Fractional Wavelet Transform (FRWT) and Its Applications

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Fractional Wavelet Transform (FRWT) and Its Applications Fractional Wavelet Transform and Its Ap- plications Fractional Wavelet Transform (FRWT) Motivation Introduction and Its Applications MRA Orthogonal Wavelets Jun Shi Applications Communication Research Center Harbin Institute of Technology (HIT) Heilongjiang, Harbin 150001, China June 30, 2018 Outline Fractional Wavelet Transform and Its Ap- plications 1 Motivation for using FRWT Motivation Introduction 2 A short introduction to FRWT MRA Orthogonal Wavelets 3 Multiresolution analysis (MRA) of FRWT Applications 4 Construction of orthogonal wavelets for FRWT 5 Applications of FRWT Motivation for using FRWT Fractional Wavelet The fractional wavelet transform (FRWT) can be viewed as a unified Transform time-frequency transform; and Its Ap- plications It combines the advantages of the well-known fractional Fourier trans- form (FRFT) and the classical wavelet transform (WT). Motivation Definition of the FRFT Introduction The FRFT is a generalization of the ordinary Fourier transform (FT) MRA with an angle parameter α. Orthogonal Wavelets F (u) = F α ff (t)g (u) = f (t) K (u; t) dt Applications α α ZR 2 2 1−j cot α u +t j 2 cot α−jtu csc α 2π e ; α 6= kπ; k 2 Z Kα(u; t) = 8δq(t − u); α = 2kπ > <>δ(t + u); α = (2k − 1)π > :> When α = π=2, the FRFT reduces to the Fourier transform. Motivation for using FRWT Fractional Wavelet Transform and Its Ap- plications Motivation Introduction MRA Orthogonal Wavelets Applications Fig. 1. FRFT domain in the time-frequency plane The FRFT can be interpreted as a projection in the time- frequency plane onto a line that makes an angle of α with respect to the time axis. Motivation for using FRWT Fractional Wavelet Transform Time domian waveform and Its Ap- plications 2 1 Motivation 0 Introduction Amplitude -1 MRA -2 Orthogonal Wavelets -40 -20 0 20 40 t Applications Fig. 2. A chirp signal (left) and its FRFTs (right) As the angle α increases continuously from 0 to π=2, the FRFT is able to exhibit all the features of signals from the time domain to the frequency domain. Motivation for using FRWT Fractional Wavelet The FRFT has one major drawback due to using global kernel; Transform and Its Ap- It only provides fractional spectral content with no indication plications about the time localization of the fractional spectral components. Motivation 1 Introduction 1 MRA 0.5 0.8 Orthogonal Wavelets 0 0.6 Amplitude Applications Amplitude 0.4 -0.5 0.2 -1 0 0 5 10 15 0 0.1 0.2 0.3 0.4 t u Fig. 3. A multicomponent chirp signal (left) and its FRFT (right) At what time the fractional frequency components occur? The FRFT can not tell! Motivation for using FRWT Fractional Wavelet Most of Real-World Signals are Non-stationary. Transform and Its Ap- (We need to know whether and also when an incident was happened.) plications Short-Time Fractional Fourier Transform (STFRFT) Motivation Window Introduction MRA Orthogonal Wavelets Applications Amplitude 0 t Definition of the STFRFT α STFRFTf (t; u) = [f(τ )g(τ − t)] Kα(u; τ )dτ ZR where g(t) denotes the window function. Motivation for using FRWT Fractional Wavelet w Transform u and Its Ap- 2DG 2Dg plications Motivation Introduction MRA a 0 Orthogonal t Wavelets Fig. 4. Tiling of the time-frequency plane for the STFRFT Applications Dilemma of Resolution Narrow window −! poor fractional frequency resolution; Wide window −! poor time resolution. Heisenberg Uncertainty Principle of the FRFT A signal cannot be simultaneously concentrated in both time and fractional-frequency domains. A short introduction to FRWT Fractional Fractional Wavelet Transform (FRWT) Wavelet Transform To overcome the resolution problem of the STFRFT. and Its Ap- plications Historical Development D. Mendlovic et al. (1997) first introduced the FRWT Motivation The FRWT was defined as a cascade of the FRFT and the WT, i.e., Introduction α 1 ∗ u−b W (a; b) = f(t)Kα(u; t)p dtdu f a a MRA ZR ZR 1 Orthogonal ∗ u−b = p Fα(u) du: Wavelets a a R Z Applications A. Prasad and A. Mahato (2012) The FRWT was defined as the FRFT-domain expression of the WT. 1 ∗ t−b Wf (a; b) = p f(t) a dt a R Z 2 csc α ∗ j u sin 2α−jbu = F (u sin α)Ψ (au sin α)e 4 du 4π2 ZR where F (u sin α) and Ψ(u sin α) denote the FTs (with arguments scaled by sin α) of f(t) and (t), respectively. A short introduction to FRWT Fractional New Definition of the FRWT Wavelet Transform J. Shi et al. (2012) introduced a new FRWT and Its Ap- plications α α ∗ Wf (a; b) , W ff(t)g (a; b) = f(t) α,a,b(t)dt ZR Motivation where the superscript ∗ denotes complex conjugate, and Introduction 2 2 1 t−b −j t −b cot α + α,a,b(t) , p e 2 ; a 2 R ; b 2 R: MRA a a Orthogonal π 1 Wavelets 2 ∗ t−b When α = π=2;Wf (a; b) = p f(t) a dt Applications a R Z The classical WT 1. J. Shi, N. Zhang, and X. Liu, \A novel| fractional wavelet{z transform} and its applications," Sci. China Inf. Sci., vol. 55, no. 6, pp. 1270{1279, Jun. 2012. 2. J. Shi, X. Liu, and N. Zhang, \Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform," Signal, Image Video Process., vol. 9, no. 1, pp. 211{220, Aug. 2015. 3. J. Shi, X. Liu, X. Sha, Q. Zhang, and N. Zhang, \A sampling theorem for fractional wavelet transform with error estimates," IEEE Trans. Signal Process., vol. 65, no. 18, pp. 4797{4811, Sep. 2017. A short introduction to FRWT Fractional Wavelet The FRWT Structure 2 Transform j t cot α and Its Ap- Considering a chirped signal f(t)e 2 , the FRWT of f(t) can be plications viewed as the ordinary WT of the chirped signal, which contains a chirp 2 −j b cot α factor e 2 . Motivation 2 2 α −j b cot α j t cot α 1 ∗ t−b W (a; b) = e 2 f(t)e 2 p dt Introduction f a R a Z MRA Orthogonal f ()t Wavelet Wabf(,) Wavelets f()t Waba(,) Transform f Applications 2 2 e(/2)cotjt a e-(/2)cotjb a Fig. 5. The FRWT structure. 2 j t cot α 1. a product by a chirp signal, i.e., f(t) ! f~(t) = f(t)e 2 ~ 2. a classical WT, i.e., f(t) ! Wf~(a; b). 3. another product by a chirp signal, i.e., b2 α −j 2 cot α Wf (a; b) ! Wf~(a; b)e A short introduction to FRWT Fractional Wavelet The inverse FRWT Transform 1 da and Its Ap- α f(t) = Wf (a; b) α,a,b(t) 2 db plications 2πC + a ZR ZR where C is a constant that depends on the wavelet used. Motivation Admissibility condition Introduction Ψ(Ω) 2 MRA C = j j dΩ < Ω 1 Orthogonal ZR Wavelets where Ψ(Ω) denotes the FT of (t). Applications α 1 t W (a; b) = f(t) ∗ (t)dt =( fΘ )(b); (t) ∗ f α,a,b α a a , pa − a ZR Fractional convolution 2 2 α j t cot α j t cot α (xΘ y)(t) e− 2 x(t)e− 2 y(t) F p2πX (u)Y (u csc α) α , ∗ −! α A short introduction to FRWT Fractional Wavelet Physical interpretation of the FRWT Transform The FRWT can be rewritten in terms of FRFT-domain representations as and Its Ap- plications p α ∗ ∗ Wf (a; b) = 2πaFα(u)Ψ (au csc α)Kα(u; b)du ZR Motivation where Fα(u) and Ψ(u csc α) denote the FRFT of f(t) and the FT (with Introduction its argument scaled by csc α) of (t), respectively. MRA Orthogonal It follows from the admissibility condition, C < 1, that Wavelets Applications Ψ(0) = 0; i:e:; (t)dt = 0 ZR which implies that fractional wavelet bases must oscillate and behave as bandpass filters in the FRFT domain. From a signal processing point of view, if we treat Ψ∗(u csc α) as the transfer function of a bandpass filter in the FRFT domain, then the FRWT can be viewed as the output of a bandpass filter bank. A short introduction to FRWT Fractional Wavelet Tiling of the time-frequency plane Transform The classical WT is based on rectangular tessellations of the time- and Its Ap- plications frequency plane; The FRWT tiles the time-frequency plane in a parallelogram fashion, which makes it as a unified time-frequency transform. Motivation 2aD Introduction w y w u 2aD a y MRA D DY 2 Y 2 a a Orthogonal Wavelets Applications a 0 t 0 t Fig. 6. Tiling of the time-frequency plane: WT (left) and FRWT (right) a > 1: dilate the signal; a < 1: compress the signal; Low fractional frequency ) High scale ) Non-detailed global view; High fractional frequency ) Low scale ) Detailed view. A short introduction to FRWT Fractional Basic Properties of the FRWT Wavelet Transform and Its Ap- Linearity plications f(t) = k f (t) + k f (t) W α(a; b) = k W α (a; b) + k W α (a; b) 1 1 2 2 , f 1 f1 2 f2 Fractional time shift Motivation jτ(t τ ) cot α α α f(t) = f1(t τ)e− − 2 W (a; b) = W (a; b τ) Introduction − , f f1 − Time scaling MRA α 1 β cot α Orthogonal f(t) = f1(ct); c > 0 W (a; b) = W (ac; bc); β = arccot 2 , f pc f1 c Wavelets Fractional convolution Applications b f(t) = f (t)Θ f (t) W α(a; b) = f (t) W α (a; b) 1 α 2 , f 2 ∗ f1 Inner product α α da ∗ Wx (a; b) Wy (a; b) 2 db = 2πC x(t); y(t) ;C < + + a h i 1 ZR ZR Parseval's relation 2 1 α 2 da f(t) dt = Wf (a; b) 2 db; C < + j j 2πC + a 1 ZR ZR ZR Multiresolution analysis (MRA) of FRWT Fractional Wavelet 2 Transform t2− n k and Its Ap- α k k j 2 cot α V = span φ (t) 2 2 φ(2 t − n)e 2 ; k 2 plications k 8 k,n,α , 9 Z < = Motivation Multiresolution: Analysis (MRA) of FRWT ; Introduction A MRA of the FRWT is a sequence of closed subspaces V α 2 k k2Z MRA L2( ) R such that Orthogonal α α α 2 α Wavelets 1) Vk ⊆ Vk+1, k2Z Vk = L (R), and k2Z Vk = f0g; (2t)2−t2 Applications α j cot α α 2) f(t) 2 Vk ifS and only if f(2t)e 2 T 2 Vk+1; 2 2 −j t cot α α 3) There exists a function φ(t) 2 L (R) with φ(t)e 2 2 V0 α such that φ0,n,α(t) forms a Riesz basis of V .
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