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 {f (t)} (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 ∈ Z Kα(u, t) = δq(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)√ ψ dtdu f a a MRA ZR ZR 1   Orthogonal ∗ u−b = √ Fα(u)ψ 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) = √ f(t)ψ 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 ψ(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 {f(t)} (a, b) = f(t)ψα,a,b(t)dt ZR Motivation where the superscript ∗ denotes complex conjugate, and

Introduction 2 2 1 t−b −j t −b cot α + ψα,a,b(t) , √ ψ e 2 , a ∈ R , b ∈ R. MRA a a   Orthogonal π 1 Wavelets 2 ∗ t−b When α = π/2,Wf (a, b) = √ f(t)ψ 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 √ ψ 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)ψα,a,b(t) 2 db plications 2πC + a ψ ZR ZR where Cψ is a constant that depends on the wavelet used. Motivation Admissibility condition Introduction Ψ(Ω) 2 MRA C = | | dΩ < ψ Ω ∞ Orthogonal ZR Wavelets where Ψ(Ω) denotes the FT of ψ(t). Applications

α 1 t W (a, b) = f(t)ψ∗ (t)dt =( fΘ ψ )(b), ψ (t) ψ∗ f α,a,b α a a , √a − 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 √2πX (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 √ 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 ψ(t), respectively. MRA

Orthogonal It follows from the admissibility condition, Cψ < ∞, that Wavelets Applications Ψ(0) = 0, i.e., ψ(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 √c 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 = 2πCψ x(t), y(t) ,Cψ < + + a h i ∞ ZR ZR Parseval’s relation  2 1 α 2 da f(t) dt = Wf (a, b) 2 db, Cψ < + | | 2πC + a ∞ ZR ψ 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 ∈ plications k  k,n,α ,  Z  

Motivation Multiresolution Analysis (MRA) of FRWT  Introduction A MRA of the FRWT is a sequence of closed subspaces V α ∈ k k∈Z MRA L2( ) R such that  Orthogonal α α α 2 α Wavelets 1) Vk ⊆ Vk+1, k∈Z Vk = L (R), and k∈Z Vk = {0}; (2t)2−t2 Applications α j cot α α 2) f(t) ∈ Vk ifS and only if f(2t)e 2 T ∈ Vk+1; 2 2 −j t cot α α 3) There exists a function φ(t) ∈ L (R) with φ(t)e 2 ∈ V0 α such that φ0,n,α(t) forms a Riesz basis of V . n∈Z 0 Riesz basis condition 0 < A ≤ |Φ(u csc α + 2kπ)|2 ≤ B < +∞ k∈ XZ where Φ(u csc α) is the scaled FT of φ(t). Construction of orthogonal wavelets for FRWT

Fractional Wavelet Fractional Wavelet Subspaces Transform α and Its Ap- For every k ∈ Z, define Wk to be the orthogonal complement of plications α α Vk in Vk+1. It follows that

α α α α α Motivation Wk ⊥Vk ,Vk+1 = Vk Wk . Introduction M MRA From the MRA definition of the FRWT, it is easy to verify that α α Orthogonal a) Wk ⊥Wl , ∀ k 6= l; Wavelets L2( ) = W α b) R k∈Z k ; Applications 2 2 α j (2t) −t cot α α g(t) ∈ W ⇔ g(2t)e 2 ∈ W ∀ k ∈ c) kL k+1, Z.   Two-Scale Equations α α Since φ0,n,α(t) ∈ V0 ⊆ V1 holds for all n ∈ Z, there exists a 2 sequence h[n] ∈ ` (Z) such that

φ0,0,α(t) = h[n]φ1,n,α(t) n∈ XZ Construction of orthogonal wavelets for FRWT

Fractional α Wavelet On the other hand, since fractional wavelet function ψ0,0,α(t) ∈ W0 ⊆ Transform V α {g[n]} ∈ `2( ) and Its Ap- 1 , there must be coefficients n∈Z Z , such that plications

ψ0,0,α(t) = g[n]φ1,n,α(t) n∈Z Motivation X

Introduction FRFT-Domain Expression of Two-Scale Equations

MRA u csc α u csc α Φ(u csc α) = Λ( 2 )Φ( 2 ) Orthogonal u csc α u csc α Wavelets Ψ(u csc α) = Γ( 2 )Φ( 2 ) Applications where Φ(u csc α) and Ψ(u csc α) denote the FTs (with their argument scaled by csc α) of φ(t) and ψ(t), respectively, and

2 1 j n cot α −jnu csc α Λ(u csc α) = √ h[n]e 8 e 2 n∈ XZ 2 1 j n cot α −jnu csc α Γ(u csc α) = √ g[n]e 8 e 2 n∈ XZ Construction of orthogonal wavelets for FRWT

Fractional Wavelet Orthogonal Wavelets of the FRWT Transform √ 2 j n cot α and Its Ap- ψ(t) = 2 g[n]e 8 φ(2t − n) If n∈Z , then the set of func- plications α tions {ψ0,n,α(t)}n∈ forms an orthonormal basis for W if and only if P Z 0 M(u csc α) is unitary matrix, or Motivation † Introduction M(u csc α)M (u csc α) = I, a.e. u ∈ R

MRA where † in the superscript denotes conjugate transpose, I is identity Orthogonal Wavelets matrix, and Applications Λ(u csc α)Λ(u csc α + π) M(u csc α) = . Γ(u csc α)Γ(u csc α + π)  

This is due to the fact that {φ0,n,α(t)}n∈Z and {ψ0,m,α(t)}m∈Z are α α orthonormal bases for V0 and W0 , respectively, and {φ0,n,α(t)}n∈Z and {ψ0,m,α(t)}m∈Z are orthogonal. Applications of FRWT

Fractional Wavelet 哈尔滨工业大学工学博士学位论文 Transform Signal Denoising and Its Ap- plications 2 2 s 0 x 0 -2 -2 0 10 20 30 0 10 20 30 Motivation 2 2

Introduction 3 3

a 0 d 0 MRA -2 -2 0 10 20 30 0 10 20 30 Orthogonal 2 2

Wavelets 2 2

a 0 d 0 Applications -2 -2 0 10 20 30 0 10 20 30 2 2 1 1

a 0 d 0 -2 -2 0 10 20 30 0 10 20 30

Fig.图 7.4-11 Three含噪信号 levelsx( oft) the的角度 FRWTπ/4 分数阶小波变换的 decomposition3 for层分解结果 an interfered Fig.4-11 Three levels of FRWT decompositionchirp signal with α = π/4 for interfered signal x(t)

在图 4-11 中，s 和 x 分别为期望信号及其含噪信号的时域波形， a1、a2 和

a3 分别表示 π/4 角度分数阶小波变换在 1 至 3 层上分解得到的信号概貌部分，

相应的细节信息分别为 d1、d2 和 d3。可以看出，反映噪声的细节部分幅度较 小，可以利用所提出的方法将其剔除，从而达到有效抑制噪声的目的。 4.4.9.2 基于分数阶小波变换的含噪线性调频信号的时延估计 由于线性调频信号在分数域呈现能量最佳聚集特性，且时延在分数域反映 为信号分数阶频率的搬移，因此分数傅里叶变换是用于线性调频信号时延估计 的有效工具。然而，由于分数傅里叶变换缺乏信号局部化表征功能，基于该变 换的时延估计方法通常需要迭代搜索，计算量很大，往往无法满足实际应用需 求。注意到分数阶小波变换具有刻画信号分数域局部特征的功能，能够有效地 克服基于分数傅里叶变换时延估计方法的缺陷。 假设源信号 s(t) 为一线性调频信号，具体表达式为 2 (t t0) k 2 − j t +jω0t s(t) = e− 2σ2 e− 2 (4-293)

式中， t0, σ, k, ω0 R。观测信号 y(t) 由该源信号 s(t) 的多径时延信号和加性高 ∈ 斯白噪声 n(t) 组成，即 N

y(t) = cn(t) + n(t) (4-294) n=0 ∑ 式中， N 为路径条数， c (t) def= λ s(t τ ) 表示第 n 径时延信号，其中 0 τ < n n − n ≤ 0 τ < τ < < τ ，λ > 0。这里， τ 和 λ 分别表示第 n 径信道的传播增益和 1 2 ··· N n n n 时延。于是，利用分数阶小波变换对含噪信号 y(t) 进行时延估计的过程如下： 步骤一 ：利用提出的分数阶小波变换去噪方法对含噪观测信号 y(t) 进行降

- 162 - Applications of FRWT

Fractional Wavelet Link with Chirplet Transform Transform and Its Ap- The chirplet transform (CT) is defined as plications

Cf (tc, fc, log σ, ξ) = f(t)ct∗c,fc,log σ,ξ(t)dt Motivation ZR Introduction 2 MRA 1 t tc jπξ(t tc) +j2πfc(t tc) ctc,fc,log σ,ξ(t) = √σ g −σ e − − Orthogonal Wavelets where tc is the time center, fc the
frequency center, σ > 0 the Applications effective time spread, ξ the chirp rate, and g(t) a window function.

2 2 jπξtc jπξt 1 t tc j2π(fc ξtc)(t tc) Cf (tc, fc, log σ, ξ) = e− f(t)e √σ g −σ e − − dt Z R
which implies that the CT is identical to a FRWT, with cot α = 2πξ, t tc t tc j2π(fc ξtc)(t tc) b = tc, a = σ, and ψ −σ = g −σ e − − .

Therefore, the FRWT is a very promising tool for signal analysis and processing.

Applications of FRWT

Fractional Wavelet Link with Shift-Invariant Subspaces Transform and Its Ap- Let V0 be the shift-invariant subspace or the multiresolution sub- plications space of the ordinary WT, which is generated by the L2-closure of the linear combination of {φ(t−n)}n∈Z. The relationship between α Motivation V0 and V0 is given by Introduction t2 α j 2 cot α MRA f(t) ∈ V0 ⇔ f(t)e ∈ V0. Orthogonal Wavelets Consequently, {φ (t)} is a Riesz basis for V α if and only 0,n,α n∈Z 0 Applications {φ(t − n)} V if n∈Z is a Riesz basis for 0, for which there are many known results, e.g., nonuniform sampling and reconstruction, over- sampling, compressed sensing, and frames of translates. Applications of FRWT

Fractional Wavelet Link with Shift-Invariant Subspaces Transform and Its Ap- Let V0 be the shift-invariant subspace or the multiresolution sub- plications space of the ordinary WT, which is generated by the L2-closure of the linear combination of {φ(t−n)}n∈Z. The relationship between α Motivation V0 and V0 is given by Introduction t2 α j 2 cot α MRA f(t) ∈ V0 ⇔ f(t)e ∈ V0. Orthogonal Wavelets Consequently, {φ (t)} is a Riesz basis for V α if and only 0,n,α n∈Z 0 Applications {φ(t − n)} V if n∈Z is a Riesz basis for 0, for which there are many known results, e.g., nonuniform sampling and reconstruction, over- sampling, compressed sensing, and frames of translates.

Therefore, the FRWT is a very promising tool for signal analysis and processing. References

Fractional Wavelet Transform and Its Ap- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with plications Applications in Optics and Signal Processing, New York: Wiley, 2000. R. Tao, Y. Lei, and Y. Wang, “Short-time fractional Fourier transform and its ap- plications,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2568–2580, May 2010. Motivation J. Shi, X. Liu, and N. Zhang, “On uncertainty principle for signal concentrations Introduction with fractional Fourier transform,” Signal Process., vol. 92, pp. 2830–2836, 2012.

MRA D. Mendlovic, Z. Zalevsky, D. Mas, J. Garc´ıa,and C. Ferreira, “Fractional wavelet transform,” Appl. Opt., vol. 36, pp. 4801–4806, 1997. Orthogonal Wavelets A. Prasad and A. Mahato, “The fractional wavelet transform on spaces of type S,” Applications Integral Transform Spec. Funct., vol. 23, no. 4, pp. 237–249, 2012. J. Shi, N. Zhang, and X. Liu, “A novel fractional wavelet transform and its applica- tions,” Sci. China Inf. Sci., vol. 55, no. 6, pp. 1270–1279, June 2012.

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.

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. Fractional Wavelet Transform and Its Ap- plications

Motivation

Introduction

MRA

Orthogonal Thank you! Wavelets

Applications