fractional wavelet transform (frwt) and its applications

FractionalWavelet

Transformand Its Ap-plications

Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Fractional Wavelet Transform (FRWT)and Its Applications

Jun Shi

Communication Research CenterHarbin Institute of Technology (HIT)Heilongjiang, Harbin 150001, China

June 30, 2018

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Outline

1 Motivation for using FRWT

2 A short introduction to FRWT

3 Multiresolution analysis (MRA) of FRWT

4 Construction of orthogonal wavelets for FRWT

5 Applications of FRWT

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Motivation for using FRWT

The fractional wavelet transform (FRWT) can be viewed as a unifiedtime-frequency transform;It combines the advantages of the well-known fractional Fourier trans-form (FRFT) and the classical wavelet transform (WT).

Definition of the FRFT

The FRFT is a generalization of the ordinary Fourier transform (FT)with an angle parameter α.

Fα (u) = Fα f (t) (u) =

∫

Rf (t)Kα (u, t) dt

Kα(u, t) =

√1−j cotα

2πej

u2+t2

2cotα−jtu cscα, α 6= kπ, k ∈ Z

δ(t− u), α = 2kπ

δ(t+ u), α = (2k − 1)π

When α = π/2, the FRFT reduces to the Fourier transform.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

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 respectto the time axis.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


-40 -20 0 20 40

-2

-1

0

1

2

t

Am

plit

ude

Time domian waveform

Fig. 2. A chirp signal (left) and its FRFTs (right)

As the angle α increases continuously from 0 to π/2, the FRFTis able to exhibit all the features of signals from the time domainto the frequency domain.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


The FRFT has one major drawback due to using global kernel;It only provides fractional spectral content with no indicationabout the time localization of the fractional spectral components.

0 5 10 15-1

-0.5

0

0.5

1

t

Am

plitu

de

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

u

Am

plitu

deFig. 3. A multicomponent chirp signal (left) and its FRFT (right)

At what time the fractional frequency components occur? TheFRFT can not tell!

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Most of Real-World Signals are Non-stationary.(We need to know whether and also when an incident was happened.)

Short-Time Fractional Fourier Transform (STFRFT)

Am

plitu

de

Window

t0

Definition of the STFRFT

STFRFTαf (t, u) =

∫

R[f(τ )g(τ − t)]Kα(u, τ )dτ

where g(t) denotes the window function.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


a

u

w

t0

2 gD2 GD

Fig. 4. Tiling of the time-frequency plane for the STFRFT

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 andfractional-frequency domains.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

A short introduction to FRWT

Fractional Wavelet Transform (FRWT)

To overcome the resolution problem of the STFRFT.

Historical Development

D. Mendlovic et al. (1997) first introduced the FRWT

The FRWT was defined as a cascade of the FRFT and the WT, i.e.,

Wαf (a, b) =

∫

R

∫

Rf(t)Kα(u, t)

1√aψ∗(u−ba

)dtdu

=1√a

∫

RFα(u)ψ∗

(u−ba

)du.

A. Prasad and A. Mahato (2012)

The FRWT was defined as the FRFT-domain expression of the WT.

Wf(a, b) =1√a

∫

Rf(t)ψ∗

(t−ba

)dt

=cscα

4π2

∫

RF (u sinα)Ψ∗(au sinα)ej

u2

4sin 2α−jbudu

where F (u sinα) and Ψ(u sinα) denote the FTs (with argumentsscaled by sinα) of f(t) and ψ(t), respectively.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


New Definition of the FRWT

J. Shi et al. (2012) introduced a new FRWT

Wαf (a, b) ,Wα f(t) (a, b) =

∫

Rf(t)ψ∗α,a,b(t)dt

where the superscript ∗ denotes complex conjugate, and

ψα,a,b(t) ,1√aψ(t−ba

)e−j

t2−b22

cotα, a ∈ R+, b ∈ R.

When α = π/2, Wπ2f (a, b) =

1√a

∫

Rf(t)ψ∗

(t−ba

)dt

︸︷︷︸The classical WT

1. J. Shi, N. Zhang, and X. Liu, “A novel fractional wavelet transform and itsapplications,” Sci. China Inf. Sci., vol. 55, no. 6, pp. 1270–1279, Jun. 2012.

2. J. Shi, X. Liu, and N. Zhang, “Multiresolution analysis and orthogonalwavelets associated with fractional wavelet transform,” Signal, Image VideoProcess., vol. 9, no. 1, pp. 211–220, Aug. 2015.

3. J. Shi, X. Liu, X. Sha, Q. Zhang, and N. Zhang, “A sampling theoremfor fractional wavelet transform with error estimates,” IEEE Trans. SignalProcess., vol. 65, no. 18, pp. 4797–4811, Sep. 2017.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


The FRWT Structure

Considering a chirped signal f(t)ejt2

2cotα, the FRWT of f(t) can be

viewed as the ordinary WT of the chirped signal, which contains a chirp

factor e−jb2

2cotα.

Wαf (a, b) = e−j

b2

2cotα

∫

R

[f(t)ej

t2

2cotα

]1√aψ∗(t−ba

)dt

Wavelet Transform

( )f t( )f t ( , )fW a b

( , )fW a ba

2( /2) cotj te a 2( /2) cotj be a-

Fig. 5. The FRWT structure.

1. a product by a chirp signal, i.e., f(t)→ f(t) = f(t)ejt2

2cotα

2. a classical WT, i.e., f(t)→Wf(a, b).3. another product by a chirp signal, i.e.,

Wαf (a, b)→Wf(a, b)e−j

b2

2cotα

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


The inverse FRWT

f(t) =1

2πCψ

∫

R

∫

R+

Wαf (a, b)ψα,a,b(t)

da

a2db

where Cψ is a constant that depends on the wavelet used.

Admissibility condition

Cψ =

∫

R

|Ψ(Ω)|2Ω

dΩ <∞

where Ψ(Ω) denotes the FT of ψ(t).

Wαf (a, b) =

∫

Rf(t)ψ∗α,a,b(t)dt = (fΘαψa) (b), ψa(t) ,

1√aψ∗(− ta

)

Fractional convolution

(xΘαy) (t) , e−jt2

2cotα

[(x(t)e−j

t2

2cotα

)∗ y(t)

]Fα←−→√

2πXα(u)Y (u cscα)

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Physical interpretation of the FRWT

The FRWT can be rewritten in terms of FRFT-domain representations as

Wαf (a, b) =

∫

R

√2πaFα(u)Ψ∗(au cscα)K∗α(u, b)du

where Fα(u) and Ψ(u cscα) denote the FRFT of f(t) and the FT (withits argument scaled by cscα) of ψ(t), respectively.

It follows from the admissibility condition, Cψ <∞, that

Ψ(0) = 0, i.e.,

∫

Rψ(t)dt = 0

which implies that fractional wavelet bases must oscillate and behave asbandpass filters in the FRFT domain.

From a signal processing point of view, if we treat Ψ∗(u cscα) as thetransfer function of a bandpass filter in the FRFT domain, then theFRWT can be viewed as the output of a bandpass filter bank.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Tiling of the time-frequency plane

The classical WT is based on rectangular tessellations of the time-frequency plane;The FRWT tiles the time-frequency plane in a parallelogram fashion,which makes it as a unified time-frequency transform.

2aYD

2a yDw

t

w

t

u

0a

a2aYD

2a yD

0

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.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Basic Properties of the FRWT

Linearity

f(t) = k1f1(t) + k2f2(t)⇔Wαf (a, b) = k1W

αf1

(a, b) + k2Wαf2

(a, b)

Fractional time shift

f(t) = f1(t− τ)e−jτ(t−τ2) cotα ⇔Wα

f (a, b) = Wαf1

(a, b− τ)

Time scaling

f(t) = f1(ct), c > 0⇔Wαf (a, b) = 1√

cW βf1

(ac, bc), β = arccot(cotαc2

)

Fractional convolution

f(t) = f1(t)Θαf2(t)⇔Wαf (a, b) = f2(t)

b∗Wαf1

(a, b)

Inner product∫

R

∫

R+

Wαx (a, b)

[Wαy (a, b)

]∗ daa2db = 2πCψ 〈x(t), y(t)〉 , Cψ < +∞

Parseval’s relation∫

R|f(t)|2dt =

1

2πCψ

∫

R

∫

R+

∣∣Wαf (a, b)

∣∣2 daa2db, Cψ < +∞

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Multiresolution analysis (MRA) of FRWT

V αk = span

φk,n,α(t) , 2

k2φ(2kt− n)ej

t2−(n2k

)2

2cotα

, k ∈ Z

Multiresolution Analysis (MRA) of FRWT

A MRA of the FRWT is a sequence of closed subspacesV αk

k∈Z ∈

L2(R) such that

1) V αk ⊆ V α

k+1,⋃k∈Z V

αk = L2(R), and

⋂k∈Z V

αk = 0;

2) f(t) ∈ V αk if and only if f(2t)ej

(2t)2−t22

cotα ∈ V αk+1;

3) There exists a function φ(t) ∈ L2(R) with φ(t)e−jt2

2cotα ∈ V α

0

such thatφ0,n,α(t)

n∈Z forms a Riesz basis of V α

0 .

Riesz basis condition

0 < A ≤∑

k∈Z|Φ (u cscα+ 2kπ)|2 ≤ B < +∞

where Φ (u cscα) is the scaled FT of φ(t).

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Construction of orthogonal wavelets for FRWT

Fractional Wavelet Subspaces

For every k ∈ Z, define Wαk to be the orthogonal complement of

V αk in V α

k+1. It follows that

Wαk ⊥V

αk , V

αk+1 = V α

k

⊕Wαk .

From the MRA definition of the FRWT, it is easy to verify that

a) Wαk ⊥Wα

l , ∀ k 6= l;b) L2(R) =

⊕k∈ZW

αk ;

c) g(t) ∈Wαk ⇔

(g(2t)ej

(2t)2−t2

2 cotα)∈Wα

k+1, ∀ k ∈ Z.

Two-Scale Equations

Since φ0,n,α(t) ∈ V α0 ⊆ V α

1 holds for all n ∈ Z, there exists asequence h[n] ∈ `2(Z) such that

φ0,0,α(t) =∑

n∈Zh[n]φ1,n,α(t)

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


On the other hand, since fractional wavelet function ψ0,0,α(t) ∈Wα0 ⊆

V α1 , there must be coefficients g[n]n∈Z ∈ `2(Z), such that

ψ0,0,α(t) =∑

n∈Zg[n]φ1,n,α(t)

FRFT-Domain Expression of Two-Scale Equations

Φ(u cscα) = Λ(u cscα2

)Φ(u cscα2

)

Ψ(u cscα) = Γ(u cscα2

)Φ(u cscα2

)

where Φ(u cscα) and Ψ(u cscα) denote the FTs (with their argumentscaled by cscα) of φ(t) and ψ(t), respectively, and

Λ(u cscα) = 1√2

∑

n∈Zh[n]ej

n2

8cotαe−jnu cscα

Γ(u cscα) = 1√2

∑

n∈Zg[n]ej

n2

8cotαe−jnu cscα

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Orthogonal Wavelets of the FRWT

If ψ(t) =√

2∑n∈Z g[n]ej

n2

8cotαφ(2t − n), then the set of func-

tions ψ0,n,α(t)n∈Z forms an orthonormal basis for Wα0 if and only if

M(u cscα) is unitary matrix, or

M(u cscα)M†(u cscα) = I, a.e. u ∈ R

where † in the superscript denotes conjugate transpose, I is identitymatrix, and

M(u cscα) =

[Λ (u cscα) Λ (u cscα+ π)Γ (u cscα) Γ (u cscα+ π)

].

This is due to the fact that φ0,n,α(t)n∈Z and ψ0,m,α(t)m∈Z areorthonormal bases for V α

0 and Wα0 , respectively, and φ0,n,α(t)n∈Z

and ψ0,m,α(t)m∈Z are orthogonal.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Applications of FRWT

Signal Denoising 哈尔滨工业大学工学博士学位论文

0 10 20 30-2

0

2

s0 10 20 30

-2

0

2

x

0 10 20 30-2

0

2

a3

0 10 20 30-2

0

2

d3

0 10 20 30-2

0

2

a2

0 10 20 30-2

0

2

d2

0 10 20 30-2

0

2

a1

0 10 20 30-2

0

2

d1

图 4-11 含噪信号 x(t)的角度 π/4分数阶小波变换的 3层分解结果

Fig.4-11 Three levels of FRWT decomposition 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

s(t) = e− (t−t0)2

2σ2 e−j k2t2+jω0t (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

y(t) =N∑

n=0

cn(t) + n(t) (4-294)

式中，N 为路径条数，cn(t)def= λns(t− τn)表示第 n径时延信号，其中 0 ≤ τ0 <

τ1 < τ2 < · · · < τN，λn > 0。这里，τn 和 λn 分别表示第 n径信道的传播增益和

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 -

Fig. 7. Three levels of the FRWT decomposition for an interferedchirp signal

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Link with Chirplet Transform

The chirplet transform (CT) is defined as

Cf (tc, fc, log σ, ξ) =

∫

R

f(t)c∗tc,fc,log σ,ξ(t)dt

ctc,fc,log σ,ξ(t) = 1√σg(t−tcσ

)ejπξ(t−tc)

2+j2πfc(t−tc)

where tc is the time center, fc the frequency center, σ > 0 theeffective time spread, ξ the chirp rate, and g(t) a window function.

Cf (tc, fc, log σ, ξ) = e−jπξt2c

∫

R

f(t)ejπξt2 1√

σg(t−tcσ

)ej2π(fc−ξtc)(t−tc)dt

which implies that the CT is identical to a FRWT, with cotα = 2πξ,b = tc, a = σ, and ψ

(t−tcσ

)= g

(t−tcσ

)ej2π(fc−ξtc)(t−tc).

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications


Link with Shift-Invariant Subspaces

Let V0 be the shift-invariant subspace or the multiresolution sub-space of the ordinary WT, which is generated by the L2-closure ofthe linear combination of φ(t−n)n∈Z. The relationship betweenV α0 and V0 is given by

f(t) ∈ V α0 ⇔ f(t)ej

t2

2cotα ∈ V0.

Consequently, φ0,n,α(t)n∈Z is a Riesz basis for V α0 if and only

if φ(t− n)n∈Z is a Riesz basis for V0, for which there are manyknown results, e.g., nonuniform sampling and reconstruction, over-sampling, compressed sensing, and frames of translates.

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

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

References

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform withApplications 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.

J. Shi, X. Liu, and N. Zhang, “On uncertainty principle for signal concentrations

with fractional Fourier transform,” Signal Process., vol. 92, pp. 2830–2836, 2012.

D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcıa, and C. Ferreira, “Fractional wavelet

transform,” Appl. Opt., vol. 36, pp. 4801–4806, 1997.

A. Prasad and A. Mahato, “The fractional wavelet transform on spaces of type S,”

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.

FractionalWavelet


Motivation

Introduction

MRA

OrthogonalWavelets

Applications

Thank you!

fractional wavelet transform (frwt) and its applications

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