Generalized wavelet transform based on the convolution operator in the linear canonical transform domain

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Generalized wavelet transform based on the convolution operator in the linear canonical transform domain Deyun Wei ∗ , Yuan-Min Li Department of Mathematics, Xidian University, Xi’an 710071, China

a r t i c l e

i n f o

Article history: Received 7 August 2013 Accepted 12 February 2014 Available online xxx Keywords: Wavelet transform Time-frequency analysis Linear canonical transform

a b s t r a c t The wavelet transform (WT) and linear canonical transform (LCT) have been shown to be powerful tool for optics and signal processing. In this paper, firstly, we introduce a novel time-frequency transformation tool coined the generalized wavelet transform (GWT), based on the idea of the LCT and WT. Then, we derive some fundamental results of this transform, including its basis properties, inner product theorem and convolution theorem, inverse formula and admissibility condition. Further, we also discuss the timefractional-frequency resolution of the GWT. The GWT is capable of representing signals in the timefractional-frequency plane. Last, some potential applications of the GWT are also presented to show the advantage of the theory. The GWT can circumvent the limitations of the WT and the LCT. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction The wavelet transform (WT), which has had a growing importance in optics and signal processing, has been shown to be a successful tool for time-frequency analysis and image processing [1]. It has found many applications in time-dependent frequency analysis of short-transient signals, data compression, optical correlators, sound analysis, representation of fractal aggregates and many others [1–7]. However, the signal analysis capability of the WT is limited in the time-frequency plane. Therefore, the WT is inefficient for processing signals whose energy is not well concentrated in the frequency domain. Now, many of novel signal processing tools have been proposed to rectify the limitations of the WT, and can provide signal representation in the fractional domain. Such as fractional Fourier transform (FRFT) [8], the Radon–Wigner transform [9], the fractional wavelet transform (FRWT) [10–14], fractional wave packet transform (FRWPT) [15], the short-time FRFT [16], the LCT [17–20] and so on. In the past decade, although the FRFT has attracted much attention of the signal processing community, it cannot obtain information about local properties of the signal. Therefore, the FRWT fails in obtaining information about local properties of the signal [10–12]. Simultaneously, the drawback of the short-time FRFT is that its time and fractional domain resolutions cannot simultaneously be arbitrarily high [16]. The FRWPT did not receive

∗ Corresponding author. E-mail addresses: [email protected], [email protected] (D. Wei), [email protected], [email protected] (Y.-M. Li).

much attention for the lack of physical interpretation and high computation complexity [15]. The LCT [17–20], which was introduced during the 1970s with four parameters, has been proven to be one of the most powerful tools for non-stationary signal processing. The well-known signal processing operations, such as the Fourier transform (FT), the FRFT, the Fresnel transform, and the scaling operations are all special cases of the LCT [18,19]. The LCT has also found many applications in the solution of optical systems, filter design, time-frequency analysis and many others [21–31]. This transform, however, has one major drawback due to using global kernel, i.e., the LCT representation only provides such LCT spectral content with no indication about the time localization of the LCT spectral components [19,20]. Therefore, the analysis of nonstationary signals whose LCT spectral characteristics change with time requires joint signal representations in both time and LCT domains, rather than just a LCT domain representation. As a generalization of the WT, a novel FRWT can combine the advantages of the WT and the FRFT, i.e., it is a linear transformation without cross-term interference and is capable of providing multiresolution analysis and representing signal in the fractional domain [13,14]. Simultaneously, comparing to the FRFT with one extra degree of freedom, LCT is more flexible for its extra three degree of freedom, and has been used frequently in time-frequency analysis and non-stationary signal processing. Inspired of FRWT, we introduce the concept of the generalized wavelet transform (GWT), combining the idea of LCT and WT. the proposed transform not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the LCT domain which is similar to LCT. Compared with the existing FRWT, the GWT can offer signal representations in the time-fractional-frequency

http://dx.doi.org/10.1016/j.ijleo.2014.02.021 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

Please cite this article in press as: D. Wei, Y.-M. Li, Generalized wavelet transform based on the convolution operator in the linear canonical transform domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.021

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plane in LCT domain. Besides, it has explicit physical interpretation, low computation complexity and usefulness for practical application. The rest of this paper is organized as follows. Section 2 presents the theoretical basis of WT, LCT and convolution theory. In Section 3, the GWT is proposed. Moreover, some fundamental results of this transform are presented, including its basis properties, theorems, inverse formula and admissibility condition. In Section 4, the time-fractional-frequency analysis of the GWT is discussed. Potential applications for GWT are presented in section 5. Finally, Section 6 concludes this paper.

energy is not well concentrated in the frequency domain. Thus, signal analysis associated with it is limited to the time-frequency plane.

2. Preliminaries

FT (u) = LT [f (x)](u) =

2.2. Linear canonical transform and convolution theorem The linear canonical transform (LCT) provides a mathematical model of paraxial propagation though quadratic phase systems. The output light field FT (u) a quadratic phase systems is related to its input field f(x) through [19]

⎧ ⎨

∞

/ 0, b=

f (x)KT (u, x) dx,

⎩ √−∞j(1/2)CDu2 De

(9)

b = 0,

f (Du),

2.1. Fourier transform and wavelet transform where Fourier transform (FT) is a tool widely applied for signal processing. In this paper, the FT is defined as follows [28]:



F(u) = Ᏺ(f (x))(u) = f (x) =

1 2

∞

f (x) e−jux dx

(1)

−∞



∞

F(u) ejux du

(2)

−∞

where Ᏺ denotes the FT operator and is also used in the subsequent sections. In the following section, Asymmetric definition of the FT is utilized in this paper. The conventional convolution of two signals f(x) and g(x) is defined as





∞

f (x) ∗ g(x) =



¯ − .) f ()g(x − ) d = f (.), g(x

(3)

−∞

where * and the bar-in the subscript denote the conventional convolution operator and the complex conjugate, respectively, and ., . indicates the inner product. To be specific, the convolution theorem of the FT for the signals f(x) and g(x) with associated FTs, F(u) and G(u), respectively is given by:

Ᏺ

f (x) ∗ g(x)←→F(u)G(u)

(4)

A one-dimensional wavelet transform (WT) of a signal f(x) is defined as [1]



∞

f (x) h¯ ab (x) dx

Wf (a, b) = W (f (x))(u) =

(5)

−∞

It can be also defined as a conventional convolution, i.e.,





¯ Wf (a, b) = f (x) ∗ (a−1/2 h(−x/a)) = f (.), hab (.)

(6)

where the kernel hab (x) is a continuous affine transformation of the mother wavelet function h(x), 1 hab (x) = √ h a

x − b a

(7)

√ where b is the shift amount, a is the scale parameter, and a is the normalization factor. Based on the conventional convolution theorem and inverse FT, the WT of the signal f(x) can be also expressed as:



∞

Wf (a, b) =

√ ¯ aF(u) H(au) ejub du

(8)

−∞

where F(u) and H(u) denote the FT of f(x) and h(x), respectively.  +∞ Since H(0) = −∞ h(x) dx = 0, each wavelet component is actually a differently scaled bandpass filter, the wavelet transform is a localized transformation and thus is efficient for processing transient signals. However, WT is inefficient for processing signals whose



KT (u, x) =

1 2 2 ej(1/2)[(A/B)x −(2/B)xu+(D/B)u ] , (j2B)

(10)

where LT is the unitary LCT operator with parameter matrix T = (A, B; C, D), A,B,C,D are real numbers satisfying AD − BC = 1. The inverse transform for LCT is given by a LCT having parameter T −1 = (D, −B; −C, A), that is



∞

FT (u)K¯ T (u, x) du

f (x) =

(11)

−∞

The transform matrix T is useful in the analysis of optical systems because if several systems are cascaded, the overall system matrix can be found by multiplying the corresponding matrices. It should be noted that, when B = 0, the LCT of a signal is essentially a chirp multiplication and it is of no particular interest to our objective in this work, so it will not be discussed in this paper. The LCT family includes the FT and FRFT, coordinate scaling, and chirp multiplication and convolution operations as its special cases. For further details about the definition and properties of LCT, [17–20] can be referred. Contrast with FT, the LCT has a number of unique properties, and it has been widely applied in optics and signal processing. However, the LCT is a global transformation, it cannot obtain information about local properties of the signal. In other word, the LCT tells us the fractional frequencies that exist across the whole duration of the signal but not the fractional frequencies which exist only at a particular time. A convolution and product structures of the LCT is introduced in [28] f (x) g(x) = e−jAx

2 /(2B)



(f (x) ejAx

2 /(2B)

) ∗ g(x)

(12)

where is the generalized convolution operation for the LCT. Then, the convolution theorem of the LCT for the signal f(x) and g(x)is given by LT

f (x) g(x)←→FT (u)G(u/B)

(13)

where FT (u) and G(u) denotes the LCT of f(x) and the FT of g(x), respectively. Particularly, when T = (0, 1 ; −1, 0), (12) reduces to the conventional convolution as given by (3). 3. Generalized wavelet transform 3.1. Definition of generalized wavelet transform In this subsection, we defined a generalized wavelet transform (GWT) based on the convolution operation in LCT domain. The GWT of a square integrable signal f(x) is defined as

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¯ WfT (a, b) = f (x) ∗ (a−1/2 h(−x/a)) 2 /(2B)

= e−jAb





2

f (.)e−jA(.)

/(2B)

3.2.2. Scaling property If f (x) ↔ WfT (a, b), then we have





, hab (.)

f (x)h¯ T,ab (x) dx

(14)

−∞

where 2 −b2

)/(2B) h (x) ab



∞

2 /(2B)

WfT (a, b) = e−jAb



f (x) ejAx

2 /(2B)



h¯ ab (x) dx

(16)

−∞



−jAb2 /(2B)

∞

 f (x) e

jAx2 /(2B)



2 /(2B)

Wf˜ (a, b)

(17)

2

where f˜ (x) = f (x) e−jAb /(2B) . The computation of the GWT corresponds to the following steps:

∞

−∞

(18)

3.2. Some properties and theorems of GWT

ac

dx

2 /(2B)

(f˜ ∗ g)(x)

(21)

where * is the conventional convolution operation for the FT. Theorem 1: Let z(x) = (fg)(x) and WfT (a, b) denote the GWT of f, Wg (a, b) denotes the WT of g. Then WzT (a, b) = WfT (a, b)Wg (a, b)

(22)

Proof: From the definition of the GWT and the relationship between the GWT and WT (17), we have ∞

(z (x) ejAx

2 /(2B)

)h¯ ab (x) dx

 2 /(2B)

= e−jAb

1

∞

((f (x)ejAx

2 /2B

) ∗ g(x))h¯ ab (x) dx

−∞ 2 /(2B)

= e−jAb

W [f˜ (x) ∗ g(x)]

(23)

According to convolution for WT and (17), we have WzT

2 /(2B)

(a, b) = e−jAb





W f˜ (x) W [g (x)] = WfT (a, b) Wg (a, b) (24)

and this complete the proof of (22). When T = (0, 1 ; −1, 0), Eq. (21) reduce to the convolution of the FT. Eq. (22) states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for GWT and WT. Therefore, Eq. (22) is useful in filter design. 3.2.4. Inner product theorem and parseval’s relation Theorem 2: Let WfT (a, b) and WgT (a, b) denote the GWT of f(x), g(x) with parameter matrix T in LCT domain, and let H(ω) denotes the FT of h(x). If H(ω) satisfies



+∞

Ch =

  H(ω)2 ω

−∞

then,



+∞



dω < ∞

+∞

−∞

¯ gT (a, b) WfT (a, b)W

(25)

  dadb = Ch f (.), g(.) 2 a

(26)

Proof: According to the expression of GWT (18), we have

3.2.1. Linearity property If f(x) = kf1 (x) + lf2 (x), and f1 (x) ↔ WfT (a, b), f2 (x) ↔ WfT (a, b), we =

 x − bc 

3.2.3. Convolution theorem In this subsection, we introduce a convolution structure for GWT. The convolution operation is similar to the generalized convolution in the LCT domain [28]. For any function f(x), let us define 2 the function f˜ (x) by f˜ (x) = f (x) ejAx /(2B) . For any two functions f and g, we define the convolution operation  by

−∞

kWfT (a, b) + lWfT (a, b) 1 2

1 √ h¯ ac

√ x − b ah¯ dx a

−∞

where H(u/B) denote the FT of h(x). (18) states that each generalized wavelet component is actually a differently scaled bandpass filter in the fractional domain. The GWT is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane. Thus, it can circumvent the limitations of the WT and the LCT.

WfT (a, b)

A /2B

2 −b2 )A/2B

(20)

2 /(2B)

√ ¯ aFT (u)H(au/B)K T −1 (u, b) du

can easily derive that



−∞

WzT (a, b) = e−jAb

Detailed steps are as follows: first, multiply the signal f(x) with 2 a chirp signal ejAx /(2B) , and sample the multiplied signal. Next, discrete the scaling and translation parameter a and b. The most popular approach of discretizing a and b is using a = 2m , b = n2m where m and n are integers. Then, perform a discrete WT on the samples of multiplied signal. Finally, after multiplying another 2 chirp signal e−jAb /(2B) , the GWT of the signal f(x) is obtained. From above steps, we see that the GWT can be implemented by the discrete algorithm of the WT. Moreover, it follows that the computational complexity of the GWT depends on that of the WT which has O(N) (N is the length of the sequence) implementation time. So, the computational complexity of the GWT is O(N). Thus, the GWT has low complexity and is easy to implement for practical application. Based on the convolution theorem (13) and the inverse LCT, the GWT of the signal f(x) can be expressed as WfT (a, b) =

j

−∞ x 2 −(bc)2



(1) a product by a chirp signal (2) a traditional WT (3) another product by a chirp signal



f (x ) e



z(x) = (fg)(x) = e−jAx

h¯ ab (x) dx

−∞

= e−jAb

+∞

f (cx) ej(x

where T = (A, B ; C, D), T = (A , B ; C , D ) and A /B = A/Bc2 .

Noted that the GWT is a function of time, frequency and scale. Note also that for T = (0, 1; −1, 0), the GWT reduces to the WT. Based on the definition of the WT (5), the relationship between the GWT and the WT is given below =e

+∞

1  = √ WfT (ac, bc) c

(15)

where hab (x) is expressed in (7). According to (15), the GWT can also expressed as

WfT (a, b)



1 = √ c

−∞

hT,ab (x) = e−jA(x



+∞

f (cx)h¯ T,ab (x) dx =

+∞

=

3



WfT (a, b) =

(19)

−∞



2

WgT (a, b)

∞

∞

= −∞

√ ¯ aFT (u)H(au/B)K T −1 (u, b) du

(27)

√ ¯ aGT (u)H(au/B)K T −1 (u, b) du

(28)

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Then, substituting Eqs. (27) and (28) into Eq. (26), we obtain





+∞

−∞

+∞

−∞



+∞

=

dadb a2

¯ gT (a, b) WfT (a, b)W

  H(au/B)2 a

−∞



∞

¯ T (u) du FT (u)G

da

(29)

−∞

Based on the inner product theorem of the LCT and (29), we can get (26). When g(x) = f(x) in theorem 1, we can derive the following corollary-parseval’s relation for GWT. Corollary 1: For any function f(x) ∈ L2 (R), let WfT (a, b) denotes the GWT of f(x). We have 1 Ch



+∞

−∞



+∞

−∞

 T  W (a, b)2 dadb = f 2 a



+∞

 2 f (x) dx

(30)

−∞

In the following, we will give the detail analysis about the window function of GWT. To be specific, h(x) and H(u) must have sufficiently fast decay so that they can be used as window functions. So, we assume that h(x) and H(u) are functions with finite centers Eh and EH and finite radii h and H . Let E [.] and [.] denote the expectation and deviation operator, then, we can give the center and radii of the time-domain window function hT,ab (x) of the GWT as follows

2 2  +∞   +∞  xhT,ab (x) dx xhab (x) dx −∞ −∞ E[hT,ab (x)] =  =    +∞  +∞  hT,ab (x)2 dx hab (x)2 dx −∞

−∞

= E[hab (x)] = b + aEh

  +∞ −∞

[hT,ab (x)] =

 2   where WfT (a, b) called a fractional scalegram. This corollary

  +∞

states that the fractional scalegram with parameter T show how the energy of the signals is distributed in the time-scale plane associated with parameter T.

−∞

=

(32)



2

(x − b − aEh )hT,ab (x) dx

1/2

  +∞  hT,ab (x)2 dx −∞



2

(x − b − aEh )hab (x) dx

1/2

  +∞  hab (x)2 dx −∞

= [hab (x)] = ah

(33)

3.3. Inverse formula and admissibility condition for GWT In the following, we will introduce the inverse formula for GWT based on the inner product theorem of GWT. Corollary 2: For f(x), h(x) ∈ L2 (R), and H(ω) satisfies (25), we obtain the inverse formula of the GWT as f (x) =

1 Ch



+∞

−∞



+∞

 T  W (a, b)2 hT,ab (x) dadb

−∞

f

a2

(31)

This corollary can be easily derived by setting g(x) = ı(x) in theorem 1 and is omitted here. where Ch is a constant satisfying (24) which is called the admissibility condition for the GWT. It follows from the condition that H(0) = 0 and therefore, the GWT is intrinsically a bank of multiscale bandpass filters in the fractional domain with parameter T 4. Time-fractional-frequency analysis with parameter T

Similarly, we can easily derive the center and radii of the fractional domain window function H(au/B) of the GWT. E[H(au/B)] =

EH B, a

H B a

[H(au/B)] =

(34)

Then, the Q-factor of the fractional-domain window function of the GWT is given by Q =

[H(au/B)]



=

E H(au/B)

H (H /a)B = EH (EH /a)B

(35)

which is independent of the parameter T and the scaling parameter a. This is the constant-Q property of the GWT. 4.2. Time-fractional-frequency resolution with parameter T According to (14) and (32), (33), we see that the GWT WfT (a, b) of a signal f(x) localizes the signal with a time window as follows

In this section, we will study the Q-factor (or the ratio between the width and the center) of the fractional domain window function of the GWT and time-fractional-frequency resolution of the GWT with parameter T. From the following constant-Q property and time-fractional-frequency resolution of the GWT, we see that the GWT can offer signal representations in the time-fractionalfrequency plane. It can inherits the advantage of multiresolution analysis of the WT, and also has the capability of signal representations in the fractional domain with parameter T.

In signal analysis, it is called time localization. Similarly, from the (18) and (34), it follows that the GWT WfT (a, b) gives localized information of the LCT spectrum FT (u) of the signal f(x), with a fractional frequency window with parameter B.

4.1. Constant-Q property

This is called fractional frequency localization in signal analysis. By equating the quantities WfT (a, b) from (14) and (18), we get a time-fractional-frequency window with parameter B as follows

From (14) and (18), whether the time domain or fractional domain GWT, if the kernel of the GWT hT,ab (x) is supported in the time domain, WfT (a, b) is also supported in the time domain. Thus, ¯ W T (a, b) contains information about f(x) near b. Similarly, if H(u/B) f

is bandpass and H(ω) satisfied the admissibility condition, then the ¯ multiplication of FT (u) and H(au/B) can provide fractional-domain local properties of f(x) with parameter T. Since each generalized wavelet component is actually differently scaled bandpass filter in the fractional domain with parameter T, the GWT is capable of providing the time-and fractional domain information simultaneously and representing signals in the time-fractional-frequency plane.

[b + aEh − ah ,

E

H

a

−

H , a

b + aEh + ah ]

EH H + a a

(36)

[b + aEh − ah , b + aEh + ah ] ×

(37)

E

H

a

−

H EH H , + a a a

B

(38)



with constant window area 2ah × 2 aH B = 4h H B in the t − u plane. From (38), this window area only depends on the mother wavelet function h(x) and the transform parameter B. However, for a given transform parameter B, the shape of the timefractional-frequency window varies with the scaling parameter a. Simultaneously, from (38), it can be seen that how the time and fractional-frequency resolutions change with the width and height of the time-fractional-frequency window. Therefore, the GWT is

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capable of providing the time-and fractional-domain information simultaneously and representing signals in the time-fractional- frequency plane with transform parameter T. 5. Some potential applications for GWT In this paper, in order to rectify the limitations of the WT and the LCT, we have proposed a new time-frequency analysis tool-GWT. The GWT not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to LCT. In this section, we will present some potential applications show the advantages of GWT. To be specific, we will discuss the filter design, signal denoising and multipath chirp signal separation associated with GWT. From the convolution structure of GWT (21) and (22), we see that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for GWT and WT. Therefore, Eq. (22) is useful in filter design. For example, if we are interested only in the frequency spectrum of the GWT in the region [u1 , u2 ] of a signal f, we choose the filter impulse response (WT), g, so that Wg is constant over [u1 , u2 ], and zero or of rapid decay outside that region. Passing the output of the filter, yields that part of the spectrum of f over [u1 , u2 ]. Since the WT is actually a differently scaled bandpass filter in the frequency domain, the signal analysis capability of the WT is limited in the time-frequency plane. Simultaneously, the conventional wavelet denoising methods are often based on the assumption that the energy of the analyzed signal is well concentrated in the frequency domain. Therefore, the WT is inefficient for processing signals whose energy is not well concentrated in the frequency domain. For example, chirp-like signals, which are ubiquitous in nature, are this kind of signals. In this paper, the GWT is introduced. We known that each generalized wavelet component is actually a bank of multiscale bandpass filters in the fractional domain with specific parameter. This means that the GWT can overcome the weakness of the WT whose analysis is limited to the time-frequency plane and circumvent the drawback of the LCT. Thus, the GWT can analyze the signal at different fractional-frequency bands with different scales by successive decomposition into coarse approximation and detail information. If the details are small, they might be omitted without substantially affecting the main features of the signal. The idea of thresholding is to set to zero all coefficients that are less than a particular threshold. The modified coefficients are used in an inverse GWT to reconstruct the derived signal. Since the chirp signal is highly concentrated in the fractional domain and a time delay leads to a fractional-frequency shift, the LCT is an efficient processing tool for separating multipath chirp signals [30,31]. However, since the LCT tells us the fractional frequencies that lasts for the total duration of the signal rather than for a particular time, the fractional spectrum of the signal cannot be ascertained when those fractional frequencies exist. Therefore, the GWT fails in obtaining information about local properties of the signal. Simultaneously, the separation based on LCT often requires a large number of iterations and is inefficient for practical applications. However, the proposed GWT has the capability of obtaining fractional-domain local properties of a signal. Besides, it has low computational complexity. Therefore, GWT can overcome the deficiency of the LCT based separation. 6. Conclusion In this paper, a novel time-frequency transformation was defined and coined the generalized wavelet transform (GWT). This transform combines the advantage of WT and LCT. Some fundamental results of this transform are derived, including its

5

basis properties, inner product theorem and convolution theorem, inverse formula and admissibility condition. Moreover, the constant-Q property and time-fractional-frequency resolution of the GWT are also discussed. The GWT can offer signal representations in the time-fractional-frequency plane. It can overcome the limitations of the WT and LCT. Besides, the GWT has explicit physical interpretation, i.e., each generalized wavelet component is actually a differently scaled bandpass filter in the LCT domain, and it has low complexity and is easy to implement in practical applications. Further study can be done the relationships of the GWT with other time-frequency representations such as Wigner distribution, the ambiguity function and the spectrogram. Acknowledgement This work was supported by National Natural Science Foundation of China under Grant 61301283, and also sponsored by the Fundamental Research Funds for the Central Universities under Grants K5051370011, BDY111407, K5051370024, and K5051370006. References [1] I. Daubechies, The wavelet transform time–frequency localization and signal analysis, IEEE Trans. Inf. Theory 36 (1990) 961–1005. [2] J. Caulfield, H. Szu, Parallel discrete and continuous Wavelet transforms, Opt. Eng. 31 (1992) 1835–1839. [3] H. Szu, Y. Sheng, J. Chen, Wavelet transform as a bank of matched filters, Appl. Opt. 31 (1992) 3267–3277. [4] X.J. Lu, A. Katz, E.G. Kanterakis, N.P. Caviris, Joint transform correlation using wavelet transforms, Opt. Lett. 18 (1993) 1700–1703. [5] E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Optical wavelet transform of fractal aggregates, Phys. Rev. Lett. 64 (1990) 7745–7748. [6] H.M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transforms, J. Opt. Soc. Am. A 11 (1994) 547–559. [7] S.-Y. Lee, H. Szu, Fractional Fourier transforms, wavelet transforms, and adaptive neural networks, Opt. Eng. 33 (1994) 1159–1161. [8] L.B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process. 42 (1994) 3084–3091. [9] J.C. Wood, D.T. Barry, Linear signal synthesis using the Radon-Wigner transform, IEEE Trans Signal Process. 42 (1994) 2105–2111. [10] D. Mendlovic, Z. Zalevsky, D. Mas, et al., Fractional wavelet transform, Appl. Opt. 36 (1997) 4801–4806. [11] G. Bhatnagar, B. Raman, Encryption based robust watermarking in fractional wavelet domain, Rec. Adv. Mult. Sig. Proc. Commun. 231 (2009) 375–416. [12] L. Chen, D. Zhao, Optical image encryption based on fractional wavelet transform, Opt. Commun. 254 (2005) 361–367. [13] J. Shi, N. Zhang, X. Liu, A novel fractional wavelet transform and its applications, Sci. China Inf. Sci. 55 (2012) 1270–1279. [14] A. Prasad, S. Manna, A. Mahato, V.K. Singh, The generalized continuous wavelet transform associated with the fractional Fourier transform, J. Comput. Appl. Math. 259 (2014) 660–671. [15] Y. Huang, B. Suter, The fractional wave packet transform, Multidim Syst. Signal Process. 9 (1998) 399–402. [16] R. Tao, Y.L. Li, Y. Wang, Short-time fractional Fourier transform and its applications, IEEE Trans. Signal. Process. 58 (2010) 2568–2580. [17] M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971) 1772–1783. [18] K.B. Wolf, Construction and properties of canonical transforms, in: Integral Transforms in Science and Engineering, Plenum, 1979 (Chap. 9). [19] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, Chichester, UK, 2000. [20] S.C. Pei, J.J. Ding, Relations between fractional operations and time-frequency distributions, and their applications, IEEE Trans. Signal Process. 49 (2001) 1638–1655. [21] B. Barshan, M.A. Kutay, H.M. Ozaktas, Optimal filtering with linear canonical transformations, Opt. Commun. 135 (1997) 32–36. [22] K.K. Sharma, S.D. Joshi, Signal separation using linear canonical and fractional Fourier transform, Opt. Commun. 265 (2006) 454–460. [23] B.M. Hennelly, D.P. Kelly, R.F. Patten, J.E. Ward, U. Gopinathan, F.T. O’Neill, J.T. Sheridan, Metrology and the linear canonical transform, J. Mod. Opt. 53 (2006) 2167–2186. [24] D. Wei, Q. Ran, Y. Li, New convolution theorem for the linear canonical transform and its translation invariance property, Optik 123 (2012) 1478–1481. [25] D. Wei, Y. Li, Linear canonical Wigner distribution and its applications, Optik 125 (2014) 89–92. [26] A. Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Am. A 25 (2008) 647–652.

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Please cite this article in press as: D. Wei, Y.-M. Li, Generalized wavelet transform based on the convolution operator in the linear canonical transform domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.021