Convolution for The Offset Linear Canonical Transform with Gaussian Weight and Its Application

47

Original Article

Convolution for The Offset Linear Canonical Transform

with Gaussian Weight and Its Application

Quan Thai Ha

1,*

_{, Lai Tien Minh}

2

_{, Nguyen Minh Tuan}

3

1_{Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science,}

334 Nguyen Trai, Hanoi, Vietnam

2

Department of Mathematics, Hanoi Architectural University, Hanoi, Vietnam

3_{Department of Mathematics, VNU University of Education, 144 Xuan Thuy, Hanoi, Vietnam}

Received 26 November 2018

Revised 25 February 2019; Accepted 15 March 2019

Abstract: This paper presents the convolution for the offset linear canonical transform (OLCT) with the Gaussian weight and its applications. The product theorem is also studied. In applications, some ways to design the filters in the OLCT domain as well as the multiplicative filter and the Gaussian filter are introduced.

Keywords: Reconstruction, Shannon theorem, convolution, filter, signal, offset linear canonical transform, fractional Fourier transform, Fourier transform.

1.Introduction

Throughout this paper we shall consider parameters a b c d u, , , , 0,0 and i will be denoted the

unit imaginary number. The Offset Linear Canonical Transform (OLCT) (see [1]) of a signal f t

 

with real parameters A



a b c d u, , , , ₀,₀



, (adbc1) is defined as

 



 



 

   









2 0 0

( ) 2

0

, 0

: :

, 0

,

A

cd

A A

i u u i u

u t

u f

f t dt b

F

d e f d u u

t u

b

  

 _



 _{ }

 _{ }





, (1.1)

________

_{Corresponding author.}

E-mail address: [email protected]

where

2 1 2 ( 0 0) 0

2 2

( , ) :

b du u

d a

u tu t u t

b b b b b

i A

u t

K e

A



 _{   } 

 

 



, and

2 0

2

2

idu b

A e K

bi



 .

The inverse OLCT expression is given by



 

 

 

1 1

( ) _A A( ) A _A ,

f t   F u t C



F u  u t du, (1.2)

where 1





0 0 0 0

, , , , ,

A  d  b c a b du cu a , and

 2 2

0 2 0 0 0

1

2cdu adu ab i

Ce    . (1.3)

In this paper, we only consider b0 since the OLCT becomes a chirp multiplication operation otherwise.

The OLCT is generalization of many operations, as follows: the Linear Canonical Transform (LCT), the Fractional Fourier Transform (FRFT), the Fourier Transform (FT). When u0 0 0, we

back to the definition of the Linear Canonical Transform (see [2]).

The Fractional Fourier Transform (FRFT) (see [3]) is considered a special case of the OLCT when parameters A have the form A



cos ,sin , sin ,cos ,0,0    



. For any real angle , the FRFT is defined as



 

1 cot cot2 2 sin cot2 2

( ) , sin 0

2

ut

i u t

i

f u f t e dt

 



 _  

 _{ } 

 

 







 . (1.4)

When the angle 2



 , the FRFT becomes the Fourier Transform (FT) (see [4]). In this paper, we

will use the Fourier Transform and its inverse defined by

 





 

:

 

iut

FT f t u f t e dt



 

_

, (1.5)

 

1



 



 

2

iut FT

f t f t u e du







  , (1.6)

respectively. If _{f h,} _L1_{( ), the classic Fourier convolution operation in the time domain is}

defined as



f *h t

 





f

  

 h t 



d . (1.7) It is easy to see that



f*h

 

t  f

   

t *h t ,   , (1.8) and



 



*



 



 



 

 

 

 

FT f h t u FT f t u FT h t u

     . (1.9)

•

We also have the Young’s inequality (see [5]). If p

 

fL , hLq

 

, and1 1 1 1

p   q r ,



p q r, , 1



. Then the following inequality holds

1

* _{r p q}

f h C f  h , (1.10)

Now we will exemplify some basic properties of A (see [6]).

Suppose _f_L1

 

_{, and}

 

_,  _{, we have}

•

Time shift:









2





0 0

( ) 2

ac

i c u u a

A f t e F uA a

  

      

   .

•

Modulation:

 





 

2 ( 0) 0





2

bd

i d u u b

i t

A e f t u e F uA b

  

 _      _{  .}

•

Time shift/modulation:









2 2 2 2   0   0

_

_

ac bd

i bc

i c d u u i a b y

A e f t e e e F uA a b

   _{    }

 _{ }        _{   .}

•

A is a linear, continuous and one-to-one map from the Schwartz space onto (whose

inverse is obviously also continuous).

Let C₀

 

be the Banach space of all continuous functions on that vanish at infinity and being

endowed with the supremum norm  _, and let ₁: 1 ( ) 2

f f t dt







be the norm in L1

 

.

•

(Riemann-Lebesgue type lemma for the OLCT). If fL1

 

, then _AfC₀

 

, and

1

1 | |

Af f

b  .

•

(Plancherel type theorem for the OLCT). Let f be a complex-valued function in the space

 

2

L and let

 

   

| |

, : ,

Af u k 



_tk A u t f t dt.

Then, as k , _Af u k

 

, converges strongly (over ) to a function, say _Af L2

 

, and, reciprocally,

 

1

 

 

| |

, : , _A

A t k

f u k C  u t f t dt

 

_

converges strongly to f u

 

, where C is the same as in (1.3).

•

(Parseval type identity for the OLCT). For any f h, L2

 

the following identity holds

, ,

Af Ah  f h ,

where ,  is denoting the usual inner product in L2

 

. In the special case when h f , it holds

2 2

Af  f

.

For convenience, we denote 



du₀b₀





2 2



,

 

2 0

2

u a

i t t

b b

A t e

 _ 

 

 



,

f t

 

 _A

   

t f t , and

the Gaussian function

 

 

2 2

1 2

1 t

b

t e

b





 

 



 



 

2 2  0 0

 

b du

d _iut

i u u

b b _b

A A A

F u f t u K e f t e dt



  



  _

 

 

 

_

. (1.11)

There are many different types of convolutions for the OLCT. Most of them have the weight

functions in the form  

2

i u u

e   (see [1]). In [7], some convolutions for the FRFT with the Hermite

weights in the form i u2

 

n

e  u , and the Gaussian weight in the form

2 2 1

2u

i u

e e , are also obtained. In this paper, we focus on studying the convolution for the OLCT with the Gaussian weight in the form

2

1 2u

e , and its applications.

The paper is divided into two sections and organized as follows. In the next section, we provide the convolution for the OLCT with the Gaussian weight function and study its product theorem. Some special cases of this convolution are also deduced.

2.Convolution for the OLCT with the Gaussian weight function and product theorem

Definition 2.1. Let f h, L1

 

, the convolution for the OLCT of two signals f t

 

and h t

 

with the Gaussian weight function

 

t is defined by

 

fh t

 

K_A



_A( )t



1



f * *h



 

2t . (2.1)

It easily seen that if _{f h,} _L1

 

_then

 

 

1

 

fh t L . Moreover, ₁

1 2 1

fh C f  h ,

where C2 is a positive constant.

Theorem 2.1. Assume that

 

1

,

f hL

, z t

 



 

fh t

 

and F uA

 

_, ZA

 

u _, HA

 

u _denote

the OLCT of the signals f t

 

, z t

 

, h t

 

with a set of parameters A, respectively. The factorization following identity is fulfilled

 

1 2

4

2 2

u

A A A

u u

Z u e F _{ }H _ _

   .

Moreover, if



 



2

1

1 4

2 2

u

A A A

u u

e F _ _H _ _ L

    then

 

2

 

1

1 4

2 2

u

A A

A

u u

z t  e F H t



    

 _  _{ } _{ }

   

  . (2.2)

Proof. Based on classic Fourier convolution (1.7), the convolution (2.1) can be expressed as

 

 





1

   





( ) 2

A A

fh t K t 

 

f u h v t u v dudv. (2.3)

Since (1.11), we realize that

 

 

2 0 0

   

2 2 2( )

1 1

2

2 2

b du

d iu iuv

i u u

u u

b b b b

A A A

e F u H u e K e f h v e e d dv

 _

 



 _ 

 

  _ _

 

   

0 0 2 2 2 1 2 2 1 2 b du

d _{iu iuv}

i u u

t iut _{b b}

b b

A

e dtK e f h v e e d dv

 _          _{ }   _ _ 

_

_{ }

   

2 1 ( 0 0) 12

2 _{( ) ( 2 )}

2

2

b du d

i u v bt u u _t

b b b

A K

e f h v e d dvdt

        _{    }       

_{  }

.

By making s    v  bt, we obtain

   

2

1 2u

A A

e F u H u

   2  _{0 0} 

1

1

2 2 2

2 2 2

2 2

b du

d s A A

i u u u

b b b

A A s K s K e b     _  _        _{ }           _{ }  _{ }       _      



   

2 2

1 2b s v

f  h v e     d dv ds _



 

1

2

2, ( ) ( ) ( )

2 2 A A A s K s s

u f h v s v d dv d

b    

  _{ }   _{ }            _     _{ }           _{ }  



 

 

2,

2 2 2

A

s s s

u f h d

           _      



 

 

 

2

A f h u

  .

The proof is completed.

Remark 2.1. Furthermore, using the fomula (1.8) the convolution (2.1) can also be rewritten as

 

 



 



1



     



2 _{A A} 2 * 2 * 2

fh t  K t  f t h t t . (2.4)

Remark 2.2. In particular, if we chosse h t( )( )t , where



( )t is the Dirac delta function, we then have



1



 



 



1





 



 



1



   



* 2 2 2 * 2

A A A A A

f  t K t  f t  K t  f t t . (2.5)

3.Applications

The output signal rout

 

t can be expressed as following

 



 



1



   



2 2 * 2

out t A A t in

r  K  r t t . (3.1)

The method to achieve the multiplicative filter in the OLCT domain through the convolution (2.1) is shown in Fig 1.

In this following example, our objective is using the proposed filters to restore an observed signal

 

   

in

r t y t n t where y t n t

   

, denote the desired signal and the additive noise, respectively.

Example 3.1. Let 2 1, , 1, 3,0,0

7 7

A _   _

 ,

 





 

2 ₂

21

20 2t _{sin 1.5} i t

in

r t e  t e  ,

 

212





2t _{sin 1.5}

y t e  t ,

 

 

2

20

i t

n t e  , and the Gaussian function

 

2

49 2

7 t

t e





 .

 

7 2



   



2 * 2

it

out t in

r e r t t

i



 .

Then the results of Gaussian filter is given in Fig. 2

Figure 1. The method to achieve Gaussian filter in the OLCT domain.

3.2. The multiplicative filter in the OLCT domain.In this subsection, rin

 

t and rout

 

t are denoted

as the input signal and output signal, respectively.

The output signal of OLCT can be obtained as following

 

 

 

2



 



 

1

1 4

2 2

u

out t in A A in A

u u

r r h t  e r t H t

    

  

 

   _   _

     _{. (3.2)}

Let

 

 

2

1 2u

A u e HA u

H   , since the OLCT-frequency spectrum is usually interested only in the

region



u u₁, ₂



, then the filter impulse response h t

 

can be selected such that H_A

 

u is constant over



u u1, 2



, and zero or rapid decay outside that region. In paticular, we then have

 

1



 



 

, ₁ 2, ₂ 2

2

out t _A A in t

u

r  r t u u u

 

 

 

  _{ }

 _{ } _{ }

  .

Moreover, H_A

 

u can also be chosen equal the constant over



u u₁, ₂



, and zero outside that region. Thus, we can get

 

2



 



 

1

1 4

1 2

, 2, 2

2

u

out t A A in t

u

r  e r t u u u

  

 

 

  _{ }

 _  _ _{ }

  .

By denoting

 



 



2

1 4

2 2

u

A in t A

u u

E u e r _ _ H _ _

  

 

, the realization method is given by Fig.3.

Therefore, when the OLCT becomes the LCT or the FRFT, it is easy to implement in the designing of multiplicative filters through the product in the OLCT domain (see [2]).

References

[1] Q. Xiang, K.Y. Qin, Convolution, correlation, and sampling theorems for the offset linear canonical transform, K. SIViP 8(3) (2014) 433:442. https://doi.org/10.1007/s11760-012-0342-0.

[2] A. Koc, H.M. Ozaktas, Cagatay Candan, and M. Alper Kutay, Digital computation of linear canonical transforms, IEEE Transactions on Signal Processing 56(6) (2008) 2383-2394. https://doi.org/10.1109/TSP.2007.912890 [3] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in

Optics and Signal Processing, Wiley, New York, 2001.

[4] R.N. Bracewell, The Fourier Transform and its Applications, 3rd ed, McGraw-Hill Press, New York, 2000. [5] W. Beckner, Inequalities in Fourier analysis, Annals of Mathematics 102(1) (1975) 159-182.

https://doi.org/10.2307/1970980.

[6] L.P. Castro, L.T. Minh, N.M. Tuan, New convolutions for quadratic-phase Fourier integral operators and their applications, Mediterr. J. Math. (2018) 15:13. https://doi.org/10.1007/s00009-017-1063-y.