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47
<i>1</i>
<i>Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science, </i>
<i>334 Nguyen Trai, Hanoi, Vietnam </i>
<i>2<sub>Department of Mathematics, Hanoi Architectural University, Hanoi, Vietnam </sub></i>
<i>3<sub>Department of Mathematics, VNU University of Education, 144 Xuan Thuy, Hanoi, Vietnam </sub></i>
Received 26 November 2018
<i>Revised 25 February 2019; Accepted 15 March 2019 </i>
<b>Abstract: This paper presents the convolution for the offset linear canonical transform (OLCT) </b>
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.
<i>Keywords and phrase: Reconstruction, Shannon theorem, convolution, filter, signal, offset linear </i>
canonical transform, fractional Fourier transform, Fourier transform.
<b>1. Introduction</b>
Throughout this paper we shall consider parameters <i>a b c d u</i>, , , , 0,0 and <i>i will be denoted the </i>
unit imaginary number. The Offset Linear Canonical Transform (OLCT) (see [1]) of a signal <i>f t </i>
with real parameters <i>A</i>
2
0 0
( )
2
0
, 0
: :
, 0
,
<i>A</i>
<i>A</i> <i>A</i>
<i>i</i> <i>u u</i> <i>i</i> <i>u</i>
<i>u t</i>
<i>u</i> <i>f</i>
<i>f t</i> <i>dt</i> <i>b</i>
<i>F</i>
<i>d e</i> <i>f d u</i> <i>u</i>
<i>t</i> <i>u</i>
<i>b</i>
<sub> </sub>
<sub></sub> <sub></sub>
, (1.1)
________
<sub>Corresponding author. </sub>
<i> E-mail address: </i>
where
2 1 2 ( 0 0) 0
2 2
<i>b</i> <i>du</i> <i>u</i>
<i>d</i> <i>a</i>
<i>u</i> <i>tu</i> <i>t</i> <i>u</i> <i>t</i>
<i>b</i> <i>b</i> <i>b</i> <i>b</i> <i>b</i>
<i>i</i>
<i>A</i>
<sub></sub> <sub></sub> <sub></sub> <sub></sub>
2
0
2
2
<i>idu</i>
<i>b</i>
<i>A</i>
<i>e</i>
<i>K</i>
<i>bi</i>
.
The inverse OLCT expression is given by
1 1
( ) <i><sub>A</sub></i> <i>A</i>( ) <i>A</i> <i><sub>A</sub></i> ,
<i>f t</i> <i>F u</i> <i>t</i> <i>C</i>
where 1
0 0 0 0
, , , , ,
<i>A</i> <i>d</i> <i>b</i> <i>c a b</i> <i>du cu</i> <i>a</i> , and
2 2
0 2 0 0 0
1
2<i>cdu</i> <i>adu</i> <i>ab</i>
<i>i</i>
<i>C</i><i>e</i> . (1.3)
In this paper, we only consider <i>b</i>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 <i>u</i>000, 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
<i>parameters A have the form A</i>
( ) , sin 0
2
<i>ut</i>
<i>i</i> <i>u</i> <i>t</i>
<i>i</i>
<i>f</i> <i>u</i> <i>f t e</i> <i>dt</i>
<sub></sub>
<sub></sub> <sub></sub>
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
<i>FT</i> <i>f t</i> <i>u</i> <i>f t</i> <i>e</i> <i>dt</i>
2
<i>iut</i>
<i>FT</i>
<i>f t</i> <i>f t</i> <i>u</i> <i>e du</i>
respectively. If <i><sub>f h</sub></i><sub>,</sub> <i><sub>L</sub></i>1<sub>(</sub> <sub>)</sub><i><sub>, the classic Fourier convolution operation in the time domain is defined </sub></i>
as
<i>FT</i> <i>f</i> <i>h t</i> <i>u</i> <i>FT</i> <i>f t</i> <i>u</i> <i>FT</i> <i>h t</i> <i>u</i>
. (1.9)
<i>f</i><i>L</i> , <i>h</i><i>Lq</i>
<i>p</i> <i>q</i> <i>r</i> ,
1
* <i><sub>r</sub></i> <i><sub>p</sub></i> <i><sub>q</sub></i>
<i>f</i> <i>h</i> <i>C</i> <i>f</i> <i>h</i> , (1.10)
Now we will exemplify some basic properties of <i>A</i> (see [6]).
Suppose <i><sub>f</sub></i><i><sub>L</sub></i>1
0 0
( )
2
<i>ac</i>
<i>i</i> <i>c</i> <i>u u</i> <i>a</i>
<i>A</i> <i>f t</i> <i>e</i> <i>F uA</i> <i>a</i>
.
2
<i>bd</i>
<i>i</i> <i>d</i> <i>u u</i> <i>b</i>
<i>i t</i>
<i>A</i> <i>e</i> <i>f t</i> <i>u</i> <i>e</i> <i>F uA</i> <i>b</i>
<sub></sub> <sub></sub> <sub></sub> <sub>. </sub>
<i>ac</i> <i>bd</i>
<i>i</i> <i>bc</i>
<i>i c</i> <i>d</i> <i>u u</i> <i>i a</i> <i>b</i>
<i>y</i>
<i>A</i> <i>e f t</i> <i>e</i> <i>e</i> <i>e</i> <i>F uA</i> <i>a</i> <i>b</i>
<sub> </sub> <sub> </sub>
<sub></sub><sub></sub> <sub></sub> <sub></sub> <sub></sub><sub></sub> <sub></sub> <sub>. </sub>
inverse is obviously also continuous).
Let <i>C</i><sub>0</sub>
2
<i>f</i> <i>f t dt</i>
1
1
| |
<i>Af</i> <i>f</i>
<i>b</i>
.
2
<i>L</i> and let
| |
, : ,
<i>Af u k</i>
<i>Then, as k</i> , <i><sub>A</sub>f u k converges strongly (over </i>
| |
, : , <i><sub>A</sub></i>
<i>A</i>
<i>t</i> <i>k</i>
<i>f u k</i> <i>C</i> <i>u t</i> <i>f t dt</i>
converges strongly to <i>f u , where C is the same as in (1.3). </i>
, ,
<i>Af</i> <i>Ah</i> <i>f h</i> ,
where , is denoting the usual inner product in <i>L</i>2
2 2
<i>Af</i> <i>f</i>
For convenience, we denote
2 0
2
<i>u</i>
<i>a</i>
<i>i</i> <i>t</i> <i>t</i>
<i>b</i> <i>b</i>
<i>A</i> <i>t</i> <i>e</i>
<sub></sub>
2
2
1
2
1 <i>t</i>
<i>b</i>
<i>t</i> <i>e</i>
<i>b</i>
<i>b</i> <i>du</i>
<i>d</i> <i><sub>iut</sub></i>
<i>i</i> <i>u</i> <i>u</i>
<i>b</i> <i>b</i> <i><sub>b</sub></i>
<i>A</i> <i>A</i> <i>A</i>
<i>F</i> <i>u</i> <i>f t</i> <i>u</i> <i>K e</i> <i>f t e</i> <i>dt</i>
<sub></sub>
There are many different types of convolutions for the OLCT. Most of them have the weight
functions in the form
2
<i>i</i> <i>u</i> <i>u</i>
<i>e</i> (see [1]). In [7], some convolutions for the FRFT with the Hermite
weights in the form <i>i u</i>2
<i>n</i>
<i>e</i> <i>u</i> , and the Gaussian weight in the form
2
2 1
2<i>u</i>
<i>i u</i>
<i>e</i> <i>e</i> , are also obtained. In
this paper, we focus on studying the convolution for the OLCT with the Gaussian weight in the form
2
1
2<i>u</i>
<i>e</i> , 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.
<b>2. Convolution for the OLCT with the Gaussian weight function and product theorem </b>
<b>Definition 2.1. Let </b> <i>f h</i>, <i>L</i>1
with the Gaussian weight function
<i>f</i><i>h t</i> <i>L</i> . Moreover, <sub>1</sub>
1 2 1
<i>f</i><i>h</i> <i>C</i> <i>f</i> <i>h</i> ,
where <i>C</i>2 is a positive constant.
<b>Theorem 2.1. Assume that </b> <i>f h</i>, <i>L</i>1
the OLCT of the signals <i>f t , </i>
following identity is fulfilled
4
2 2
<i>u</i>
<i>A</i> <i>A</i> <i>A</i>
<i>u</i> <i>u</i>
<i>Z</i> <i>u</i> <i>e</i> <i>F</i> <sub></sub> <sub></sub><i>H</i> <sub></sub> <sub></sub>
<i>. </i>
Moreover, if
2
1
1
4
2 2
<i>u</i>
<i>A</i> <i>A</i> <i>A</i>
<i>u</i> <i>u</i>
<i>e</i> <i>F</i> <sub></sub> <sub></sub><i>H</i> <sub></sub> <sub></sub> <i>L</i>
then
1
1
4
2 2
<i>u</i>
<i>A</i> <i>A</i>
<i>A</i>
<i>u</i> <i>u</i>
<i>z t</i> <i>e</i> <i>F</i> <i>H</i> <i>t</i>
<sub></sub> <sub></sub> <sub></sub> <sub></sub> <sub></sub><sub></sub>
<i>. </i> (2.2)
<i>Proof. Based on classic Fourier convolution (1.7), the convolution (2.1) can be expressed as </i>
( ) 2
<i>A</i> <i>A</i>
<i>f</i><i>h t</i> <i>K</i> <i>t</i>
Since (1.11), we realize that
2 2 2( )
1 1
2
2 2
<i>b</i> <i>du</i>
<i>d</i> <i>iu</i> <i>iuv</i>
<i>i</i> <i>u</i> <i>u</i>
<i>u</i> <i>u</i>
<i>b</i> <i>b</i> <i>b</i> <i>b</i>
<i>A</i> <i>A</i> <i>A</i>
<i>e</i> <i>F</i> <i>u</i> <i>H</i> <i>u</i> <i>e</i> <i>K e</i> <i>f</i> <i>h v e</i> <i>e</i> <i>d dv</i>
<sub></sub>
<sub></sub> <sub></sub>
<i>d</i> <i><sub>iu</sub></i> <i><sub>iuv</sub></i>
<i>i</i> <i>u</i> <i>u</i>
<i>t</i> <i>iut</i> <i><sub>b</sub></i> <i><sub>b</sub></i>
<i>b</i> <i>b</i>
<i>A</i>
<i>e</i> <i>dtK e</i> <i>f</i> <i>h v e</i> <i>e</i> <i>d dv</i>
<sub></sub>
2 1 ( 0 0) 12
2 <sub>(</sub> <sub>)</sub> <sub>( 2 )</sub>
2
2
<i>b</i> <i>du</i>
<i>d</i>
<i>i</i> <i>u</i> <i>v</i> <i>bt u</i> <i>u</i> <i><sub>t</sub></i>
<i>b</i> <i>b</i> <i>b</i>
<i>A</i>
<i>K</i>
<i>e</i> <i>f</i> <i>h v e</i> <i>d dvdt</i>
<i>By making s</i> <i>v</i> <i>bt</i>, we obtain
2
1
2<i>u</i>
<i>A</i> <i>A</i>
<i>e</i> <i>F</i> <i>u H</i> <i>u</i>
2 <sub>0</sub> <sub>0</sub>
1
1
2 2 2
2 2 2
2 2
<i>b</i> <i>du</i>
<i>d</i> <i>s</i> <i>A</i> <i>A</i>
<i>i</i> <i>u</i> <i>u</i> <i>u</i>
<i>b</i> <i>b</i> <i>b</i>
<i>A</i> <i>A</i>
<i>s</i>
<i>K</i>
<i>s</i>
<i>K</i> <i>e</i>
<i>b</i>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub>
<sub></sub>
1
2<i>b</i> <i>s</i> <i>v</i>
<i>f</i> <i>h v e</i> <i>d dv ds</i> <sub></sub>
1
2
2, ( ) ( ) ( )
2 2
<i>A</i> <i>A</i>
<i>A</i>
<i>s</i>
<i>K</i>
<i>s</i> <i>s</i>
<i>u</i> <i>f</i> <i>h v</i> <i>s</i> <i>v d dv d</i>
<i>b</i>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub>
<sub></sub>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub>
2 2 2
<i>A</i>
<i>s</i> <i>s</i> <i>s</i>
<i>u</i> <i>f</i> <i>h</i> <i>d</i>
<sub></sub>
<i>A</i> <i>f</i> <i>h</i> <i>u</i>
.
The proof is completed.
<b>Remark 2.1. Furthermore, using the fomula (1.8) the convolution (2.1) can also be rewritten as </b>
2 <i><sub>A</sub></i> <i><sub>A</sub></i> 2 * 2 * 2
<i>f</i><i>h t</i> <i>K</i> <i>t</i> <i>f</i> <i>t</i> <i>h</i> <i>t</i> <i>t</i> . (2.4)
<b>Remark 2.2. In particular, if we chosse </b><i>h t</i>( )( )<i>t</i> , where
* 2 2 2 * 2
<i>A</i> <i>A</i> <i>A</i> <i>A</i> <i>A</i>
<i>f</i> <i>t</i> <i>K</i> <i>t</i> <i>f</i> <i>t</i> <i>K</i> <i>t</i> <i>f</i> <i>t</i> <i>t</i> . (2.5)
<b>3. Applications </b>
<i>3.1. The Gaussian filter in the OLCT domain. The Gaussian filter is of importance in the signal </i>
The output signal <i>rout</i>
2 2 * 2
<i>out</i> <i>t</i> <i>A</i> <i>A</i> <i>t</i> <i>in</i>
<i>r</i> <i>K</i> <i>r</i> <i>t</i> <i>t</i> . (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
<i>in</i>
<i>r</i> <i>t</i> <i>y</i> <i>t</i> <i>n</i> <i>t</i> where <i>y t n t denote the desired signal and the additive noise, respectively. </i>
7 7
<i>A</i> <sub></sub> <sub></sub>
,
2 <sub>2</sub>
21
20
2<i>t</i> <sub>sin 1.5</sub> <i>i t</i>
<i>in</i>
<i>r</i> <i>t</i> <i>e</i> <i>t</i> <i>e</i> ,
2<i>t</i> <sub>sin 1.5</sub>
<i>y t</i> <i>e</i> <i>t</i> ,
2
20
<i>i t</i>
<i>n t</i> <i>e</i> <i>, and the Gaussian function </i>
2
49
2
7 <i>t</i>
<i>t</i> <i>e</i>
<i> . </i>
2 * 2
<i>it</i>
<i>out</i> <i>t</i> <i>in</i>
<i>r</i> <i>e</i> <i>r</i> <i>t</i> <i>t</i>
<i>i</i>
<b>. </b>
<i>Then the results of Gaussian filter is given in Fig. 2 </i>
Figure 1. The method to achieve Gaussian filter in the OLCT domain.
<i><b>3.2. The multiplicative filter in the OLCT domain. In this subsection, </b>rin</i>
<b>as the input signal and output signal, respectively. </b>
The output signal of OLCT can be obtained as following
1
1
4
2 2
<i>u</i>
<i>out</i> <i>t</i> <i>in</i> <i><sub>A</sub></i> <i>A</i> <i>in</i> <i>A</i>
<i>u</i> <i>u</i>
<i>r</i> <i>r</i> <i>h t</i> <i>e</i> <i>r</i> <i>t</i> <i>H</i> <i>t</i>
<sub></sub> <sub></sub>
. (3.2)
Let
2
1
2<i>u</i>
<i>A</i> <i>u</i> <i>e</i> <i>HA</i> <i>u</i>
<i>H</i> , since the OLCT-frequency spectrum is usually interested only in the
region
2
<i>out</i> <i>t</i> <i>A</i> <i>A</i> <i>in</i> <i>t</i>
<i>u</i>
<i>r</i> <i>r</i> <i>t</i> <i>u</i> <i>u</i> <i>u</i>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub> <sub></sub> <sub></sub>
.
Moreover, <i>H<sub>A</sub></i>
1
1
4
1 2
, 2, 2
2
<i>u</i>
<i>out</i> <i>t</i> <i>A</i> <i>A</i> <i>in</i> <i>t</i>
<i>u</i>
<i>r</i> <i>e</i> <i>r</i> <i>t</i> <i>u</i> <i>u</i> <i>u</i>
<sub></sub> <sub></sub>
<sub></sub> <sub></sub> <sub></sub> <sub></sub>
.
By denoting
2
1
4
2 2
<i>u</i>
<i>A</i> <i>in</i> <i>t</i> <i>A</i>
<i>u</i> <i>u</i>
<i>E u</i> <i>e</i> <i>r</i> <sub></sub> <sub></sub> <i>H</i> <sub></sub> <sub></sub>
<i>, the realization method is given by Fig.3. </i>
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]).
<b>References </b>
[1] Qiang Xiang, KaiYu Qin, Convolution, correlation, and sampling theorems for the offset linear canonical
transform, K. SIViP 8(3) (2014) 433:442.
[2] Aykut Koc, Haldun M. Ozaktas, Cagatay Candan, and M. Alper Kutay, Digital computation of linear canonical
transforms, IEEE Transactions on Signal Processing 56(6) (2008) 2383-2394.
[3] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in
[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.
[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.