# Convolution for the offset linear canonical transform with gaussian weight and its application

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Convolution for the offset linear canonical transform with gaussian weight and its application. 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.
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VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 47-54
Original article
Convolution for The Offset Linear Canonical Transform
with Gaussian Weight and Its Application
Quan Thai Ha1,*, Lai Tien Minh2, Nguyen Minh Tuan3
1Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science,
334 Nguyen Trai, Hanoi, Vietnam
2Department of Mathematics, Hanoi Architectural University, Hanoi, Vietnam
3Department 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 and phrase: 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,u0,ω0 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,u0,ω0 ), (ad bc =1) is defined as
FA (u):=
f (t) A (u,t)dt, b 0
A f t u := deicd (uu0 )2 +iω0u f (d(u u0 )),
b =0
,
(1.1)
________
Corresponding author.
https//doi.org/ 10.25073/2588-1124/vnumap.4300
47
i
u tu+ t +
u+ t
2
.
1
2 2
ot
u
t
2
π
1
1
q
1
1 1
1
48
Q.T. Ha et al. / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 47-54
where
d 2 1 a 2 (bω0 du0 ) u0
A(u,t):= KAe 2b b 2b b b , and KA =
idu0
e 2b
2πbi
The inverse OLCT expression is given by
f (t) =
A1 FA (u)(t)=C
FA (u)
A1 (u,t)du ,
(1.2)
where A1 =(d,b,c,a,bω0 du0,cu0 aω0 ), and
C =ei2(cdu0 2adu0ω0 +abω0 ) .
(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 =ω0 = 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
(
f )(u)=
1icot
2π
f (t)ei c 2 u2 sin +cot t2 dt,
sin 0.
(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
FT (f (t))(u):=
f (t)eiutdt ,
(1.5)
f (t)= 2π FT (f (t))(u)eiutdu,
(1.6)
respectively. If
f ,hL(
), 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
f Lp (
), hL (
), and p + q = r +1,
(p,q,r 1). Then the following inequality holds
f *h r C
f
p h q ,
(1.10)
where C1 is a positive constant.
1
2
i
2
2 2
b
1
1
1
2
2
2
2
u
a
0
1
Q.T. Ha et al. / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 47-54
49
Now we will exemplify some basic properties of
Suppose f L ( ), and ,, we have
A (see [6]).
Time shift:
A f (t )= ei ac2 c (uu0 )aω0 FA (u a ).
Modulation:
A et f (t)(u)=eibd2 d(uu0 )b ω0 FA (u b ).
Time shift/modulation:
A ey f (t )=eiac2 +bc+bd2 ei(c+d)(uu0 )ei(a+b)ω0 FA (u a ).
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
f 1 :=
2π
f (t) dt be the norm in L (
).
(Riemann-Lebesgue type lemma for the OLCT). If
f L (
), then
A f C0 (
), and
A f
1
|b|
f 1 .
(Plancherel type theorem for the OLCT). Let
f
be a complex-valued function in the space
L (
) and let
A f (u,k):= |t|<k
A (u,t) f (t)dt .
Then, as k ,
A f (u,k) converges strongly (over
) to a function, say
A f L (
), and,
reciprocally,
f (u,k):=C |t|<k
A1 (u,t)
A 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 ,hL (
) the following identity holds
A f ,
Ah
=
f ,h ,
where
,
is denoting the usual inner product in L (
). In the special case when h = f , it holds
A f 2 =
f 2 .
For convenience, we denote = (du0 bω0 )(2
2),
A (t)= ei 2bt2 + b t ,
f (t)=
A (t) f (t), and
the Gaussian function
(t)= b 1π e2b2 (t)2 . The OLCT (1.1) becomes

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VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 47-54 Original article Convolution for The Offset Linear Canonical Transform with Gaussian Weight and Its Application Quan Thai Ha1,*, Lai Tien Minh2, Nguyen Minh Tuan3 1Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam 2Department of Mathematics, Hanoi Architectural University, Hanoi, Vietnam 3Department 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 and phrase: 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,u0,ω0 ∈ 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,u0,ω0 ), (ad −bc =1) is defined as FA (u):=  f (t) A (u,t)dt, b  0 A f t u :=  deicd (u−u0 )2 +iω0u f (d(u −u0 )), , (1.1) b =0 ________ ∗Corresponding author. E-mail address: haqt80@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4300 47 48 Q.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 47-54  d 2 1 a 2 (bω0 −du0 ) u0  where A(u,t):= KAe  2b b 2b b b  , and KA = idu0 e 2b 2πbi The inverse OLCT expression is given by f (t) = A−1 FA (u)(t)=C FA (u) A−1 (u,t)du , (1.2) where A−1 =(d,−b,−c,a,bω0 −du0,cu0 −aω0 ), and C =ei2(cdu0 −2adu0ω0 +abω0 ) . (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 =ω0 = 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 (  f )(u)= 1−icot 2π f (t)ei c 2 u2 −sin +cot t2 dt, sin  0. (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 FT (f (t))(u):=  f (t)e−iutdt , (1.5) f (t)= 2π  FT (f (t))(u)eiutdu, (1.6) respectively. If f ,h∈L( ), 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 f ∈Lp ( ), h∈L ( ), and p + q = r +1, (p,q,r 1). Then the following inequality holds f *h r C f p  h q , (1.10) where C1 is a positive constant. Q.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 47-54 49 Now we will exemplify some basic properties of A (see [6]). Suppose f ∈L ( ), and ,∈ , we have • Time shift: A f (t −)= e−i ac2 −c (u−u0 )−aω0 FA (u −a ). • Modulation: A et f (t)(u)=e−ibd2 −d(u−u0 )−b ω0 FA (u −b ). • Time shift/modulation: A ey f (t −)=e−iac2 +bc+bd2 ei(c+d)(u−u0 )ei(a+b)ω0 FA (u −a − ). • 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 f 1 := 2π  f (t) dt be the norm in L ( ). • (Riemann-Lebesgue type lemma for the OLCT). If f ∈L ( ), then A f ∈C0 ( ), and A f   |b| f 1 . • (Plancherel type theorem for the OLCT). Let f be a complex-valued function in the space L ( ) and let A f (u,k):= |t|