Convolution Theorem For Canonical Sine Transform And Its Properties

Convolution Theorem For Canonical Sine Transform And Its Properties

A. S. Gudadhe # A. V. Joshi*

#Govt. Vidarbha Institute of Science and Humanities, Amravati. (M. S.)

* Shankarlal Khandelwal College, Akola – 444002 (M. S.)

Abstract: The Canonical Sine transform, has many applications in several areas, including signal processing and optics. In this paper we have introduced convolution theorem, linearity property, derivative property, modulation property and Parsevals identity for the generalized canonical sine transform.

Keywords: Fractional Fourier Transform, LCT.

Introduction: Linear canonical transform (LCT) is a four parameter linear transform which is a generalization of the fractional Fourier transform and is given by,

[LCT f (t)](s) f (t), K(t, s)

i d 2

i a 2

s

where,

1 s

2

.t i.

t .

KM (t, s)

2ib

e 2 b

• e b

e b

Hence the generalized canonical cosine transform of

f E(Rn )

c)an be defined by,

i d 2

i a 2

s

1 s

2 .t i.

t .

LCT f (t) (s) e 2 b

2 ib

e b

e b f (t) dt,

The Fractional Fourier Transform (FRFT) which is the generalization of conventional Fourier transform is a special case of LCT. Just as Fourier cosine and Fourier Sine are defined from Fourier transform similarly Canonical Cosine Transform (CCT) and Canonical Sine Transform (CST) are defined from LCT by Pei and Ding in [4]. We have already studied Operation Transform Formulae for the Generalized Half Canonical Sine Transform in [2] and analyticity theorem and operational properties of CST [3].

Convolution plays a very important role in the theory of integral transform. Almeida [1] had defined convolution for fractional Fourier transform. Zayed [5] had revised the definition in order to follow the standard Convolution theorem. This paper emphasizes on defining canonical sine transform and deriving its convolution theorem, then some properties of the canonical sine transform are discussed and finally conclusions are given.

1.1 Testing Function Space E :

An infinitely differentiable complex valued function on Rn belongs to E (Rn), if for

each compact set, I S

where S

{t :t Rn, t , 0}

sup

and for k Rn ,

E, k (t)

t I

Dk(t) .

Note that space E is complete and a Frechet space, let E denotes the dual space of E.

1. Definition: The generalized Canonical Sine Transform

   {CST f (t)} (s) = < f(t), KS(t, s) > where,

   f E(Rn ) can be defined by,

   i d 2

   i a 2

   s

   1 2 b

   2 b t

   s

   KS (t, s) (i)

   e

   e

   sin t

   2 .ib

   b

   Hence the generalized canonical sine transform of

   f E(Rn ) can be defined by,

   1

   i d s 2

   i a t 2

   s

   CST f (t)

   (s) (i)

   2ib

   e 2 b

   e 2 b

   sin

   b

   t f (t) dt

   (1.1)

2. The Generalized Canonical Sine Transform of Convolution

   Now we introduced a special type of convolution and product for canonical sine transform.

3. Definition: For any function , let us define the functions and by

   ~

   ~

   i a (v y)2

   g (| v y |) e2 b

   g(v y),

   i a (v y)2

   ~

   ~

   g (v y) e2 b

   g(v y) and

   For any two functions f and g, we define the Convolution operation by

   (1.2)

   Now we state and prove convolution theorem.

4. Convolution theorem for canonical Sine Transform:

If and denote the Canonical Sine transform of

respectively, then

Proof: From the definition of the Canonical Sine transform, we have

1 i d s2

i a ( y 2 t 2 )

s

s

e b

e 2 b

{2 sin

y sin

t } f ( y) g(t) dy dt

ib

1

0 0

i d s2

i a ( y 2 t 2 )

b

s

b

s

e b e 2 b

{cos ( y t) cos ( y t) } f ( y) g(t) dy dt

ib 0 0

b b

1 i d s2

i a ( y 2 t 2 ) s

ib

e b

e 2 b

0 0

cos

b

( y t) f ( y) g(t) dy dt

1 i d s2

i a ( y 2 t 2 ) s

ib

e b

e 2 b

0 0

cos

b

( y t)

f ( y) g(t) dy dt

For I1, putting For I2, putting

I1 I 2

y t v dt dv

y t v dt dv

for limit when t o v y , when t v

for limit when t o v y , when t v

(1.3)

1 i d s2

i a y 2

i a (v y )2

s

(1.3) [FS ( f ( y))](s)[GS (g(t))](s)

e b

e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y0 v y

b

1 i d s2

i a y 2

i a (v y )2 s

e b

e 2 b

e 2 b

cos (v) f ( y) g(v y) dy dv

ib

y0 v y

b

1 i d s 2

i a y 2

i a (v y )2

s

[FS ( f ( y))](s)[GS (g(t))](s)

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

1 i d s 2 0

i a y 2

b

i a (v y )2 s

• e b

e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v y

b

1 i d s 2 i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 0 i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v y

b

1 i d s 2

i a y 2

i a (v y )2

s

[FS ( f ( y))](s)[GS (g(t))](s)

e b

e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 y i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 0 i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

[FS ( f ( y))](s)[GS (g(t))](s) I1

ib

• I 2

• I3

  y 0 v y

• I 4

b

(1.4)

For I4, putting

v 0 v 0

v v dv dv

for limit when v y v y , when

1 i d s2 0

i a y 2

i a ( y v)2 s

I 4

e b

e 2 b

e 2 b

cos (v) f ( y) g( y v) dy (dv)

ib

y0 v y

b

1 i d s2 y i a y 2

i a ( y v)2

s

e b e 2 b

e 2 b

cos v f ( y) g( y v) dy dv

ib

y0 v0

1 i d s 2

b

i a y 2

i a (v y )2

s

(1.4) [FS ( f ( y))](s)[GS (g(t))](s)

e b

e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 y i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2 y i a y 2

i a (v y )2

s

e b e 2 b

e 2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v0

b

1 i d s 2

i a y 2

i a (v y )2

s

[FS ( f ( y))](s)[GS (g(t))](s)

e b

e2 b

e2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v 0

b

1 i d s 2 i a y 2

i a (v y )2

s

e b e2 b

e2 b

cos v f ( y) g(v y) dy dv

ib

y 0 v 0

b

1 2 i d s2

i a v2 ~

s

i a v2

e b

e 2 b

f ( y)( g~( y v) g~(| v y |))dy cos v e 2 b dv

2ib ib

i d s2 i a v2

0 0

b

2

2

2 b 2 b

i d 2

i a 2

e e

s

2 b

v

2 b

s ~ ~

(i)

e e

sin

v {( f * g )(v)}dv

(i) 2ib ib 0

i d s2 i a v2

2

b

e 2 b e 2 b

~ ~

[FS ( f ( y))](s)[GS (g(t))](s)

2ib

FS f * g (v) (s)

3. Some operational results

1. Linearity property of canonical sine transformations:

   If {CST f(t)}(s), {CST g(t)}(s) denotes generalized canonical sine transform of f(t),g(t) and P1, P2 are constants then CST (P1 f (t) P2 g(t))(s) P1 CST ( f (t)(s) P2 CST (g(t))(s)

   Proof is simple and hence omitted.

2. Modulation property of canonical sine transform:

   If {CST f(t)} (s) denotes generalized canonical sine transform of f(t) then,

   • i dbz 2

     CST cos zt. f (t)(s) e 2 CST f (t)eidsz ( s bz ) CST f (t)eidsz ( s bz )

     2 b b

     Proof: We have,

     1 i d s 2

     i a t 2

     s

     {CS T cos zt. f (t)}(s) (i)

     2ib

     e2 b

     e2 b

     .sin

     b

     t cos zt f (t) dt

     1 i d s 2

     i a t 2 2

     s

     {CS T cos zt. f (t)}(s) (i)

     e 2 b

     e 2 b sin

     t cos zt f (t) dt

     2ib

     1 1 i d s 2

     2

     i a t 2 s

     b

     s

     {CS T cos zt. f (t)}(s) (i)

     e 2 b

     e 2 b

     .sin

     • z t sin

       • z t f (t) dt

         2 2ib

         1 1

         t 2

         t 2

         i a

         e

         e

         2 b

         i d ( s bz)2

         2 b

         b

         idsz

       • i dbz 2

         b

         s

         {CS T cos zt. f (t)}(s) (i) {

         2

         2ib

         e

         e e 2

         sin

         b

     • z t. f (t) dt

       t 2

       t 2

       1

       1

       i a

       2 b

       i d ( s bz)2

       2 b

       idsz

       • i dbz 2 s

         • 2ib e

       e

       e e 2

       sin

       b

       • z t.f (t) dt}

         1 i

         dbz 2

         i a

         t 2

         t 2

         1

         1

         e

         e

         2 b

         i d ( s bz)2

         2 b

         s bz

         idsz

         {CS T cos zt. f (t)}(s) e 2

         2

         {(i)

         2ib

         e

         sin

         t. e

         b

         f (t) dt

         1 i a t 2

         i d ( s bz)2

         s bz

         (i)

         • i dbz 2

         e 2 b

         2ib

         e 2 b

         sin

         t.eidsz . f (t) dt} b

         CST cos zt. f (t)(s) e 2 CST f (t)eidsz ( s bz ) CST f (t)eidsz ( s bz )

         2 b b

   • If {CST f(t)} (s) denotes generalized canonical sine transform of f(t) then,

     • i dbz2

       CST sin zt. f (t)(s) (i) e 2 CCT f (t) eidsz ( s bz ) CCT

       f (t) eidsz ( s bz )

       Proof: We have,

       2 b b

       1 i d s 2

       i a t 2

       s

       {CS T sin zt. f (t)}(s) (i)

       2ib

       e2 b

       e2 b

       .sin

       b

       t sin zt f (t) dt

       1 1 i d s2

       i a t 2 s

       s

       {CS T sin zt. f (t)}(s)

       (i)

       e 2 b

       e 2 b

       .cos

       • z t cos

         • z t f (t) dt

           2 2ib

           b

           b

           (i)

           i a

           t 2

           t 2

           1

           1

           e

           e

           2 b

           i d ( s bz)2

           2 b

           idsz

           • i dbz 2

           s

           {CS T sin zt. f (t)}(s) {

           2

           2ib

           e

           e e 2

           cos

           b

         • z t. f (t) dt

         t 2

         t 2

         1

         1

         i a

         2 b

         i d ( s bz)2

         2 b

         idsz

         • i dbz 2

           s

           • 2ib e

         e

         e e 2

         cos

         b

       • z t.. f (t) dt}

         • i dbz 2

           i a

           i d

           (i)e 2

           1

           t 2

           ( s bz)2

           s bz

           {CS T sin zt. f (t)}(s) { 2

           e 2 b

           2ib

           e 2 b

           cos

           t.eidsz f (t) dt b

           1 i a t 2

           i d ( s bz)2

           s bz

           e 2 b

           2ib

           e 2 b

           cos

           t.eidsz . f (t) dt} b

           CST sin zt. f (t)(s) (i) e

       • i dbz2

         2

         CCT

         f (t) e idsz ( s bz ) CCT

         f (t) eidsz ( s bz )

         2 b b

   • Derivative (with respect to parameter) property of canonical sine transform:

If {CST f(t)}(s) denotes generalized canonical sine transform, then,

d {CSTf (t)}(s) is d CSTf (t)}(s) 1 {CCT[t. f (t)]}(s)

ds . b { b

Proof: We have,

d d

1 i d s 2 i a t 2

s

{CST f (t)}(s) (i)

e2 b e2 b

sin t f (t)dt

ds ds

d

2ib

1

i a t 2

b

i d s2 s

{CST

f (t)}(s) (i)

e 2 b e 2 b

.sin

t f (t)dt

ds 2ib

s

b

1 i a t 2 t

i d s 2

s

d

i d s 2

s

(i)

e 2 b

e 2 b

. cos t i

.s.e 2 b

sin t f (t)dt

2ib

b

b

b

b

1 i a t 2 1

i d s 2

s

d

1 i a t 2

i d s 2

s

(i)

e2 b

e2 b

. cos t [t. f (t)]dt s.i.

(i)

e2 b

e2 b

sin t f (t)dt

2ib

b

b

b

2ib

b

(i) 1 {CCT[t. f (t)]}(s) s.i d CSTf (t)}(s)

b . b {

d {CSTf (t)}(s) is d CSTf (t)}(s) 1 {CCT[t. f (t)]}(s)

ds . b { b

4.1 Parsevals Identity for canonical sine transform:

If f(t) and g(t) are the inversion canonical sine transform of FS(s) and GS(s) respectively, then

(1) f (t).g(t) dt 2i FS (s).GS (s) ds and (2) . f (t) dt 2i . F (s)

2

S

S

2 ds

0

Proof: By definition of CST,

1

i d s2

i a t 2

s

CST

g(t) (s) (i)

2ib

e 2 b

e 2 b

sin

b

t g(t) dt

—————- (5.1.1)

Using the inversion formula of CST

2i

i a t 2

• i d s2

s

g(t) (i)

e 2 b

b

e 2 b

sin

b

t GS (s) ds

Taking complex conjugate we get,

2i

i a t 2

i d s2

s

g(t) i

e 2 b

b

e 2 b

sin

b

t GS (s) ds

2i

i a t 2

i d s 2

s

f (t).g(t) dt

f (t)dti

e 2 b

b

e 2 b

sin

b

t GS

(s) ds

Changing the order of integration, we get,

2i

1 1

i a t 2

i d s 2

s

f (t).g(t) dt

G (s) ds (i)

e 2 b

e 2 b

sin

t f (t)dt

b

b

S

(1)

1

2ib

2ib

b

f (t).g(t) dt 2i GS (s) .FS (s)ds

f (t).g(t) dt 2i FS (s).GS (s) ds

Hence proved

——————— (5.1.2)

(ii) Putting

f (t) g(t) in equation (5.1.2), we get

f (t)

2 dt 2i F (s)

S

S

2 ds

Table for canonical sine transform

S.N.	f(t)	FS(s)
1	CST (P1 f (t) P2 g(t))(s)	P1 CST ( f (t)(s) P2 CST (g(t))(s)
2	cos zt. f (t)	i dbz 2 e 2 CST f (t) eidsz ( s bz ) CST f (t) eidsz ( s bz ) 2 b b
3	sin zt. f (t)	i dbz 2 (i) e 2 CCT f (t) eidsz ( s bz ) CCT f (t) eidsz ( s bz ) 2 b b

The generalized canonical sine transform is developed in this paper. The convolution theorem, operation transform formulae proved in this paper can be used, when this transform is used to solve ordinary or partial differential equation.

References:

1. Almeida L B (1997): Product and convolution theorems for the fractional Fourier transform. IEEE Signal Processing Letters,4(1): Page No.1517

Gudadhe A. S. and Joshi A.V. (2013): Operation Transform Formulae for the Generalized Half Canonical Sine Transform, Applied Mathematical Sciences, Volume 7, no.21, 1033-1041.

2. Gudadhe A. S. and Joshi A.V. (2012): Generalized Canonical Sine Transform,Sci. Revs. Chem. Commun. 2(3), 255-263.

3. Pie S. C., Ding J. J.( 2002): Fractional cosine, sine and Hartley transform; trans. On signal processing, Vo. 50, No. 7, P. 1661-1680.

4. Zayed A. I., (1998): A convolution and product theorem for the fractional Fourier transform, IEEE Sig. Proc. Letters, Vol. 5, No. 4, 101-103.