A017001005

RESEARCH INVENTY: International Journal of Engineering and Science ISSN: 2278-4721, Vol. 1, Issue 7 (November 2012), PP 01-05 www.researchinventy.com

Operational Calculus of Canonical Cosine Transform 1,

S. B. Chavhan, 2, V.C. Borkar

Depart ment of Mathemat ics & Statistics, Yeshwant Mahavidyalaya, Nanded-431602 (India)

Abstract: This paper studies different properties of canonical cosine transform. Modulation heorem is also proved. We have proved some important results about the Kernel of canonical cosine transform.

Keywords: Generalized function, Canonical transform, Canonical Cosine transforms, Modulation theorem, Fourier transform.

I.

Introduction

The Fourier transform is certainly one of the best known of integral transforms, since its introduction

by Fourier in early 1800's.

A much mo re general integral transform, namely, linear canonical transform was introduced in 1970 by Moshinsky and Quesne [5]. In 1980, Namias [6] introduced the fractional Fourier transform using eigen value and eigen functions.

Almeida [1], [2] had introduced it and proved many of its properties. Zayed [9] had discussed about product and convolution concerning that transform. Bhosale and Choudhary [3] had studies it as a tempered distribution.Nu mber of applications of fractional Fourier transform in signal processing,image processing filtering optics, etc are studied. Pei and ding [7] ,[8] had studied linear canonical transforms.

The defin ition canonical cosine transform is as follo ws,

CCT f t  s  

id 

i a

 s   s   t e 2  b   cos  t  e 2  b  f  t  dt  2 ib b 

1

i

 d .e 2

2

cds 2

2

for b  0

for b  0

f (d .s)

Notation and terminology of this paper is as per Zemanian [10], [11].The paper is organized as follows . Section 2 gives the definit ion of canonical cosine transform on the space of generalized function and states the property of the kernel of the canonical cosine transform. In section 3 modulat ion theorem is proved. In section 4, t ime reverse property, linear p roperty, parity are proved. In section 5, operation transforms in terms of parameter are discussed and lastly the conclusion is stated.

II.

Generalized Canonical Cosine Transform

2.1 Definiti on : The canonical cosine transform of generalized function is defined as

CCT f t  s    f t  , K Where kernel K a ,b,c , d   t , s  

id 

i a

a ,b, c , d 

t , s  

 s  t s  e 2  b  e 2  b  cos  t  2 ib b 

1

2

2

……..(1) n K t , s  E K t , s  E R Clearly Kernel  a,b,c, d    and  a ,b,c , d      the kernel (1) Satisfies the follo wing properties.

2.2 2.3

Properties of Kernel : K  a , b , c , d   t , s   K  a , b , c , d   s, t 

if

2.4

K  a , b , c , d   t , s   K  a , b , c , d   s, t 

if a = d

2.5

K a , b , c , d   t , s   K  a ,  b ,  c , d   t , s 

2.6

K a,b,c ,d   t , s   K a ,b,c ,d   t , s 

ad

1

Operational Calculus of Canonical Cosine Transform 2.7

Definit ion of canonical sine transforms as ,

CST f t   s   i

id 

i a

 s   s   t e 2  b   sin  t  e 2  b  f  t  dt  2 ib b 

1

i

2

2

for b  0 ……(2)

cds 2

 d .e 2 f (d .s) for b  0 As we have obtained few properties of canonical sine transform in [4], we establish similar propert ies of canonical cosine transform here.

III.

Modulation Theorem for Canonical Cosine Transform

  then

Theorem 3.1: If CCT f  t   s  is canonical cosine transform of f(t), f  t   E1 R1 i

CCT cos ut  f t  s  

e2

 du b 2

2

CCT f  t   s  bu  eidus  CCT f  t   s  bu  eidus 

…….(3)

Proof: Using definition of canonical cosine transform

CCT f  t   s  

1

id  e   s2  2 b 

2 ib

CCT cos ut  f t  s   





1 2 ib 1 2 ib

id 

2

i a

 s   s   t e 2  b   cos  t  e 2  b  cos ut  f t  dt  2 ib b 

1

2

2

id  2  s

2  t 1 s  2 b  f  t  dt  2cos  t  cos  ut  e 2  b 

id  2  s 2 b 

2 1  s  s   2  b t f  t  dt   cos   u   cos   u   e 2   b  b 

e2 b 

e

i a

 s  2  b t f  t  dt  cos  t  e  b 

i a

i a

id 2 id 2 i a 2    s  ub   2i  ba t 2  s   s  1 1 1  s  ub  2  b t e 2  b   cos  f  t  dt  e 2  b   cos  t e f t   t e  2  2 ib   2 ib  b    b 

id  i a 2 2 i 2     s  ub   i  dus    du b   1  1  s  ub  2  b t d 2 b  e e 2 f  t  dt   cos   t e 2   2 ib   b    id  i a 2 2 i 2  1      s  ub  iuds   du b    s  ub  2  b t  e2 b  e e 2 t e f  t  dt    cos     b   2 ib  



i



e2

 du b 2

2

CCT f  t   s  bu  eidus  CCT f  t   s  bu  eidus 

 

Theorem 3.2: If CCT f  t   s  is canonical cosine transform of f  t  , f  t  E1 R1 Then CCT sin ut f  t   s  

i 2i  du2b CST f t   s  ub  eidsu  CST f t   s  ub  e idsu  e 2 ……(4)

Proof : By using defin ition canonical cosine transform

2

Operational Calculus of Canonical Cosine Transform id 

1



2 ib 1



2 ib

e

e

2

id 

2

i a

 s   s   t e 2  b   cos  t  .e 2  b  .sin ut f  t  dt  2 ib b 

CCT sin ut f t  s   1

i a

 s   S   t e 2  b   cos  t  e 2  b  f  t  dt  2 ib b 

CCT f  t   s  

1

2

2

id  2  t

id 2  s 2 b 

1   2 b  s  f  t  dt   2cos  t  sin ut  e 2   b  

id  2  s 2 b 

1  s  s   2 b  f  t  dt   sin  t  ut   sin  t  ut   e 2    b  b 

i a 2  t

i a 2   id 2 id 2 id 2  t  s   s  1 1 1  s  ub  2  b   s  ub  2  b t 2 b  2 b    e sin t e f t dt  e sin t e f t dt            2  2 ib   2 ib  b   b   



ie





i du 2 b 2

2



CST f  t   s  bu  eidsu  CST f  t   s  ub  eidus 

 

Theorem 3.3: If CCT f  t   s  is canonical cosine transform o f f  t  , f  t   E1 R1 Then

CCT e f t   s  iut





i i  du 2b  idsu    idus   e2 e CCT f  t   s  ub   CST f  t   s  ub    e CCT f  t   s  ub   CST f  t   s  ub   2

IV.

….. (5)

Some useful Properties:

In this section we discuss some useful results.

 

4.1 Ti me Reverse Property: If CCT f  t  is canonical cosine transforms of f  t  , f  t   E1 R1 then

CCT f  t  s   CCT f t  s 

……….. (6)

Proof : Using definit ion of canonical cosine transforms of f  t 

1

CCT f  t  s   Put t  x

id 

id 

 s   s   t e 2  b   cos  t  e 2  b  f  t  dt  2 ib b 

CCT f  t   s  

2

id 

2

id 

 s   s   t e 2  b   cos  t  e 2  b  f  t  dt  2 ib b 

1

2

2

 t  x

Hence dt  dx, also t   to , x  ,  

 CCT f  t   s  

id  2 id  2  s   s     x  e 2  b   cos   x  e 2  b  f  x  dx   2 ib  b 

1

3

Operational Calculus of Canonical Cosine Transform

CCT f  t   s   

id 

id 

  s   s   x e 2  b   cos  x  e 2  b  f  x  .dx  2 ib b 

1

2

2

Replacing x by t again id 

id 

 s   s   t e 2  b   cos  t  e 2  b  f  t  dt  2 ib b 

1



2

2

CCT f  t   s   CCT f t   s 

 

4.2 Parity: If CCT f  t   s  is canonical cosine transform of f  t   E1 R1 then

CCT f  t  s  CCT f t  s 

…….. (7)

4.3 Linearity Property: If C1 , C2 are constant and f1 , f 2 are functions of t then

CCT C f t   C 1 1

V.



f t   s   C1 CCT f1 t   s   C2 CCT f 2 t   s  …… (8)

2 2

Operation Transform in Terms of Parameters

This section presents the operational formu lae fo r canonical cosine transform with shifted parameter „s „ and differential of canonical cosine transform with respect to s.

Theorem 5.1 : If CCT f  t   s  is canonical cosine transform o f f  t  in terms of parameter s, then



id 2   2 s  2 b 

CCT f t  s     e

CCT f  t   s     2 ib



1 2 ib

e

2 ib id 2  s 2 b 



e

id 2   2 s 

 e2 b 

2 ib

e2 b 

e2 b 





 



id 2   2 s  2 b 

 cos





e2 b 

id 2 2   s  2 s 

1



id  2    S   

1

id  2 2   s  2 s 

1

 1        CCT cos  t  f  t   s   CST sin  t  f t   s  i b  b     ………… (9) 

CCT f  t   s  

Proof: We know that‟



 

1 2 ib

s   b

id  2  S 

e2 b 

i a

 s  2  b t f  t  dt  cos  t  e  b  2

i a 2  t

t  e 2  b  f  t  dt id 

2 s     s      2  b t cos t cos t  sin t sin t e f  t  dt            b  b   b   b 

id 2 i a 2   s      t  s      t   cos  t  cos  t  e 2  b  f  t  dt   sin  t  sin  t  e 2  b  f  t  dt   b  b  b   b    

 

i a 2 i a 2   s     2  b t  s     2  b t f  t  dt   sin  t  sin  t  e f  t  dt    cos  t  .cos  t  .e  b  b  b   b    

 

 1       CCT cos  t  f  t   s   CST sin  t  f  t  s  i b  b    

Theorem 5.2: If CCT f  t   s  is canonical cosine transform of f(t ) in terms of parameter S

4

Operational Calculus of Canonical Cosine Transform d    ds  t CCT f  t    s   i   CCT f  t   s   i CST   f  t   s  ds b  b  

Then

 ds   d   t  CCT f  t     s   i  b  CCT f  t   s   CST  b  f  t    s   ds        

Or

……….. (10) ………… (11)

Proof: Using definition of canonical cosine transform

CCT f t  s   

1 2 ib

e

id  2  s  2 b 

i a

 s  2  b t f  t  dt  cos  t  e  b  2

id  2 i a 2   s  d d  1  s  2  b t 2 b  CCT f t s  e cos t e f  t  dt           ds ds  2 ib  b  



i a 2    2i  db  s2  t   s  2 b   d   cos  t   f  t  dt  e  e ds b 2 ib         



i a 2 id  2 i a 2  t   s  s   ds   s  2  b  s  t   2 b  2 b  e e cos t i  sin t e            f  t  dt 2 ib  b   b  b   b   

1

1





id 

id

id

i a

2 2 2 2  s   s  i 1  ds  1  s   t  s   t  t   i  e 2  b   cos  t  e 2  b  f  t  dt  e 2  b   sin  t  e 2  b  .  f t  dt i 2 ib    b  2 ib b  b  b

1   ds  t   i  .CCT f  t   s   CST   f  t    s  i  b  b 

  ds t   CST   f  t    s   CCT f  t   s  b b   

d ds 1 t  CCT f  t   s   i  b  .CCT f  t   s   i CST  b  f  t   s  ds      

d  t  ds CCT f  t   s   CST f  t  .  b   s    d  CCT f  t   s  ds     

VI.

Conclusion:

The canonical cosine transform, which is the generalization of number of transforms is itself generalized to the spaces of generalized functions as per Zemanian. This transform is used to solve ordinary or partial differential equations. This transform is an important tool in signal processing and many other branches of engineering. In this paper we have discussed some of its properties.

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