Generalization of two dimensional canonical SS-transform

(1)

Scholars Research Library

Archives of Applied Science Research, 2013, 5 (2):203-207 (http://scholarsresearchlibrary.com/archive.html)

ISSN 0975-508X CODEN (USA) AASRC9

Generalization of two dimensional canonical SS-transform

S. B. Chavhan

Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded (India)

_____________________________________________________________________________________________

ABSTRACT

The canonical transform (CT) is a widely used integral transform with many applications in applied mathematics, mathematical physics and engineering sciences. The canonical transform (CT) is related to the Fourier transform (FT). The Fourier transform can be generated into the fractional Fourier transform, linear canonical transform.

This paper is devoted for analytic study of two dimensional generalized canonical SS-transform, and parsevals identity for two dimensional canonical SS-transform.

AMS Subject code: 46F12 and 44

Keywords: 2D canonical cosine-cosine transform, 2D-canonical sine-sine transform, generalized functions.

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INTRODUCTION

The Fourier transform (FT) is undoubtedly one of the most valuable and frequently used tools in signal processing and analysis [2]. The fractional Fourier transform has been found to have several applications in the areas of optics and filtering etc [4], [5]. Fractional Fourier transforms special case of linear canonical transform, numbers of integral transform are extended in its fractional domain for example, Almeida [1] had studied fractional Fourier transform, Chavhan.S.B and Borkar.V.C [3] developed two dimensional canonical cosine-cosine transform. The linear canonical transform is defined as [6]

2 ( / )2

2 2

{ ( )}( ) 1 ( ) ,

2

i d s i s t i a b t

b b

C Tf t s e e e f t dt

πib

  ∞ − 

   

   

−∞

= ⋅

∫

⋅

This paper emphasizes define two dimensional canonical SS-transform, and deriving its analyticity theorem then some properties of two dimensional of SS-transform are discussed and finally conclusion is given. Notation and terminology as per zemanian [7],[8].

2 Definition two dimensional (2D) canonical SS- transform:

{

2DCSST f t x

( )

,

} ^{( )}

s w, = f t x K

^{( )}

, , _s1

^{( ) ( )}

t x K, _s2 x w,

(2)

{ ( ) } ^{( )}

( )

² ² ² ² ² ² ²

( )

2 , ,

1 1

1 sin sin ,

2 2

i d i d i a i a

s w t x

b b b b

DCST f t x s w

s w

e e t x e e f t x dxdt

b b

ib ib

π π

 2   ∞ ∞    

       

       

−∞ −∞

   

= − ∫ ∫    

   

Where, 1

( ) ( )

² ²

2 2

, 1 sin .

2

i d i a

s t

b b

s

K t s i e st e

ib b π

   

   

     

= −  

  when b≠0 ( )²

( )

2 ic ds

d e δ t ds

= − when b = 0

( ) ( )

² ²

2

2 2

, 1 sin .

2

i d i a

w x

b b

s

K x w i e wx e

ib b π

   

   

     

= −  

 

when b≠0 ( )²

( )

2 ic dw

d e δ x dw

= − when b = 0

where ^{E k}^,

{

^K^s¹

( ) ( )

^{t s K}^, ^s² ^{x w}^,

}

^sup ^t ^{D D K}^t^k ^x^l ^s¹

^{( ) ( )}

^{t s K}^, ^s² ^{x w}^,

x

γ = −∞ < < ∞ < ∞

−∞ < < ∞

3 Analyticity Theorem of 2D Canonical SS-Transform

Theorem 3.1 :( Analyticity) Let f∈E R¹( ⁿ) and its two canonical SS- transform be defined by,

{ }

2 2 2 2

2 ( , ) ( , )

1 1

sin sin ( , )

2 2

i d i d i a i a

s w t x

b b b b

DCSST f t x s w

s w

e e e e t x f t x dxdt

ib ib b b

π π

    ∞    

       

       

−∞

   

= −    

   

∫

Then

{

²DCSS T f t x( , ) ( , )

}

s w is analytic on Cⁿ,if the a, b, sup pf ⊂s_a^and sbwhere

{

^: ⁿ^, ^, ⁰

}

sa = t t∈R t ≤a a> ^,^s^b⁼

{

^{x x}^: ^∈^Rⁿ^, ^x ^≤^{b b}^, ^>⁰

}

^moreover

{

²DCSST f t x( , ) ( , )

}

s w is differentiable and D D_s^k ^l_w{²DCSST f t x( , ) ( , )} s w = f t x D D K( , ), _s^k _w^l _s₁( , )t s K_s₂( , )x w Proof: Let, s: {s₁, s₂,………s_j……s_n}

∈

^Cⁿ^and

w: {w₁, w₂,………w_j……w_n}

∈

^Cⁿ

We first prove that, {2 ( , )}( , ) exists,

j j

DCSS T f t x s w

s w

∂ ∂

1 2

{2 ( , )}( , ) ( , ), ( , ) ( , ) (3.1)

n n n n

s s

n n n n

j j j j

DCSST f t x s w f t x K t s K x w

s w s w

∂ ∂ = ∂ ∂

∂ ∂ ∂ ∂

we prove the result n = 1, the general result following by induction.

For fixed s_j ≠ 0 choose two concentric circles C and C¹ with centre s_j and radii r and r₁ respectively such that 0<r<r1< |sj|.

Let

∆ s

_j be a complex increment satisfying

0 < ∆ s

_j

< r

. Also for fixed w_j ≠ 0.Again choose two concentric circles C and C₁ with centre wj and radiir^' and r₁^' respectively such that 0<r^' <r₁^' < |wj|.

Let ∆w_j be a complex increment satisfying 0< ∆w_j <r^'

(3)

______________________________________________________________________________

Consider,

1 2

(2 ) ( , ) (2 ) ( , ) (2 ) ( , ) (2 ) ( , )

( , ), ( , ) ( , )

( , ), ( ) ( ) (3.2)

j j j j j j j j j j

j j

s s

j j

DCSST s s w DCSST s w DCSST s w w DCSST s w

s w

f t x K t s K x w

s w

f t x s t w x

+ ∆ − + ∆ −

∆ ∆

− ∂ ∂

∂ ∂

= Ψ∆ Ψ∆

where

1 1 2 1

( ) ( ) 1 ( , , ) ( , )

j j s j j n s

j

s t w x K t s s s s s K t s

s

 

Ψ∆ ∆ =∆  LL + ∆ L − 

2 1 2 2

1 _s ( , , _j _j _n) _s ( , )

j

K x w w w w w K x w

w

 + ∆ − 

 

∆ LL L

1( , ) 2( , )

n n

s s

n n

j j

K t s K x w

s w

− ∂ ∂

∂ ∂

For any fixed

( )

^{t x}^, ^∈^Rⁿ and any fixed integer.

1 2 0

( , _n)

k= k k LLk ∈N and l=( ,l l₁ ₂LLl_n)∈N₀

1( , ) 2( , )

k l

t x s s

D D K t s K x w is analytic inside and onC^' and C^'₁. By Cauchy integral formula.

( ) ( )

1 2

' ' 1

2 2 2

2

( , )

1 1 1 1 1

, ,

4 ( )

1 1 1 1

( )

k l

t x j j

k l

t x s s

j j j j j

c c

j j j j j

D D s w t x

D D K t s K x w

s z s s z s

i z s

w y w w y w y w

π Ψ∆ ∆

   

 

= ∆  − − ∆ − − − − 

  

− −

 

∆  − − ∆ −  −



∫ ∫

dzdy

 

 



s = (s₁………s_j-1, z, s_j+1………s_n).

w = (w₁………w_j-1, y,w_j+1………w_n).

1 2

' ' 1

2 2 2

( , ) ( , )

4 ( ) ( ) ( ) ( ) ,

k l

t x s s

j j

j j j j j j

c c

D D K t s K x w s w

z s s z s y w w y w dzdy

π

=∆ ∆

−

∫ ∫

− − ∆ − − − ∆ −

But for all z∈C^'^andy∈C₁^'^and

( )

^{t x}^, restricted to a compact subset of Rⁿ,

1( , ) 2( , )

k l

t x s s

D D K t s K x w is bounded by constant Q.

' ' 1

2 ' '

1 1 1 1

( , )

4 ( ) ( )( ) ( )

j j

k l

t x j j

c c

s w Q

D D s w t x dz dy

r r r r r r

π Ψ∆ ∆ ≤ ∆ ∆

− −

∫ ∫

2 ' '

1 1 1 1

4 ( ) ( )( ) ( )

j j

s w Q

r r r r r r

π

∆ ∆

≤ − −

Thus as ∆ →s_j 0,^and ∆w_j →0, D Dt^k ^lxΨ∆ ∆sj w t xj^{( , )}tends to zero uniformly on the compact subset of R^n.,therefore it follows that Ψ∆ ∆s_j w t x_j( , ) converges in E(Rⁿ) to zero. Since f∈E¹, we conclude (3.2) tends to zero. Therefore {²DCSST f t x( , ) ( , )} s w is differentiable with respective s_jand w_j. But this is true for all i, j=1,2,……n .Hence

{ }

(4)

and, D D_s^k _w^l{2DCSST f t x( , ) ( , )} s w = f t x( , ),D D K_s^k _w^l _s1( , )t s K_s2( , )x w . 4 Parsevals Identities for Two dimensional Canonical SS-Transform.

If ^{f t x}

( )

^, ^and^{g t x}

( )

^, are inversion of two dimensional Canonical SS-Transform{²^DCSST}^{( , )}^{s w}

and {²^DCSST}^{( , )}s w respectively, then

4.1 ^{∞ ∞} f t x g t x dxdt( ) ( )^, ^, ⁼⁴π²^{∞ ∞}{²DCSST}( ){s w^, ²DCSST}( )s w dsdw^,

−∞ −∞

∫ ∫

−∞ −∞

∫ ∫

4.2 ^{∞ ∞} ^{f t x}( )^, ²^dxdt⁼⁴^π²^{∞ ∞} {²^DCSST}( )^{s w}^, ²^dsdw

−∞ −∞

∫ ∫

−∞ −∞

∫ ∫

Proof: By definition two dimensional Canonical SS-Transform,

{ ( )}( )

( ) ² ² ² ² ² ² ² ( )

2 , ,

1 1

1 sin sin , ....(4.1)

2 2

i d i d i a i a

s w t x

b b b b

DCST g t x s w

s w

e e t x e e g t x dxdt

b b

ib ib

π π

 2      

    ∞ ∞    

       

−∞ −∞

   

= − ∫ ∫    

   

Using inversion formulae of two dimensional Canonical SS-Transform, ^{g t x}

( )

^, ^is

( )

{ }( )

2 2 2 2

,

2 2

sin sin 2 , ..(4.2)

i a i a i d i d

t x s w

b b b b

g t x

i i s w

e e t x e e DCSST s w dsdw

b b b b

π π ⁻^{ }^{ }  ⁻^{ }^{ }  ^{∞ ∞} ⁻^{ }^{ }  ⁻^{ }^{ } 

−∞ −∞

   

= − ∫ ∫    

   

Taking complex conjugates we get

( )

{ }( )

2 2 2 2

,

2 2

sin sin 2 , .

i a i a i d i d

t x s w

b b b b

g t x

i i s w

e e t x e e DCSST s w dsdw

b b b b

π π ^{ }^{ }  ^{ }^{ }  ^{∞ ∞} ^{ }^{ }  ^{ }^{ } 

−∞ −∞

− −    

= − ∫ ∫    

   

( ) ( )

^, ^,

f t x g t x dxdt

∞ ∞

−∞ −∞

∫ ∫

( )^, ² ⁱ ² ⁱ ²^{i a}^b^t² ²^{i a}^b^x² ^sin ^s ^sin ^w ²^{i d}^b ^s² ²^{i d}^b^w²{² }( )^, ^.

f t x dxdt e e t x e e DCSST s w dsdw

b b b b

π π ^{ } ^{ } ^{ } ^{ }

∞ ∞     ∞ ∞    

       

−∞ −∞ −∞ −∞

 − −     

= ∫ ∫ − ∫ ∫     

   

 

 

By changing the order of the integration

( ) ( )

{ }( )

2 2 2 2 ( )

2 2 2 2

, ,

2 2

= 2 ,

1 1

2 2 sin sin ,

2 2

i d i d i a i a

s w t x

b b b b

f t x g t x dxdt

i i

DCSST s w dsdw

b b

s w

ib ib e e t x e e f t x dxdt

b b

ib ib

π π

∞ ∞

−∞ −∞

∞ ∞

−∞ −∞

    ∞ ∞    

       

       

−∞ −∞

− −

−

     

     

     

 

∫ ∫

( ) ( )^, ^, ⁼⁴ ² {² }( ){^, ² }( )^, ^...4.3

f t x g t x dxdt π DCSST s w DCSST s w dsdw

∞ ∞ ∞ ∞

−∞ −∞

∫ ∫

−∞ −∞

∫ ∫

Putting ^{f t x}

( )

^, ⁼^{g t x}

( )

^, in 4.3 we get

( )^, ² ⁼⁴ ² {² }( )^, ²

f t x dxdt π DCSST s w dsdw

∞ ∞ ∞ ∞

−∞ −∞

∫ ∫

−∞ −∞

∫ ∫

(5)

______________________________________________________________________________

5 Properties of kernel

Kernels of 2D canonical SS- transform satisfied following property (1) k_s1

( ) ( )

t s k, _s2 x w, =k_s1

( ) ( )

s t k, _s2 w x,

if a = d (2) ks₁

( ) ( )

t s k^, s₂ x w^, ≠ks₁

( ) ( )

s t k^, s₂ w x^, if a≠d

(3) k_s1

(

−t s k,

) ( )

_s2 x w, =k_s1

( ) (

t s k, _s2 −x w,

)

(4) ks₁

(

− −t^, s k

) ( )

s₂ x w^, =ks₁

( ) (

t s k^, s₂ − −x^, w

)

Above properties of kernel are simple to prove.

CONCLUSION

In this paper 2D canonical SS- transform is generalized in the distributional sense. The analyticity and parsevals identity are proved, and also properties of kernel are mentioned.

REFERENCES

[1] Almeida Lufs B., “The fractional Fourier transform and time frequency,” representation IEEE trans. On Signal Processing, Vol. 42, No. 11, Nov. 1994.

[2] Bracewell R.N., “The Hartley Transform”, Oxford U.K. Oxford Univ. Press, 1986.

[3] Chavhan.S.B. and Borkar.V.C., Two dimensional generalized canonical cosine-cosine transform IJMES. Volume 1Issue 4 April (2012).

[4] D.Mendlovic and H.M.Ozactas, Fractional Fourier transform and their optical Implementation-I J.OPT.Soc.Am.A, 10, 1875-1881(1993).

[5] H.M.Ozactas and David Mendlovic, “Fourier transforms of Fractional order. and their optical interpitation,opt.commun.,101,163-169(1993).

[6] K.B.Walf, “Integral transforms in Science and engineering,” ch.9, New York plenum press, 1979.

[7] Zemanian A.H., “Distribution theory and transform analysis,” McGraw Hill, New York, 1965.

[8] Zemanian A.H., Generalized integral transform,” Inter Science Publisher’s New York, 1968.

Generalization of two dimensional canonical SS-transform

Generalization of two dimensional canonical SS-transform

∫

{

( )

} ( )

( )

( ) ( )

{ ( ) } ( )

( )

( )

( ) ( )

( )

( ) ( )

( )

{

( ) ( )

}

( ) ( )

∫

{

}

{

}

{

}

{

}

∈

∈

∆ s

0 < ∆ s

< r

( )

∫ ∫

∫ ∫

( )

∫ ∫

( )

( )

∫ ∫

∫ ∫

∫ ∫

∫ ∫

( )

( ) ( )

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

( )

( )

∫ ∫

∫ ∫

( ) ( )

( ) ( )

( ) ( )

( ) ( )

(

) ( )

( ) (

)

(

) ( )

( ) (

)

} ^{( )}

^{( )}

^{( ) ( )}

{ ( ) } ^{( )}

^{( ) ( )}