Generalized Two Dimensional Half Canonical Cosine Transform

Generalized Two Dimensional Half Canonical Cosine Transform

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

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

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

Abstract: This paper is devoted for the analytic study of two dimensional generalized half canonical cosine transform and some properties of two dimensional half canonical cosine transform.

Keywords: 2-D cosine-cosine transform, 2-D Canonical transforms.

Introduction: Integral transforms had provided a well establish and valuable method for solving problems in several areas of both Physics and Applied Mathematics. This method proved to be of great importance, in the initial and final value problems for partial differential equations. Due to wide spread applicability of this method for partial differential equations involving distributional boundary conditions, many of the integral transforms are extended to generalized functions.

In the past decade, FRFT has attracted much attention of the signal processing community. As the generalization of FT, the relevant theory has been developed including uncertainty principle, sampling theory, convolution theorem. However, FRFT can be further generalized to obtain the linear canonical transform LCT [3]. In fact LCT is not only the generalization of the FRFT, but also the generalization of the many other integral transforms, like Fresnel transform, Chirp transform etc. Later on numbers of integral transforms are extended in its fractional domain. For examples Almeida [2] had studied fractional Fourier transform, Akay [1] developed fractional Mellin transform, A. S. Gudadhe, A.V. Joshi [4], On Generalized Half Canonical Cosine Transform.

This paper emphasizes defining two dimensional half canonical cosine transform, and deriving its inversion theorem, then some properties of the two dimensional half canonical cosine transform are discussed and finally conclusions are given.

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}

   and for k Rn ,

   E,k (t)

   sup

   t I

   Dk(t) .

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

2. Two Dimensional Half Canonical Cosine Transform:

   2.1 Definition:

   The two dimensional generalized Half Canonical Cosine Transform f E(Rn) can be defined by, {HCCT f (t, z)} (s, u) = < f(t), K HC1 (t, s) K HC2 (z, u) > where,

   i d 2

   i a 2

   s

   2 2 b

   2 b t

   s

   KHC (t, s)

   1

   e

   .ib

   e

   cos

   b

   t ,

   when b 0

   and

   i cds2

   d .e 2

   (t ds)

   , whenb 0

   i d 2

   i a 2

   u

   2 2 b

   2 b z

   s

   KHC2

   (z,u)

   e

   .ib

   e

   cos

   b

   z ,

   when b 0

   and

   i cdu2

   d .e 2

   (z du)

   , when b 0

   Hence the two dimensional generalized half canonical cosine transform of f E(Rn) can be defined by,

   2

   2 i d s2

   i d u 2

   i a t 2

   i a z 2

   s

   u

   HCCTf (t, z)

   (s, u)

   e 2 b e 2 b e 2 b

   e 2 b

   cos t cos

   z f (t, z) dtdz

   ib

   ib 0

   b

   b

   The two dimensional generalized half canonical cosine transform.

   2.1.1 Inversion theorem for two dimensional Half Canonical Cosine Transform:

   If HCCT f (t, z)(s, u) two dimensional half canonical cosine transform of f(t ,z) is

   2 2 i d s 2

   i d u 2 i a t 2

   i a z 2

   s u

   given by, HCCTf (t, z) (s,u)

   e2 b

   e2 b e2 b e2 b

   cos t cos z f (t, z) dtdz

   ib

   ib

   0

   b b

   i i

   • i a t 2

   i a z 2

   • i d s 2

   • i d u 2

     s

     u

     then f (t, z)

     e 2 b

     e 2 b

     e

     2 b

     e 2 b

     cos

     t cos z

     HCCT

     f (t, z)(s,u) dsdu

     2b 2b 0 0

     b

     b

     Proof: The two dimensional half canonical cosine transform of f(t, z) is given by

     2

     2 i d s 2

     i d u 2 i a t 2

     i a z 2

     s u

     HCCTf (t, z) (s, u)

     e 2 b

     e 2 b e 2 b

     e 2 b

     cos t cos

     z f (t, z) dtdz

     ib

     2 2

     ib

     i d s 2

     0

     i d u 2

     i a t 2

     b

     i a z 2

     b

     s u

     FHCC (s, u)

     e 2 b e 2 b e 2 b

     e 2 b

     cos t cos z

     f (t, z) dt.dz

     ib

     ib 0 0

     b

     b

     ib

     ib

     • i d s 2

     • i d u 2

       i a t 2

       i a z 2

       s u

       FHCC (s, u)

       e 2 b e 2 b e 2 b

       e 2 b

       cos t cos z f (t, z)dtdz

       2 2 0 0

       b b

       s

       u

       B1 (s, u)

       g(t, z) cos t cos z dtdz

       b b

       0 0

       i d 2

       i d 2

       i a t 2 i a zt 2

       ib

       ib

       s

       u

       2 b 2 b

       where,

       B1 (s, u) FHCC

       (s, u)

       2

       e 2 b

       2

       s

       e 2 b

       u

       and

       g(t, z) e

       e

       f (t, z).

       B1 (s, u) g(t, z)cos( t) cos( z)dtdz g(t, z)cos(t) cos(z)

       dd .

       0 0 b b 0 0

       s

       and u d 1 ds

       and d 1 du

       where,

       b

       b b b

       By using inverse formula,

       g(t, z)

       B1

       0 0

       (s, u) cos(.t) cos(z)

       dd

       i a t 2

       i a z 2

       ib

       ib

     • i d s 2

     • i d u 2

       e 2 b

       e 2 b

       f (t, z) B1 (s, u)

       2

       2

       0 0

       e 2 b

       2

       p>e 2 b

       cos(.t) cos(z)

       d d

       • i a t 2

       i a z 2

       ib

       ib

     • i d s 2

     • i d u 2

       f (t, z) e

       2 b

       e 2 b

       FHCCC (s, u)

       2

       2

       0 0

       e 2 b

       2

       e 2 b

       cos(.t) cos(z)

       d d .

       i a t 2

       i a z 2

       ib

       ib

       i d s2

       i d u 2

       s

       u 1 1

       2 2

       HCC

       b

       b b b

       f (t, z) e

       2 b

       e 2 b

       0 0

       e 2 b

       e 2 b F

       (s, u) cos

       t cos

       z ds

       du.

       i i

       • i a t 2

       i a z 2

     • i d s 2

     • i d u 2

       s

       u

       f (t, z)

       e 2 b

       e 2 b

       e

       2 b

       e 2 b

       cos

       t cos

       z HCCT

       f (t, z)

       (s, u) dsdu

       2b 2b 0 0

       b

       b

3. Property of two dimensional half canonical cosine transform.

   1. Linearity property of two dimensional half canonical cosine transform:

      If P1 ,P2 are constants and {HCCT f1(t ,z)}(s, u), {HCCT f2(t ,z)}(s, u) denotes generalized two dimensional half canonical cosines transform of f1(t, z) , f2(t, z) respectively then

      HCCT (P1 f1 (t, z) P2 f2 (t, z))(s,u) P1 {HCCT f (t, z)}( s,u) P2 HCCT f (t, z)(s,u)

   2. Shifting property of two dimensional half canonical cosine transform:

      If {HCCT f(t,z)}(s,u) denotes generalized two dimensional half canonical cosine transform, then,

      i a 2 2

      a

      a

      p q

      2 b

      s

      u

      i b ( xp yq)

      s

      u

      i b ( xp yq)

      {HCCT [ f (t p, z q)]}(s, u) e

      [cos

      p cos

      q HCCTf (x, y)e 1

      (s, u) i sin

      p cos

      q HCSCTf (x, y)e 1

      (s, u)

      a

      b

      b

      a

      b

      b

      s

      u

      i b ( xp yq)

      s

      u

      i b ( xp yq)

      • i cos p sin

        q HCCSTf (x, y)e 1

        (s, u) sin p sin

        q HCSTf (x, y)e 1 (s, u)}]

        b

        b

        b

        b

   3. Scaling property of two dimensional half canonical cosine transform:

   If {HCCT f(t)}(s) denotes generalized two dimensional half canonical cosine transform,

   then,

   1 i d 2

   1 s

   1 s

   1 i d 2

   1 i a 2 1 i a 2

   bk

   bk

   1 t 1

   t

   {HCCT [ f (k t, k

   z)]}( s, u) 1 e

   1 u

   1 u

   k1 2 b e

   k2 2 b

   HCCT f (t, z)e k

   2 bk e k

   2 bk

   ( s , u )

   1 2

   1 2

   k1 k 2

   1 1

   2 2

   1

   bk2

   Proof: We have,

   2 2 i d s 2

   i d u 2 i a t 2

   i a z 2

   s u

   {HCCT f (k1t, k 2 z)}(s, u)

   e 2 b e 2 b e 2 b

   e 2 b

   cos t cos z f (k1t, k 2 z)dtdz

   ib

   ib 0 0

   b b

   Putting,

   k t T dt 1 dT , k z Z dz 1 dZ

   1

   1

   k

   k

   k

   k

   2

   1 2

   i d

   i d

   i a T 2

   i a Z 2

   2 2

   s 2

   u 2

   2 2

   s

   u

   dT dZ

   {HCCT

   f (k1t, k2 z)}( s, u)

   e 2 b

   e 2 b

   e 2 b k1

   e 2 b k2

   cos

   T cos

   Z f (T , Z )

   ib

   ib 0 0

   bk1

   bk2

   k1 k2

   i d 2

   u

   2 bk21

   i d 2

   i d 2 i d 2

   i d 2

   i d 2

   s

   s s u

   u

   2 2

   e 2 b

   e 2 bk1 e 2 bk1

   e 2 b

   e 2 bk21 e

   ib

   ib

   i a 2

   i a 2

   i a 2

   i a 2

   i a 2

   i a

   2

   T

   T

   T T

   Z

   Z Z 2

   2 bk

   2 bk 2 bk

   2 bk 2

   2 bk 2 bk

   s u

   dT dZ

   e 1 e

   1 e 1

   e 2 e

   2 e 2

   cos T cos Z

   f (T , Z )

   0 0

   bk1 bk2

   k1 k2

   1 i d 2

   1 s

   1 s

   1 i d 2

   1 i a 2 1 i a 2

   bk

   bk

   1 t 1

   t

   {HCCT [ f (k t, k

   z)]}( s, u) 1 e

   1 u

   1 u

   k1 2 b e

   k2 2 b

   HCCT f (t, z)e k

   2 bk e k

   2 bk

   ( s , u )

   1 2

   1 2

   k1 k 2

   1 1

   2 2

   1

   bk2

4. Parsevals Identity for two dimensional half canonical cosine transforms:

If f(t, z) and g(t, z) are the inversion canonical two dimensional half cosine transform of F2DHCC(s, u) and G2DHCC(s, u) respectively, then

(1)

2

and

f (t, z).g(t, z) dtdz

0 0

FHCC (s, u)GHCC (s, u) .dsdu

4

4

0 0

(2) f (t, z)

0 0

2

dtdz

2

4

4

FHCC (s,u)

0 0

2dsdu

Proof: By definition of two dimensional HCCT,

2

2 i d s 2

i d u 2 i a t 2

i a z 2

s u

HCCT g(t, z) (s, u)

e 2 b e 2 b e 2 b

e 2 b

cos t cos z g(t, z) dt.dz

(4.1.1)

ib

ib 0 0

b b —

Using the inversion formula

i i

• i d t 2

i d z 2

• i d s 2

• i d u 2

s

u

g(t, z)

e 2 b

e 2 b

e

2 b

e 2 b

cos

t cos

z GHCC (s, u) dsdu

2b 2b 0 0

b

b

Taking complex conjugate we get,

• i

• i

i d t 2

i d z 2 i d s2

i d u 2

s u

g(t, z)

e 2 b e 2 b e 2 b

e 2 b

cos t cos z GHCC (s, u) dsdu

2b 2b 0 0

b b

i

i

i d t 2

i d z 2

i d s 2

i d u 2

s

u

f (t, z).g(t, z) dtdz

f (t, z)dtdz e2 b

e2 b

e2 b

e2 b

cos

t cos

z G

(s,u) dsdu

2b

2b

2b

2b

0 0 0 0

0 0

b

b

HCC

Changing the order of integration, we get,

i i

1 2

2 i d t 2

i d z 2

i d s 2

i d u 2

s

u

f (t, z).g(t, z) dtdz

G (s,u) dsdu

e2 b e2 b e2 b e2 b

cos t cos

z f (t, z)dtdz

2b

2b

2b

2b

0 0 0 0

HCC

2

ib

2 ib ib

0 0

0 0

ib

b

b

2

f (t, z).g(t, z) dtdz

0 0

GHCC (s,u) .FHCC (s,u)dsdu

4

4

0 0

2

f (t, z).g(t, z) dtdz

0 0

Hence proved

FHCC (s,u)GHCC (s,u) .dsdu

4

4

0 0

———(4.1.2)

(ii) Putting

2

f (t, z) g(t, z) in equation (4.1.2), we get

2

f (t, z)

0 0

dtdz

FHCC (s, u)

4

4

0 0

2 dsdu

Table for two dimensional half canonical cosine transform

f(t, z)	FHCCT(s, u)
P1 f1 (t, z) P2 f2 (t, z)(s,u)	P1 {HCCT f (t, z)}( s,u) P2 HCCT f (t, z)(s,u)
{HCCT [ f (t p, z q)]}(s,u)	s u i a ( xp yq) s u i a ( xp yq) cos p cos q HCCTf (x, y)e b1 (s, u) i sin p cos q HCSCTf (x, y)e b1 (s, u) i a p 2 q 2 b b b b e 2 b i a ( xp yq) i a ( xp yq) s u b s u b i cos p sin q HCCSTf (x, y)e 1 (s, u) sin p sin q HCSTf (x, y)e 1 (s, u) b b b b
[ f (k1t, k2 z)]}( s,u)	1 1 s 2 1 u 2 1 t 2 1 t 2 s u 1 i d 1 i d 1 i a 1 i a k 2 b k 2 b k 2 bk k 2 bk e 1 e 2 HCCT f (t, z)e 1 1 e 2 2 ( , ) k1k2 bk1 bk2

Conclusion: In this paper, brief introduction of the generalized two dimensional half canonical cosine transform is given and Inversion theorem for two dimensional half canonical cosine transform proved. Properties of two dimensional half canonical cosine transform are also

obtained which will be useful in solving differential equations occurring in signal processing and many other branches of engineering.

References:

1. Akay O. and Bertels, (1998): Fractional Mellin Transformation: An extension of fractional frequency concept for scale, 8th IEEE, Dig. Sign. Proc. Workshop, Bryce Canyan, Utah.

2. Almeida, L.B., (1994): The fractional Fourier Transform and time- frequency representations, IEEE. Trans. on Sign. Proc., Vol. 42, No.11, 3084-3091.

3. Bhosale B.N., Choudhary M.S. (2002): Fractional Fourier transform of distributions of compact support, Bull. Cal. Math. Soc., Vol. 94, No.5, 349-358.

4. Gudadhe A. S., Joshi A.V.(2013): On Generalized Half Canonical Cosine Transform, IOSR- JM Volume X, Issue X.