MODULATION AND PARSEVALS RELATION OF GENERALIZED TWO DIMENSIONAL FRACTIONAL COSINE TRANSFORM

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Available Online at www.ijpret.com 110

INTERNATIONAL JOURNAL OF PURE AND

APPLIED RESEARCH IN ENGINEERING AND

TECHNOLOGY

A PATH FOR HORIZING YOUR INNOVATIVE WORK

MODULATION AND PARSEVAL’S RELATION OF GENERALIZED TWO

DIMENSIONAL FRACTIONAL COSINE TRANSFORM

V. D. SHARMA1_{, S. A. KHAPRE}2

1. Mathematics Department, Arts, Commerce and Science College, Amravati- 444606(M.S), India.

2. Department of Mathematics P. R. Patil College of Engineering and Technology, Amravati 444604 (M.S.), India.

Accepted Date: 05/03/2015; Published Date: 01/05/2015

\

Abstract: Transforms with cosine and sine functions as the transform kernels represent an

important area of analysis. It is based on the so-called half-range expansion of a function over a set of cosine or sine basis functions. Because the cosine and the sine kernels lack the nice properties of an exponential kernel, many of the transform properties are less elegant and more involved than the corresponding ones for the Fourier transform kernel. As the sine transform, cosine transform and Hartley transform are widely use in signal processing, the application of their fractional version in signal/image processing is very promising. This paper concerned with generalized two dimensional fractional Cosine transforms and here we discuss Modulation theorem, Parseval’s identity and shifting property for generalized two dimensional fractional Cosine transform.

Keywords: Fractional Fourier Transform, Fractional Cosine Transform, Fractional Sine

Transform.

Corresponding Author: MR.V. D. SHARMA

Access Online On:

www.ijpret.com

How to Cite This Article:

V. D. Sharma, IJPRET, 2015; Volume 3 (9): 110-125

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INTRODUCTION

In recent years, fractional order signal processing has received great attentions in many engineering applications. The research topics include fractional Fourier transform, fractional stochastic processes and fractional calculus [1]-[4].In the research area of fractional calculus, the integer order n of derivative 𝐷𝑛𝑥(𝑡) of function x(t ) is generalized to fractional order 𝐷𝜗x(t ) , where 𝜗 is a real number. So far, fractional calculus have extensively used in the engineering applications including electromagnetic theory, automatic control, system identification, and biomedical applications [3]-[4]. Some typical applications of fractional calculus to digital signal processing are also described below: Firstly, fractional differential operator has been used to enhance the performance of linear prediction of speech signal [5]. Second is the signature verification where fractional differential operator is applied to extract the dynamic feature from the handwritten signature [6]. Third is the one-dimensional (1-D) and two-dimensional (2-D) linear-phase filtering designs [7] [8]. The fractional signal processing leads to a more general formulation of the problems that were solved by the integral transforms in the early days, because of the additional degree of freedom available with the designers. This is turn allows better performance or greater generality based on possibility of optimizations over a fractional variable parameter a. The FrFT found several applications in signal processing. One dimensional fractional signal processing can be further extended to two dimensional and multi dimensional fractional signal processing. Their application includes image compression, image encryption, and beam forming, digital watermarking, tomography, image restoration etc. The properties and implementation of two dimensional FrFT has been reported by Sahin et al[11] and Edren et al have generalized the two dimensional FT into the two dimensional separable FrFT . In another reported work Sahin et al also generalize it into the two dimensional non separable FrFt with four parameters. Pei and Ding have introduced the two dimensional affine generalized FrFT and Pei Soo-Chang redefined the fractional cosine and sine transform based on fractional Fourier transform.

Fractional Cosine Transform (FRCT) is a generalization of the ordinary cosine transform and it has similar relationship with Fractional Fourier Transform (FRFT) as the ordinary cosine and sine transforms have with the Fourier Transform (FT) [9]. Fractional domain is useful for solving some problems, which cannot be solved in the original domain [10].

Motivated by the above, we have extended generalized two dimensional fractional cosine transform and tried to give some properties.

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generalized two dimensional fractional cosine transform are proved like Parseval’s identity in section 3 and shifting property in section.

2 Modulation Property.

2.1. Theorem-1 If 𝐹_𝑐𝛼_{(𝑓(𝑥, 𝑦))(𝑢, 𝑣)}_{is generalized two dimensional fractional cosine transform}

of 𝑓(𝑥, 𝑦) then

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦}(𝑢, 𝑣)

=𝑒

𝑖

4((𝑐𝑠𝑐2𝜃 .𝑃𝑅+𝑎2)+(𝑐𝑠𝑐2𝜃𝑄𝑆+𝑏2))𝑠𝑖𝑛2𝛼

4

[

𝑒−𝑖2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑃, 𝑄)}

+ 𝑒−𝑖2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑃, 𝑆)}

+𝑒−𝑖2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑅, 𝑄)}

+𝑒−𝑖2(𝑅2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑅, 𝑆) ]}

Solution:

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦}(𝑢, 𝑣) = √

1 − 𝑖𝑐𝑜𝑡𝛼

2𝜋 𝑒

𝑖(𝑢2+𝑣2)𝑐𝑜𝑡𝛼 2

∫ ∫ 𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦

∞

0

𝑒𝑖(𝑥

2_+𝑦2_{)𝑐𝑜𝑡𝛼}

2 _{cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) .}

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

∞

0

Let 𝐴 = √1−𝑖𝑐𝑜𝑡𝛼

2𝜋 , 𝐵 = 𝑒

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦}(𝑢, 𝑣) = 𝐴𝐵 ∫ ∫ 𝑒

𝑖𝑥2_{𝑐𝑜𝑡𝛼}

2 _{𝑐𝑜𝑠𝑎𝑥. cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥)}

∞

0 ∞

0

𝑒𝑖𝑦

2_{𝑐𝑜𝑡𝛼}

2 _{. 𝑐𝑜𝑠𝑏𝑦cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦}

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= 𝐴𝐵 ∫ ∫ 𝑒𝑖𝑥

2 (𝑒

𝑖𝑐𝑠𝑐𝛼𝑢𝑥_+𝑒−𝑖𝑐𝑠𝑐𝛼𝑢𝑥₎

2

∞

0 ∞

0

(𝑒𝑖𝑎𝑥+𝑒−𝑖𝑎𝑥)

2

𝑒𝑖𝑦

2 (𝑒

𝑖𝑐𝑠𝑐𝛼𝑣𝑦_+𝑒−𝑖𝑐𝑠𝑐𝛼𝑣𝑦₎

2

(𝑒𝑖𝑏𝑦_+𝑒−𝑖𝑏𝑦₎

2 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑌

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦}(𝑢, 𝑣)

=𝐴𝐵

16∫ ∫ 𝑒

𝑖𝑥2𝑐𝑜𝑡𝛼

2 _{( 𝑒}𝑖(𝑐𝑠𝑐𝛼𝑢+𝑎)𝑥+ 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑢+𝑎)𝑥

+𝑒𝑖(𝑐𝑠𝑐𝛼𝑢−𝑎)𝑥_{+ 𝑒}−𝑖(𝑐𝑠𝑐𝛼𝑢−𝑎)𝑥)

∞

0 ∞

0

𝑒𝑖𝑦

2 _{( 𝑒}𝑖(𝑐𝑠𝑐𝛼𝑣+𝑏)𝑦+ 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑣+𝑏)𝑦

+𝑒𝑖(𝑐𝑠𝑐𝛼𝑣−𝑏)𝑦_{+ 𝑒}−𝑖(𝑐𝑠𝑐𝛼𝑣−𝑏)𝑦)

𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

𝐹_𝑐𝛼_{{𝑓(𝑥, 𝑦)𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑏𝑦}(𝑢, 𝑣)}

=𝐴𝐵

16∫ ∫ 𝑒

2 ₍2cos ((𝑐𝑠𝑐𝛼𝑢 + 𝑎)𝑥

+2𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑢 − 𝑎)𝑥)

∞

0 ∞

0

𝑒𝑖𝑦

2 ₍2cos ((𝑐𝑠𝑐𝛼𝑣 + 𝑏)𝑦

+2𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑣 − 𝑏)𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

=𝐴𝐵

4 ∫ ∫ 𝑒

𝑖(𝑥2+𝑦2)𝑐𝑜𝑡𝛼 2 ∞

0 ∞

0

[

cos ((𝑐𝑠𝑐𝛼𝑢 + 𝑎)𝑥. cos ((𝑐𝑠𝑐𝛼𝑣 + 𝑏)𝑦 +cos ((𝑐𝑠𝑐𝛼𝑢 + 𝑎)𝑥. 𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑣 − 𝑏)𝑦 +𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑢 − 𝑎)𝑥. cos ((𝑐𝑠𝑐𝛼𝑣 + 𝑏)𝑦 +𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑢 − 𝑎)𝑥. 𝑐𝑜𝑠(𝑐𝑠𝑐𝛼𝑣 − 𝑏)𝑦 ]

𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦

Let (𝑐𝑠𝑐𝛼𝑢 + 𝑎) = 𝑐𝑠𝑐𝜃. 𝑃 , (𝑐𝑠𝑐𝛼𝑢 − 𝑎) = 𝑐𝑠𝑐𝜃. 𝑅 ,

(𝑐𝑠𝑐𝛼𝑣 + 𝑏) = 𝑐𝑠𝑐𝜃. 𝑄 , (𝑐𝑠𝑐𝛼𝑣 − 𝑏) = 𝑐𝑠𝑐𝜃. 𝑆

And (𝑐𝑠𝑐𝛼𝑢 + 𝑎). (𝑐𝑠𝑐𝛼𝑢 − 𝑎) = 𝑐𝑠𝑐𝜃. 𝑃 𝑐𝑠𝑐𝜃. 𝑅

, 𝑐𝑠𝑐2𝛼. 𝑢2− 𝑎2 = 𝑐𝑠𝑐2𝜃 . 𝑃𝑅

𝑐𝑠𝑐2_{𝛼. 𝑢}2 _{= 𝑐𝑠𝑐}2_{𝜃 . 𝑃𝑅 + 𝑎}2

𝑢2 = 𝑠𝑖𝑛2𝛼(𝑐𝑠𝑐2𝜃 . 𝑃𝑅 + 𝑎2) 𝑣2 = 𝑠𝑖𝑛2𝛼(𝑐𝑠𝑐2𝜃 . 𝑄𝑆 + 𝑏2)

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=𝐴𝐵

4 ∫ ∫ 𝑒

𝑖(𝑥2+𝑦2)𝑐𝑜𝑡𝛼 2 ∞

0 ∞

0

[

cos (𝑐𝑠𝑐𝜃. 𝑃𝑥). cos (𝑐𝑠𝑐𝜃. 𝑄 𝑦) +cos (𝑐𝑠𝑐𝜃. 𝑃𝑥). cos (𝑐𝑠𝑐𝜃. 𝑆𝑦) +cos (𝑐𝑠𝑐𝜃. 𝑅𝑥). cos (𝑐𝑠𝑐𝜃. 𝑄)𝑦 +cos (𝑐𝑠𝑐𝜃. 𝑅𝑥). cos (𝑐𝑠𝑐𝜃. 𝑆𝑦) ]

=𝑒

𝑖

2(𝑢2+𝑣2)𝑐𝑜𝑡𝛼

4

{

∫ ∫ 𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃

cos(𝑐𝑠𝑐𝜃. 𝑃𝑥) . cos(𝑐𝑠𝑐𝜃. 𝑄𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦

+ ∫ ∫ 𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃

cos(𝑐𝑠𝑐𝜃. 𝑃𝑥) . cos(𝑐𝑠𝑐𝜃. 𝑆𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑄2+𝑅2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃

cos(𝑐𝑠𝑐𝜃. 𝑅𝑥) . cos(𝑐𝑠𝑐𝜃. 𝑄𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑅2+𝑆2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑅2+𝑆2)𝑐𝑜𝑡𝜃

cos(𝑐𝑠𝑐𝜃. 𝑅𝑥) . cos(𝑐𝑠𝑐𝜃. 𝑆𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦 }

=𝑒

𝑖

4((𝑐𝑠𝑐2𝜃 .𝑃𝑅+𝑎2)+(𝑐𝑠𝑐2𝜃𝑄𝑆+𝑏2))𝑠𝑖𝑛2𝛼

4

{ 𝑒

−𝑖

2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

+𝑒−𝑖2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑃, 𝑆)}

+ 𝑒−𝑖2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

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2.2 Theorem-2. If 𝐹_𝑐𝛼(𝑓(𝑥, 𝑦))(𝑢, 𝑣) is generalized two dimensional fractional cosine transform

of 𝑓(𝑥, 𝑦) then

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣) =

𝑒 𝑖

2(𝑢2+𝑣2)𝑐𝑜𝑡𝛼𝑒−𝑖(𝜃− 𝜋 2) 4

{ 𝑒

−𝑖

2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝐹_𝑠𝜃_{𝑒 𝑖

− 𝑒−𝑖2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑠𝜃_{𝑒 𝑖

− 𝑒−𝑖2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃_𝐹_𝑠𝜃_{𝑒 𝑖

+𝑒−𝑖2(𝑅2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑠𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑅, 𝑆) }}

Solution:

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣) = √

2𝜋 𝑒

∫ ∫ 𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦

∞

0

𝑒𝑖(𝑥

2_+𝑦2_{)𝑐𝑜𝑡𝛼}

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

∞

0

Let 𝐴 = √1−𝑖𝑐𝑜𝑡𝛼

2𝜋 , 𝐵 = 𝑒

𝐹_𝑐𝛼_{{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣) =}

𝐴𝐵 ∫ ∫ 𝑒𝑖𝑥

2 _{𝑠𝑖𝑛𝑎𝑥. cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥)}

∞

0 ∞

0

𝑒𝑖𝑦

2 _{. 𝑠𝑖𝑛𝑏𝑦 cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦}

𝐹𝑐𝛼{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣) =

𝐴𝐵 ∫ ∫ 𝑒𝑖𝑥

2 (𝑒

𝑖𝑐𝑠𝑐𝛼𝑢𝑥_+𝑒−𝑖𝑐𝑠𝑐𝛼𝑢𝑥₎

2

∞

0 ∞

0

(𝑒𝑖𝑎𝑥−𝑒−𝑖𝑎𝑥)

2𝑖

𝑒𝑖𝑦

2 (𝑒

𝑖𝑐𝑠𝑐𝛼𝑣𝑦_+𝑒−𝑖𝑐𝑠𝑐𝛼𝑣𝑦₎

2

(𝑒𝑖𝑏𝑦−𝑒−𝑖𝑏𝑦)

(7)

𝐴𝐵

−16∫ ∫ 𝑒

2 _{( 𝑒}𝑖(𝑐𝑠𝑐𝛼𝑢+𝑎)𝑥− 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑢+𝑎)𝑥

−𝑒𝑖(𝑐𝑠𝑐𝛼𝑢−𝑎)𝑥+ 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑢−𝑎)𝑥)

∞

0 ∞

0

𝑒𝑖𝑦

2 _{( 𝑒}𝑖(𝑐𝑠𝑐𝛼𝑣+𝑏)𝑦− 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑣+𝑏)𝑦

−𝑒𝑖(𝑐𝑠𝑐𝛼𝑣−𝑏)𝑦+ 𝑒−𝑖(𝑐𝑠𝑐𝛼𝑣−𝑏)𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

𝐴𝐵

−16∫ ∫ 𝑒

2 ₍2isin ((𝑐𝑠𝑐𝛼𝑢 + 𝑎)𝑥

−2𝑖𝑠𝑖𝑛(𝑐𝑠𝑐𝛼𝑢 − 𝑎)𝑥)

∞

0 ∞

0

𝑒𝑖𝑦

2 ₍ 2isin ((𝑐𝑠𝑐𝛼𝑣 + 𝑏)

−2𝑠𝑖𝑛(𝑐𝑠𝑐𝛼𝑣 − 𝑏)𝑦) 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦

Let (𝑐𝑠𝑐𝛼𝑢 + 𝑎) = 𝑐𝑠𝑐𝜃. 𝑃 , (𝑐𝑠𝑐𝛼𝑢 − 𝑎) = 𝑐𝑠𝑐𝜃. 𝑅

, (𝑐𝑠𝑐𝛼𝑣 + 𝑏) = 𝑐𝑠𝑐𝜃. 𝑄 , (𝑐𝑠𝑐𝛼𝑣 − 𝑏) = 𝑐𝑠𝑐𝜃. 𝑆

And (𝑐𝑠𝑐𝛼𝑢 + 𝑎). (𝑐𝑠𝑐𝛼𝑢 − 𝑎) = 𝑐𝑠𝑐𝜃. 𝑃 𝑐𝑠𝑐𝜃. 𝑅

𝑐𝑠𝑐2𝛼. 𝑢2− 𝑎2 = 𝑐𝑠𝑐2𝜃 . 𝑃𝑅

𝑐𝑠𝑐2𝛼. 𝑢2 = 𝑐𝑠𝑐2𝜃 . 𝑃𝑅 + 𝑎2

𝑢2 _{= 𝑠𝑖𝑛}2_{𝛼(𝑐𝑠𝑐}2_{𝜃 . 𝑃𝑅 + 𝑎}2₎

similarly 𝑣2 _{= 𝑠𝑖𝑛}2_{𝛼(𝑐𝑠𝑐}2_{𝜃 . 𝑄𝑆 + 𝑏}2₎

𝐹_𝑐𝛼_{{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣)}

=𝐴𝐵

4 ∫ ∫ 𝑒

𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃) ∞

0

𝑒2𝑖(𝑥2+𝑦2)𝑐𝑜𝑡𝜃 ∞

0

[

sin(𝑐𝑠𝑐𝜃. 𝑃𝑥) . sin(𝑐𝑠𝑐𝜃. 𝑄) 𝑦 − sin(𝑐𝑠𝑐𝜃. 𝑃) 𝑥. 𝑠𝑖𝑛(𝑐𝑠𝑐𝜃. 𝑆)𝑦

−𝑠𝑖𝑛(𝑐𝑠𝑐𝜃. 𝑅)𝑥. sin (𝑐𝑠𝑐𝜃. 𝑄)𝑦 +𝑠𝑖𝑛(𝑐𝑠𝑐𝜃. 𝑅)𝑥. 𝑠𝑖𝑛(𝑐𝑠𝑐𝜃. 𝑆)𝑦 ]

𝐹_𝑐𝛼{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣) = 𝑒

𝑖

2(𝑢2+𝑣2)𝑐𝑜𝑡𝛼_𝑒−𝑖(𝜃− 𝜋 2)

(8)

Available Online at www.ijpret.com 117 [

∫ ∫ 𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃

𝑒𝑖(𝜃−𝜋2)_{sin(𝑐𝑠𝑐𝜃. 𝑃𝑥) . sin(𝑐𝑠𝑐𝜃. 𝑄𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦}

− ∫ ∫ 𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃

𝑒𝑖(𝜃−𝜋2)_{sin(𝑐𝑠𝑐𝜃. 𝑃𝑥) . sin(𝑐𝑠𝑐𝜃. 𝑆𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦}

− ∫ ∫ 𝑒2𝑖(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑄2+𝑅2)𝑐𝑜𝑡𝜃

𝑒−𝑖2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃𝑒 𝑖(𝜃−𝜋₂)

sin(𝑐𝑠𝑐𝜃. 𝑅𝑥) . sin(𝑐𝑠𝑐𝜃. 𝑄𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦

∞

0 ∞

0

𝑒2𝑖(𝑥2+𝑦2+𝑅2+𝑆2)𝑐𝑜𝑡𝜃_𝑒 −𝑖

2(𝑅2+𝑆2)𝑐𝑜𝑡𝜃

𝑒𝑖(𝜃−𝜋2)_{sin(𝑐𝑠𝑐𝜃. 𝑅𝑥) . sin(𝑐𝑠𝑐𝜃. 𝑆𝑦)𝑓(𝑥, 𝑦) 𝑑𝑥𝑑𝑦} _]

𝐹_𝑐𝛼{𝑓(𝑥, 𝑦)𝑠𝑖𝑛𝑎𝑥. 𝑠𝑖𝑛𝑏𝑦}(𝑢, 𝑣)

=𝑒

𝑖

4((𝑐𝑠𝑐2𝜃 .𝑃𝑅+𝑎2)+(𝑐𝑠𝑐2𝜃𝑄𝑆+𝑏2))𝑠𝑖𝑛2𝛼_𝑒−𝑖(𝜃− 𝜋 2)

4

[

𝑒−𝑖2(𝑃2+𝑄2)𝑐𝑜𝑡𝜃_𝐹_𝑠𝜃_{𝑒 𝑖

− 𝑒−𝑖2(𝑃2+𝑆2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

− 𝑒−𝑖2(𝑄2+𝑅2)𝑐𝑜𝑡𝜃_𝐹_𝑐𝜃_{𝑒 𝑖

2(𝑥2+𝑦2)(𝑐𝑜𝑡𝛼−𝑐𝑜𝑡𝜃)_{𝑓(𝑥, 𝑦)} (𝑅, 𝑆) ]}

3. Parseval’sIdentity

Parseval’sidentity for two-dimensional fractional cosine transforms.

If 𝐹𝑠𝛼(𝑢, 𝑣) = 𝐺𝑠𝛼(𝑢, 𝑣) 𝐹̅̅̅̅̅̅̅̅̅̅̅ = 𝐺𝑠𝛼(𝑢, 𝑣) ̅̅̅̅̅̅̅̅̅̅̅ 𝑠𝛼(𝑢, 𝑣)

then

𝑖) ∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦

= 8𝑐𝑠𝑐𝛼

𝜋 ∫ ∫ 𝐺𝑐

𝛼_{(𝑢, 𝑣)}

̅̅̅̅̅̅̅̅̅̅̅

∞ 0 ∞

0 𝐹𝑐

(9)

ii) ∫ ∫ |𝑓(𝑥, 𝑦)|∞ 2 0

∞

0 𝑑𝑥𝑑𝑦

=8𝑐𝑠𝑐𝛼

𝜋 ∫ ∫ |𝐹𝑐

𝛼_{(𝑢, 𝑣)|}2

∞ 0 ∞ 0 𝑑𝑢𝑑𝑣 Solution:

By definition of two dimensional fractional cosine transform

𝐹𝑐𝛼{𝑔(𝑥, 𝑦)}(𝑢, 𝑣) = 𝐺𝑐𝛼(𝑢, 𝑣) = √

2𝜋 𝑒

𝑖(𝑢2+𝑣2)𝑐𝑜𝑡𝛼 2 ∫ ∫ 𝑔(𝑥, 𝑦) ∞ 0 ∞ 0 𝑒

𝑖(𝑥2+𝑦2)𝑐𝑜𝑡𝛼

2 _{cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥)}

. cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑥

𝑑𝑦

Using inversion formula of fractional cosine transform

𝑔(𝑥, 𝑦) = 4

𝜋2∫ ∫ 𝐺𝑐𝛼(𝑢, 𝑣)

∞

0 ∞

0

𝑒−

𝑖(𝑥2+𝑦2+𝑢2+𝑣2)𝑐𝑜𝑡𝛼 2

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) . cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑐𝑠𝑐2𝛼√ 2𝜋

1 − 𝑖𝑐𝑜𝑡𝛼𝑑𝑢𝑑𝑣

𝑔(𝑥, 𝑦)

̅̅̅̅̅̅̅̅̅ = 4

𝜋2 √

2𝜋

1 + 𝑖𝑐𝑜𝑡𝛼𝑒

2 _𝑐𝑠𝑐2_{𝛼 ∫ ∫ 𝐺}̅̅̅̅̅̅̅̅̅̅̅_𝑐𝛼_{(𝑢, 𝑣)}

∞

0 ∞

0

𝑒𝑖(𝑢

2_+𝑣2_{)𝑐𝑜𝑡𝛼} 2

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) . cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑢𝑑𝑣

∫ ∫ 𝑓(𝑥, 𝑦) ∞ 0 ∞ 0 𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 = ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦 ∞ 0 ∞ 0 4

𝜋2 √

2𝜋

1 + 𝑖𝑐𝑜𝑡𝛼𝑒

2 _𝑐𝑠𝑐2_𝛼

∫ ∫ 𝐺̅̅̅̅̅̅̅̅̅̅̅_𝑐𝛼(𝑢, 𝑣) ∞ 0 ∞ 0 𝑒𝑖(𝑢

2_+𝑣2_{)𝑐𝑜𝑡𝛼} 2

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) . cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑢𝑑𝑣

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 = 4

𝜋2 √

2𝜋

1 + 𝑖𝑐𝑜𝑡𝛼𝑐𝑠𝑐

(10)

Available Online at www.ijpret.com 119 ∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑒𝑖(𝑥

2_+𝑦2_+𝑢2_+𝑣2_{)𝑐𝑜𝑡𝛼} 2

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) . cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑥𝑑𝑦

∫ ∫ 𝐺_{̅̅̅̅̅̅̅̅̅̅̅}_𝑐𝛼_{(𝑢, 𝑣)} ∞

0 ∞

0

𝑑𝑢𝑑𝑣

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 =

√ 32

𝜋3_{(1 + 𝑖𝑐𝑜𝑡𝛼)}𝑐𝑠𝑐

2_𝛼

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑒

𝑖(𝑥2_+𝑦2_+𝑢2_+𝑣2_{)𝑐𝑜𝑡𝛼} 2

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥) . cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑥𝑑𝑦

∫ ∫ 𝐺̅̅̅̅̅̅̅̅̅̅̅_𝑐𝛼(𝑢, 𝑣)

∞

0 ∞

0

𝑑𝑢𝑑𝑣

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 =

√ 64

𝜋2_{(1 + 𝑐𝑜𝑡}2_𝛼)𝑐𝑠𝑐

2_{𝛼 ∫ ∫ √}1 − 𝑖𝑐𝑜𝑡𝛼

2𝜋 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑒𝑖(𝑥

2_+𝑦2_+𝑢2_+𝑣2_{)𝑐𝑜𝑡𝛼}

2 _{cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢𝑥)}

. cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦) 𝑑𝑥𝑑𝑦

∫ ∫ 𝐺̅̅̅̅̅̅̅̅̅̅̅_𝑐𝛼(𝑢, 𝑣)

∞

0 ∞

0

𝑑𝑢𝑑𝑣

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 = √ 64

𝜋2_{(1 + 𝑐𝑜𝑡}2_𝛼)𝑐𝑠𝑐2𝛼

∫ ∫ 𝐺̅̅̅̅̅̅̅̅̅̅̅𝑐𝛼(𝑢, 𝑣) ∞

0 ∞

0

(11)

Available Online at www.ijpret.com 120 ∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑔(𝑥, 𝑦) ̅̅̅̅̅̅̅̅̅̅𝑑𝑥𝑑𝑦 =

8𝑐𝑠𝑐𝛼

𝜋 ∫ ∫ 𝐺𝑐

𝛼_{(𝑢, 𝑣)}

̅̅̅̅̅̅̅̅̅̅̅

∞

0 ∞

0

𝐹_𝑐𝛼_{{𝑓(𝑥, 𝑦)}(𝑢, 𝑣)𝑑𝑢𝑑𝑣}

ii) Let 𝑓(𝑥, 𝑦) = 𝑔(𝑥, 𝑦) then

𝐹_𝑐𝛼(𝑢, 𝑣) = 𝐺_𝑐𝛼_{(𝑢, 𝑣) 𝐹}

𝑐𝛼(𝑢, 𝑣)

̅̅̅̅̅̅̅̅̅̅̅ = 𝐺_{̅̅̅̅̅̅̅̅̅̅̅}_𝑐𝛼_{(𝑢, 𝑣)}

∫ ∫ 𝑓(𝑥, 𝑦)

∞

0 ∞

0

𝑓(𝑥, 𝑦)

̅̅̅̅̅̅̅̅̅ 𝑑𝑥𝑑𝑦 =

8𝑐𝑠𝑐𝛼

𝜋 ∫ ∫ 𝐹𝑐

𝛼_{(𝑢, 𝑣)}

∞

0 ∞

0

𝐹̅̅̅̅̅̅̅̅̅̅̅𝑑𝑢𝑑𝑣_𝑐𝛼(𝑢, 𝑣)

∫ ∫ |𝑓(𝑥, 𝑦)|2

∞

0 ∞

0

𝑑𝑥𝑑𝑦 =8𝑐𝑠𝑐𝛼

𝜋

∫ ∫ |𝐹_𝑐𝛼_{(𝑢, 𝑣)|}2

∞

0 ∞

0

𝑑𝑢𝑑𝑣

4. Shifting property:

If 𝐹_𝑠𝛼_{(𝑓(𝑥, 𝑦))(𝑢, 𝑣)}_{is two dimensional fractional cosine transform of}_{𝑓(𝑥, 𝑦)}_then

𝐹_𝑐𝛼_{{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣) =}

cos(cscα. ua) cos(cscα. vb)

𝐹_𝑐𝛼_{𝑒𝑖(𝑎 2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

+ sin(cscαua) sin(cscαvb) e−i(θ−π2)

𝐹𝑠𝛼{𝑒𝑖(

𝑎2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

+ cos(cscα. ua) sin(cscαvb)

e−i(θ−π2)_𝑒 𝑖

2(𝑎2+𝑏2)𝑐𝑜𝑡𝛼√ 2𝜋

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+ cos(cscα. vb) sin(cscαua)e−i(θ−π2)

𝑒2𝑖(𝑎2+𝑏2)𝑐𝑜𝑡𝛼√ 2𝜋

𝐺𝑐𝛼{𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼. 1}(𝑣)𝐺𝑠𝛼{𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼𝑓(𝑡, 𝑠)}(𝑢)

Solution:

𝐹𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣) = √

2𝜋 𝑒

∫ ∫ 𝑒

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣𝑦)𝑓(𝑥 + 𝑎, 𝑦 + 𝑏) 𝑑𝑥 𝑑𝑦

∞

−∞ ∞

−∞

Let, A=√1−𝑖𝑐𝑜𝑡𝛼

2𝜋 B=𝑒

𝑖(𝑢2+𝑣2)𝑐𝑜𝑡𝛼

2

𝑥 + 𝑎 = 𝑡, 𝑦 + 𝑏 = 𝑠 , 𝑑𝑥 = 𝑑𝑡 , 𝑑𝑦 = 𝑑𝑠

, 𝑡 → −∞ 𝑡𝑜 ∞ ,

𝑠 → −∞ 𝑡𝑜 ∞

𝐹_𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)

= 𝐴𝐵 ∫ ∫ 𝑒

𝑖((𝑡−𝑎)2+(𝑠−𝑏)2)𝑐𝑜𝑡𝛼

2 _{cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑢(𝑡 − 𝑎))}

cos(𝑐𝑜𝑠𝑒𝑐𝛼. 𝑣(𝑠 − 𝑏))𝑓(𝑡, 𝑠) 𝑑𝑡𝑑𝑠

∞

−∞ ∞

−∞

𝐹_𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)

= 𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑎2−2𝑡𝑎)𝑐0𝑡𝛼_{cos(𝑐𝑜𝑠𝑒𝑐𝛼. (𝑢𝑡 − 𝑢𝑎))}

∞

−∞ ∞

−∞

𝑒2𝑖(𝑠2+𝑏2−2𝑠𝑏)𝑐0𝑡𝛼_{cos(𝑐𝑜𝑠𝑒𝑐𝛼. (𝑣𝑠 − 𝑣𝑏)) 𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠}

𝐹_𝑐𝛼_{{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)}

= 𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑎2)𝑐𝑜𝑡𝛼_𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼_{cos(𝑐𝑜𝑠𝑒𝑐𝛼. (𝑢𝑡 − 𝑢𝑎))}

∞

−∞ ∞

−∞

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𝐹_𝑐𝛼_{{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)}

= 𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑎2)𝑐𝑜𝑡𝛼_𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼

∞

−∞

𝑒2𝑖(𝑠2+𝑏2)𝑐𝑜𝑡𝛼 ∞

−∞

𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼

[cos(cscα. ua) cos(cscαut)

+ sin(cscαua) sin(cscαut)] [

cos(cscα. vb) cos(cscαvs)

+ sin(cscαvb) sin(cscαvs)]

𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠

𝐹𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)

=

[

𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑠2+𝑎2+𝑏2)𝑐𝑜𝑡𝛼_𝑒−𝑖(𝑡𝑎+𝑠𝑏)𝑐𝑜𝑡𝛼

∞

−∞ ∞

−∞

cos(cscα. ua) cos(cscαut) cos(cscα. vb) cos(cscαvs)𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠

+𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑠2+𝑎2+𝑏2)𝑐𝑜𝑡𝛼_𝑒−𝑖(𝑡𝑎+𝑠𝑏)𝑐𝑜𝑡𝛼

∞

−∞ ∞

−∞

cos(cscα. ua) cos(cscαut) cos(cscα. vb) cos(cscαvs) 𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠

+𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑠2+𝑎2+𝑏2)𝑐𝑜𝑡𝛼_𝑒−𝑖(𝑡𝑎+𝑠𝑏)𝑐𝑜𝑡𝛼

∞

−∞ ∞

−∞

sin(cscαua) sin(cscαut) cos(cscα. vb) cos(cscαvs) 𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠

+ 𝐴𝐵 ∫ ∫ 𝑒2𝑖(𝑡2+𝑠2+𝑎2+𝑏2)𝑐𝑜𝑡𝛼_𝑒−𝑖(𝑡𝑎+𝑠𝑏)𝑐𝑜𝑡𝛼

∞

−∞ ∞

−∞

sin(cscαua) sin(cscαut) sin(cscαvb) sin(cscαvs)

𝑓(𝑡, 𝑠)𝑑𝑡𝑑𝑠 ]

𝐹_𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣)

= cos(cscα. ua) cos(cscα. vb)

𝐹_𝑐𝛼{𝑒𝑖(𝑎

2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

𝐹_𝑠𝛼_{𝑒𝑖(𝑎 2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

+ cos(cscα. ua) sin(cscαvb)e−i(θ−π2)_𝑒

𝑖

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Available Online at www.ijpret.com 123 [

∫ √1 − 𝑖𝑐𝑜𝑡𝛼

2𝜋

∞

−∞

𝑒2𝑖(𝑡2+𝑢2)𝑐𝑜𝑡𝛼_𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼_{cos(cscαut) 𝑑𝑡}

2𝜋 e

i(θ−π₂) ∞

−∞

𝑒2𝑖(𝑠2+𝑣2)𝑐𝑜𝑡𝛼_𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼

sin(cscαvs)𝑓(𝑡, 𝑠) 𝑑𝑠 ]

+ cos(cscα. vb) sin(cscαua)e−i(θ−π2)_𝑒

𝑖

[

2𝜋

∞

−∞

𝑒2𝑖(𝑠2+𝑣2)𝑐𝑜𝑡𝛼

𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼_{cos(cscαvs) 𝑑𝑠}

2𝜋 e

i(θ−π₂) ∞

−∞

𝑒2𝑖(𝑡2+𝑢2)𝑐𝑜𝑡𝛼

𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼sin(cscαut)𝑓(𝑡, 𝑠) 𝑑𝑡 ]

𝐹_𝑐𝛼{𝑓(𝑥 + 𝑎, 𝑦 + 𝑏)}(𝑢, 𝑣) =

cos(cscα. ua) cos(cscα. vb)

𝐹_𝑐𝛼{𝑒𝑖(𝑎

2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

𝐹_𝑠𝛼_{𝑒𝑖(𝑎 2_+𝑏2

2 −(𝑡𝑎+𝑠𝑏))𝑐𝑜𝑡𝛼_{𝑓(𝑡, 𝑠)} (𝑢, 𝑣)}

+ cos(cscα. ua) sin(cscαvb)e−i(θ−π2)

𝑒2𝑖(𝑎2+𝑏2)𝑐𝑜𝑡𝛼√ 2𝜋

𝐺𝑐𝛼{𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼. 1}(𝑢)𝐺𝑠𝛼{𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼𝑓(𝑡, 𝑠)}(𝑣)

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Available Online at www.ijpret.com 124 𝑒2𝑖(𝑎2+𝑏2)𝑐𝑜𝑡𝛼√ 2𝜋

𝐺𝑐𝛼{𝑒−𝑖𝑠𝑏𝑐𝑜𝑡𝛼. 1}(𝑣)𝐺𝑠𝛼{𝑒−𝑖𝑡𝑎𝑐𝑜𝑡𝛼𝑓(𝑡, 𝑠)}(𝑢)

Where 𝐺_𝐶𝛼, 𝐺_𝑆𝛼 are one dimensional fractional cosine and sine transform respectively.

CONCLUSION

In the proposed work we have proved Parseval’s identity and shifting property for generalized two fractional Cosine transform, also Modulation Property is described in the form of theorem.

REFERENCES

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