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Convolution Theorem for Canonical Cosine Transform and Their Properties

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IOSR Journals
IOSR Journals

The document discusses the convolution theorem for the generalized canonical cosine transform. It begins with an introduction to fractional Fourier transforms and linear canonical transforms. It then defines the generalized canonical cosine transform and introduces a special type of convolution for this transform. The main result is that the convolution theorem holds for the generalized canonical cosine transform: the transform of the convolution of two functions is equal to the convolution of the individual transforms of the two functions. The theorem is stated and a proof is provided. Finally, some properties of the canonical cosine transform are mentioned.

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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728.Volume 6, Issue 1 (Mar. - Apr. 2013), PP 74-79 www.iosrjournals.org www.iosrjournals.org 74 | Page Convolution Theorem for Canonical Cosine Transform and Their 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 Cosine transform, which is a generalization of the linear canonical 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 Parseval’s identity for the generalized canonical cosine transform. Keywords: Fractional Fourier Transform, Linear Canonical Transform, Convolution. I. Introduction: As generalization of the Fourier Transform (FT), the Fractional Fourier Transform (FrFT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known. In recent years, Almeida and Zayed derived product and convolution of two functions in a usual manner and proved the convolution theorem in fractional Fourier transform domain. 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. A further generalization of FrFT is Linear Canonical Transform (LCT). Just as Fourier cosine transform and Fourier sine transform are defined from Fourier Transform, similarly canonical cosine and canonical sine transforms are defined from LCT by Pie and Ding [3]. We have discussed some properties of Half canonical cosine transform in [2]. This paper emphasizes on defining generalized canonical cosine transform and deriving its convolution theorem, then some properties of the canonical cosine transform are discussed and finally conclusions are given. Notations and terminology as per [5] II. Testing Function Space E : An infinitely differentiable complex valued function  on Rn belongs to E (Rn ), if for each compact set, SI  where }0,,:{   tnRttS and for ,n Rk  sup ( ) ( ) . , k t D t k t I      E Note that space E is complete and a Frechet space, let E’ denotes the dual space of E. 2.1Definition: The generalized Canonical Cosine Transform ( )n f RE can be defined by, {CCT f (t)} (s) = < f(t), KC(t, s) > where,                      t b s t b ai e s b di e ib st C K cos 2 2 2 2 .2 1 ),(  Hence the generalized canonical cosine transform of a regular function ( )n f RE can be defined by,                         )1.1()(cos 2 1 )()( 22 22 dttfet b s e ib stCCTf t b ai s b di  2.2 The Generalized Canonical Cosine Transform of Convolution Now we introduced a special type of convolution and product for canonical cosine transform. 2.3 Definition: For any function , let us define the functions and by )()(~,)(|)(|~ 22 )( 2 )( 2 yvgeyvgyvgeyvg yv b ai yv b ai              and )()( ~ 2 2 yfeyf y b ai       
Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 75 | Page For any two functions f and g, we define the Convolution operation by (1.2) Now we state and prove convolution theorem. 3.1 Convolution Theorem: If and denote the Canonical Cosine transform of respectively, then Proof: From the definition of the Canonical Cosine transform, we have dtdytgyft b s y b s ee bi ty b ai s b d i )()(coscos2 2 1 0 )( 2 0 222                            dtdytgyfty b s ty b s ee bi ty b ai s b d i )()(})(cos)({cos 1 0 )( 2 0 222                            dtdytgyfty b s ee bi dtdytgyfty b s ee bi ty b ai s b d i ty b ai s b d i )()()(cos 1 )()()(cos 1 0 )( 2 0 0 )( 2 0 222 222                                           21 II  (1.3) For I1, putting dvdtvty  for limit when yvot  , when  vt For I2, putting dvdtvty  for limit when yvot  , when  vt dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i yv yv b ai y b ai y s b d i CC )()()(cos 1 )()(cos 1 )))]((()[))]((([)3.1( 222 222 )( 22 0 )( 22 0                                                             dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                      
Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 76 | Page dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                       4321)))]((()[))]((([ IIIIstgGsyfF CC  (1.4) For I4, putting dvdvvv  for limit when yvyv  , when 00  vv dvdyvygyfv b s eee bi dvdyvygyfv b s eee bi I y v vy b ai y b ai y s b d i yv vy b ai y b ai y s b d i )()(cos 1 )()()()(cos 1 0 )( 22 0 0 )( 22 0 4 222 222                                                           dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([)4.1( 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                       dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 222 222                                                             dvev b s dyyvgvygee bibi v b ai v b ai s b d i yf 222 2 0 0 2 cos|))(|~)(~( 2 2 1 )( ~                                    by (1.2)                                         dvvgfv b s ee biib ee v b ai s b div b ai s b di )})(~* ~ {(cos 2 2 0 22 22 22 22  by (2.3)    )()(~* ~ 2 )))]((()[))]((([ 22 22 svgfF ib ee stgGsyfF C v b ai s b di CC              
Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 77 | Page 4.1 Some operational results: 4.1.1 Linearity property of canonical cosine transformations: If {CCT f(t)}(s), {CCT g(t)}(s) denotes generalized canonical cosine transform of f(t), g(t) and P1, P2 are constants then       )())(()()(()())()(( 2121 stgCCTPstfCCTPstgPtfPCCT  Proof is simple and hence omitted. 4.1.2 Derivative (with respect to parameter) of canonical cosine transform: If {CCT f(t)}(s) denotes generalized canonical cosine transform, then,                ))]}((.[{ 1 ))}(({.))}(({ stftCST b stCCTf b d sistCCTf ds d Proof: We have,                               dttft b s ee ibds d stfCCT ds d t b ai s b di )(cos 2 1 ))}(({ 22 22  dttft b s e s e ib stfCCT ds d s b di t b ai )(cos. 2 1 ))}(({ 22 22                                dttft b s es b d it b s e b t e ib s b di s b di t b ai )(cos..sin. 2 1 222 222                                                                                                                  dttft b s ee ibb d sdttftt b s e b e ib ii s b di t b ai s b di t b ai )(cos 2 1 .)](.[sin. 1 2 1 )( 2222 2222  ))}(({.))]}((.[[ 1 { stCCTf b d sstftCST b i                       ))]}((.[{ 1 ))}(({.))}(({ stftCST b stCCTf b d sistCCTf ds d 4.1.3 Modulation property of canonical cosine transform: If {CCT f(t)} (s) denotes generalized canonical cosine transform of f(t) then,                   )()()()( 2 )()(.cos 2 2 b bzs etfCCT b bzs etfCCT e stfztCCT idszidsz dbz i Proof: By definition of CCT, dttftz b s tz b s ee ib stfztTCC t b ai s b di )(coscos. 2 1 2 1 ))}((.cos{ 22 22                                        })(..cos 2 1 )(.cos 2 1 { 2 1 ))}((.cos{ 222 222 2 )( 22 2 )( 22 dttftz b s eeee ib dttftz b s eeee ib stfztTCC zdb i idsz bzs b di t b ai zdb i idsz bzs b di t b ai                                                   })(.. )( cos 2 1 )(. )( cos 2 1 { 2 ))}((.cos{ 22 22 2 )( 22 )( 22 2 dttfet b bzs ee ib dttfet b bzs ee ib e stfztTCC idsz bzs b di t b ai idsz bzs b di t b aizdb i                                                
Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 78 | Page                   )()()()( 2 )()(.cos 2 2 b bzs etfCCT b bzs etfCCT e stfztCCT idszidsz dbz i 4.1.4 If {CCT f(t)} (s) denotes generalized canonical cosine transform of f(t) then,                     )()()()( 2 )()()(.sin 2 2 b bzs etfCCT b bzs etfCST e istfztCCT idszidsz zdb i Proof: By definition of CCT, dttfztt b s ee ib stfztTCC t b ai s b di )(sincos. 2 1 ))}((.sin{ 22 22                        dttftz b s tz b s ee ib stfztTCC t b ai s b di )(sinsin. 2 1 2 1 ))}((.sin{ 22 22                                        })(..sin 2 1 )(.sin 2 1 { 2 1 ))}((.sin{ 222 222 2 )( 22 2 )( 22 dttftz b s eeee ib dttftz b s eeee ib stfztTCC zdb i idsz bzs b di t b ai zdb i idsz bzs b di t b ai                                                   })(..cos 2 1 )( )(.sin 2 1 ){( 2 )( ))}((.sin{ 22 22 2 )( 22 )( 22 2 dttfet b bzs ee ib i dttfet b bzs ee ib i ei stfztTCC idsz bzs b di t b ai idsz bzs b di t b aizdb i                                                                        )()()()( 2 )()()(.sin 2 2 b bzs etfCCT b bzs etfCST e istfztCCT idszidsz zdb i 5.1 Parseval’s Identity for canonical cosine transform: If )(tf and )(tg are the inversion canonical cosine transform of )(sFC and )(sGC respectively, then (1) dssGsFdttgtf CC )(.)(2)(.)(        and (2) dssFdttf C      22 )(2)(.  Proof: By definition of CCT,   dttgt b s ee ib stgCCT t b ai s b di )(cos 2 1 )()( 22 22                        ---------------- by (1.1) Using the inversion formula of CCT dssGt b s ee b i tg C s b di t b ai )(cos 2 )( 22 22                        Taking complex conjugate we get, dssGt b s ee b i tg C s b di t b ai )(cos 2 )( 22 22                         
Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 79 | Page                                     dssGt b s ee b i dttfdttgtf C s b di t b ai )(cos 2 )()(.)( 22 22 Changing the order of integration, we get,                                   dttft b s ee ib ib dssG b i dttgtf s b di t b ai C )(cos 2 1 2 1 1 )( 2 )(.)( 22 22    (1.5) (2) Now putting )()( tgtf  in equation (1.5), we get dssFdttf C      22 )(2)(  Table for canonical cosine transform S.N . f(t) FC(s) 1 ))()(( 21 tgPtfP      )())(()()(( 21 stgCCTPstfCCTP  2 )(.cos tfzt                   )()()()( 2 2 2 b bzs etfCCT b bzs etfCCT e idszidsz zdb i 3 )(.sin tfzt                   )()()()( 2 )( 2 2 b bzs etfCCT b bzs etfCST e i idszidsz zdb i III. Conclusion: The Convolution of generalized canonical cosine transform is developed in this paper. 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.15—17 [2] Gudadhe A. S., Joshi A.V. (2013): On Generalized Half Canonical Cosine Transform, IOSR-JM Volume X, Issue X. [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, pub. [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. [5] Zemanian A. H., (1965): Distribution Theory and Transform Analysis, Mc. Graw Hill, New York.

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Convolution Theorem for Canonical Cosine Transform and Their Properties

  • 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728.Volume 6, Issue 1 (Mar. - Apr. 2013), PP 74-79 www.iosrjournals.org www.iosrjournals.org 74 | Page Convolution Theorem for Canonical Cosine Transform and Their 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 Cosine transform, which is a generalization of the linear canonical 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 Parseval’s identity for the generalized canonical cosine transform. Keywords: Fractional Fourier Transform, Linear Canonical Transform, Convolution. I. Introduction: As generalization of the Fourier Transform (FT), the Fractional Fourier Transform (FrFT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known. In recent years, Almeida and Zayed derived product and convolution of two functions in a usual manner and proved the convolution theorem in fractional Fourier transform domain. 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. A further generalization of FrFT is Linear Canonical Transform (LCT). Just as Fourier cosine transform and Fourier sine transform are defined from Fourier Transform, similarly canonical cosine and canonical sine transforms are defined from LCT by Pie and Ding [3]. We have discussed some properties of Half canonical cosine transform in [2]. This paper emphasizes on defining generalized canonical cosine transform and deriving its convolution theorem, then some properties of the canonical cosine transform are discussed and finally conclusions are given. Notations and terminology as per [5] II. Testing Function Space E : An infinitely differentiable complex valued function  on Rn belongs to E (Rn ), if for each compact set, SI  where }0,,:{   tnRttS and for ,n Rk  sup ( ) ( ) . , k t D t k t I      E Note that space E is complete and a Frechet space, let E’ denotes the dual space of E. 2.1Definition: The generalized Canonical Cosine Transform ( )n f RE can be defined by, {CCT f (t)} (s) = < f(t), KC(t, s) > where,                      t b s t b ai e s b di e ib st C K cos 2 2 2 2 .2 1 ),(  Hence the generalized canonical cosine transform of a regular function ( )n f RE can be defined by,                         )1.1()(cos 2 1 )()( 22 22 dttfet b s e ib stCCTf t b ai s b di  2.2 The Generalized Canonical Cosine Transform of Convolution Now we introduced a special type of convolution and product for canonical cosine transform. 2.3 Definition: For any function , let us define the functions and by )()(~,)(|)(|~ 22 )( 2 )( 2 yvgeyvgyvgeyvg yv b ai yv b ai              and )()( ~ 2 2 yfeyf y b ai       
  • 2. Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 75 | Page For any two functions f and g, we define the Convolution operation by (1.2) Now we state and prove convolution theorem. 3.1 Convolution Theorem: If and denote the Canonical Cosine transform of respectively, then Proof: From the definition of the Canonical Cosine transform, we have dtdytgyft b s y b s ee bi ty b ai s b d i )()(coscos2 2 1 0 )( 2 0 222                            dtdytgyfty b s ty b s ee bi ty b ai s b d i )()(})(cos)({cos 1 0 )( 2 0 222                            dtdytgyfty b s ee bi dtdytgyfty b s ee bi ty b ai s b d i ty b ai s b d i )()()(cos 1 )()()(cos 1 0 )( 2 0 0 )( 2 0 222 222                                           21 II  (1.3) For I1, putting dvdtvty  for limit when yvot  , when  vt For I2, putting dvdtvty  for limit when yvot  , when  vt dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i yv yv b ai y b ai y s b d i CC )()()(cos 1 )()(cos 1 )))]((()[))]((([)3.1( 222 222 )( 22 0 )( 22 0                                                             dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                      
  • 3. Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 76 | Page dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF yv yv b ai y b ai y s b d i v yv b ai y b ai y s b d i y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                       4321)))]((()[))]((([ IIIIstgGsyfF CC  (1.4) For I4, putting dvdvvv  for limit when yvyv  , when 00  vv dvdyvygyfv b s eee bi dvdyvygyfv b s eee bi I y v vy b ai y b ai y s b d i yv vy b ai y b ai y s b d i )()(cos 1 )()()()(cos 1 0 )( 22 0 0 )( 22 0 4 222 222                                                           dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i y v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )()(cos 1 )()(cos 1 )))]((()[))]((([)4.1( 0 )( 22 0 0 )( 22 0 0 )( 22 0 0 )( 22 0 222 222 222 222                                                                                                                       dvdyyvgyfv b s eee bi dvdyyvgyfv b s eee bi stgGsyfF v yv b ai y b ai y s b d i v yv b ai y b ai y s b d i CC )()(cos 1 )()(cos 1 )))]((()[))]((([ 0 )( 22 0 0 )( 22 0 222 222                                                             dvev b s dyyvgvygee bibi v b ai v b ai s b d i yf 222 2 0 0 2 cos|))(|~)(~( 2 2 1 )( ~                                    by (1.2)                                         dvvgfv b s ee biib ee v b ai s b div b ai s b di )})(~* ~ {(cos 2 2 0 22 22 22 22  by (2.3)    )()(~* ~ 2 )))]((()[))]((([ 22 22 svgfF ib ee stgGsyfF C v b ai s b di CC              
  • 4. Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 77 | Page 4.1 Some operational results: 4.1.1 Linearity property of canonical cosine transformations: If {CCT f(t)}(s), {CCT g(t)}(s) denotes generalized canonical cosine transform of f(t), g(t) and P1, P2 are constants then       )())(()()(()())()(( 2121 stgCCTPstfCCTPstgPtfPCCT  Proof is simple and hence omitted. 4.1.2 Derivative (with respect to parameter) of canonical cosine transform: If {CCT f(t)}(s) denotes generalized canonical cosine transform, then,                ))]}((.[{ 1 ))}(({.))}(({ stftCST b stCCTf b d sistCCTf ds d Proof: We have,                               dttft b s ee ibds d stfCCT ds d t b ai s b di )(cos 2 1 ))}(({ 22 22  dttft b s e s e ib stfCCT ds d s b di t b ai )(cos. 2 1 ))}(({ 22 22                                dttft b s es b d it b s e b t e ib s b di s b di t b ai )(cos..sin. 2 1 222 222                                                                                                                  dttft b s ee ibb d sdttftt b s e b e ib ii s b di t b ai s b di t b ai )(cos 2 1 .)](.[sin. 1 2 1 )( 2222 2222  ))}(({.))]}((.[[ 1 { stCCTf b d sstftCST b i                       ))]}((.[{ 1 ))}(({.))}(({ stftCST b stCCTf b d sistCCTf ds d 4.1.3 Modulation property of canonical cosine transform: If {CCT f(t)} (s) denotes generalized canonical cosine transform of f(t) then,                   )()()()( 2 )()(.cos 2 2 b bzs etfCCT b bzs etfCCT e stfztCCT idszidsz dbz i Proof: By definition of CCT, dttftz b s tz b s ee ib stfztTCC t b ai s b di )(coscos. 2 1 2 1 ))}((.cos{ 22 22                                        })(..cos 2 1 )(.cos 2 1 { 2 1 ))}((.cos{ 222 222 2 )( 22 2 )( 22 dttftz b s eeee ib dttftz b s eeee ib stfztTCC zdb i idsz bzs b di t b ai zdb i idsz bzs b di t b ai                                                   })(.. )( cos 2 1 )(. )( cos 2 1 { 2 ))}((.cos{ 22 22 2 )( 22 )( 22 2 dttfet b bzs ee ib dttfet b bzs ee ib e stfztTCC idsz bzs b di t b ai idsz bzs b di t b aizdb i                                                
  • 5. Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 78 | Page                   )()()()( 2 )()(.cos 2 2 b bzs etfCCT b bzs etfCCT e stfztCCT idszidsz dbz i 4.1.4 If {CCT f(t)} (s) denotes generalized canonical cosine transform of f(t) then,                     )()()()( 2 )()()(.sin 2 2 b bzs etfCCT b bzs etfCST e istfztCCT idszidsz zdb i Proof: By definition of CCT, dttfztt b s ee ib stfztTCC t b ai s b di )(sincos. 2 1 ))}((.sin{ 22 22                        dttftz b s tz b s ee ib stfztTCC t b ai s b di )(sinsin. 2 1 2 1 ))}((.sin{ 22 22                                        })(..sin 2 1 )(.sin 2 1 { 2 1 ))}((.sin{ 222 222 2 )( 22 2 )( 22 dttftz b s eeee ib dttftz b s eeee ib stfztTCC zdb i idsz bzs b di t b ai zdb i idsz bzs b di t b ai                                                   })(..cos 2 1 )( )(.sin 2 1 ){( 2 )( ))}((.sin{ 22 22 2 )( 22 )( 22 2 dttfet b bzs ee ib i dttfet b bzs ee ib i ei stfztTCC idsz bzs b di t b ai idsz bzs b di t b aizdb i                                                                        )()()()( 2 )()()(.sin 2 2 b bzs etfCCT b bzs etfCST e istfztCCT idszidsz zdb i 5.1 Parseval’s Identity for canonical cosine transform: If )(tf and )(tg are the inversion canonical cosine transform of )(sFC and )(sGC respectively, then (1) dssGsFdttgtf CC )(.)(2)(.)(        and (2) dssFdttf C      22 )(2)(.  Proof: By definition of CCT,   dttgt b s ee ib stgCCT t b ai s b di )(cos 2 1 )()( 22 22                        ---------------- by (1.1) Using the inversion formula of CCT dssGt b s ee b i tg C s b di t b ai )(cos 2 )( 22 22                        Taking complex conjugate we get, dssGt b s ee b i tg C s b di t b ai )(cos 2 )( 22 22                         
  • 6. Convolution Theorem For Canonical Cosine Transform And Their Properties www.iosrjournals.org 79 | Page                                     dssGt b s ee b i dttfdttgtf C s b di t b ai )(cos 2 )()(.)( 22 22 Changing the order of integration, we get,                                   dttft b s ee ib ib dssG b i dttgtf s b di t b ai C )(cos 2 1 2 1 1 )( 2 )(.)( 22 22    (1.5) (2) Now putting )()( tgtf  in equation (1.5), we get dssFdttf C      22 )(2)(  Table for canonical cosine transform S.N . f(t) FC(s) 1 ))()(( 21 tgPtfP      )())(()()(( 21 stgCCTPstfCCTP  2 )(.cos tfzt                   )()()()( 2 2 2 b bzs etfCCT b bzs etfCCT e idszidsz zdb i 3 )(.sin tfzt                   )()()()( 2 )( 2 2 b bzs etfCCT b bzs etfCST e i idszidsz zdb i III. Conclusion: The Convolution of generalized canonical cosine transform is developed in this paper. 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.15—17 [2] Gudadhe A. S., Joshi A.V. (2013): On Generalized Half Canonical Cosine Transform, IOSR-JM Volume X, Issue X. [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, pub. [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. [5] Zemanian A. H., (1965): Distribution Theory and Transform Analysis, Mc. Graw Hill, New York.