Convolution And Rayleigh’s Theorem For Generalized Fractionla Hartley Transform EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol 2, No. 1, 2009 (162-170) ISSN 1307-5543 – www.ejpam.com *Corresponding Author. http://www.ejpam.com 162 © 2009 EJPAM All Rights Reserved Convolution And Rayleigh’s Theorem For Generalized Fractional Hartley Transform P. K. Sontakke*1 and A. S. Gudadhe2 1H.V.P.M.’S College of Engineering and Technology, Amravati, India. 2Govt. Vidarbha Institute of Science and Humanities, Amravati, India. Abstract. The fractional Hartley transform, which is a generalization of the Hartley transform, has many applications in several areas, including signal processing and optics. In this paper we have introduced convolution theorem, modulation theorem and Parseval’s identity (Rayleigh’s Theorem) for the generalized fractional Hartley transform. AMS subject code: 46F12 and 44 Key Words: Convolution theorem, Modulation theorem, Parseval’s identity, Hartley transform. 1. Introduction: The fractional Fourier transform has become the focus of many research papers, because of its recent applications in many fields, including optics and signal processing. Hence fractional Hartley transform is also useful tool in many fields, due to it’s close relation with fractional Fourier transforms. Many properties of the fractional Fourier transform are well known, including its product and convolution theorem, which have been derived by Almeida [5] and Zayed [2]. P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 163 Using the eigen value function as used in fractional Fourier transform, different integral transform in Fourier class, including Hartley transform are generalized to fractional transform by Pei [4]. He had shown that for all non negative integer m, )(2 2 tHe m t− is the eigen function of the Hartley transform and had given the formula for fractional Hartley transform as, { } ∫ ∞ ∞− = ,),()()()( dtstKtfstfH α α where [ ]).csc()1().(csc)1( 2 1. 2 cot1),( cot 2 cot 2 22 stcasiestcasieeeistK ii tisi φφ π φ φφφφ α −++− − = . (1.1) Almedia [5] had defined convolution for the fractional Fourier transform as [ ] dvevugvFeuH viui α α α α αα tan 2tan 2 22 .sec)()(sec)(        ∞ ∞− − ∫ −= (1.2) Since the convolution theorem for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transform, the one for the fractional Fourier transform does not seem as nice or as practical. The reason is that the convolution operation defined by (1.2) is not the right sort of convolution for the fractional Fourier transform. Zayed had defined fractional Fourier type convolution as follows. For any function )(tf , if 2 cot 2 )()( ti etftf φ = then for any two function f and g the convolution operation ∗ is defined by Zayed [2] as, )(th = ( f ⋆ g ) )(t = ),)((. 2 cot1 2 cot 2 tgfei ti ∗ − − φ π φ where ∗ is the convolution operation for the Fourier transform as defined by (1.2). In this paper first we have defined generalized fractional Hartley transform in section 2. We have proved convolution theorem for fractional Hartley transform in section 3. Also discussed the modulation theorem and Parseval’s identity in section 4. P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 164 2. Generalized fractional Hartley transform 2.1 The test function space )E(Rn An infinitely differentiable complex valued function ψ on nR belongs to )( nRE if for each compact set aSK ⊂ where { },0,, >≤∈= aatRtS n a ,)(sup)(, ∞<= ∈ tDk t Kt kE ψψγ ,.....3,2,1=k Note that the space E is complete and therefore a Frechet space. 2.2 The fractional Hartley transform on 'E It can be easily proved that function ),( stKα as a function of t, is a member of )( nRE , where [ ],).csc()1().(csc)1( 2 1. 2 cot1),( cot 2 cot 2 22 stcasiestcasieeeistK ii tisi φφ π φ φφφφ α −++− − = and 2 απφ = . The generalized fractional Hartley transform of ,)()( ' nREtf ∈ where )(' nRE is the dual of the testing function space, can be defined as, { } .),(,)()()( stKtfstfH α α = (2.2.1) Another simple form of fractional Hartley transform as in Sontakke [3] is { } [ ] dttfstiesteeistfH i tisi )( ).sin(csc).cos(csc 2 cot1)( )( cot 2 cot 2 22 φφ π φ φφφα − − = ∫ ∞ ∞− (2.2.2) 3. Convolution of fractional Hartley transform 3.1 Convolution theorem: We define fractional Hartley type convolution as follows. For any function )(tf , define the function 2 cot 2 )()( ti etftf φ = . Then for any two function f and g we define the convolution operation ‘∗ ’ by P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 165 )(th = ( f ⋆ g ) )(t = ).)((. 2 cot1 2 cot 2 tgfei ti ∗ − − φ π φ Now we state and prove our convolution theorem. Theorem: Let )(th = ( f ⋆ g ) )(t and { })(thH α , { })(tfH α and { })(tgH α denote the fractional Hartley transform of )(th , )(tf and )(tg respectively, then { } { } { } { }( )[ )( )()()( (s) )()( )(2 cot 2 2 stfHstfHtgHsthHe si −−= ααααφ { } { }( )] { } { } { }( ))( )()()((s) )(sin.)( )()( )(cos. stfHstfHtgHestfHstfHe ii αααφααφ φφ −−−+−++ −− Proof: From the definition of the fractional Hartley transform (2.2.2) and definition of convolution we have, { } [ ] dtthstiesteeisthH i tisi )( ).sin(csc).cos(csc 2 cot1)( )( cot 2 cot 2 22 φφ π φ φφφα − − = ∫ ∞ ∞− . [ ] ∫∫ ∞ ∞− − ∞ ∞− −      − = φφφφφ φφ π φ cot 2 cot 2 cot 2 cot 2 2 2222 )(.).sin(csc).cos(csc 2 cot1 uitii tisi eufestiesteei dudteutg uti 2 )(cot 2 )( − − φ [ ] . ).sin(csc).cos(csc )()( 2 cot1 2 )(cot 2 cotcot 2 2 222 dtdustiesteeutgufei i utiuisi φφ π φ φφφφ −−      − = −∞ ∞− ∞ ∞− ∫ ∫ By making the change of variable ,vut =− we obtain [ ]dvduvusievuseevgufei i viuisi )(.sin(csc))(.cos(csc )()( 2 cot1 2 cot 2 cotcot 2 2 222 +−+      − = ∫ ∫ ∞ ∞− ∞ ∞− φφ π φ φφφφ { } { } { } { } { }[ )()(cos )((s) )( )((s) )( 2cot 2 2 svgHiufSevgHufCthHe isi αα π φ αααφ φ       −− −= { } ])()(sin svgHi −− αφ , P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 166 where αC and αS denote fractional cosine and fractional sine transform. Using the relation of fractional cosine transform and fractional sine transform to fractional Hartley transform, we get { } { } { } { }( ) { } { }( )[ ])()(cos.)()( )()( )(2 cot 2 2 ufHufHeufHufHvgHsthHe i si −++−−= − ααφααααφ φ { } { } { }( ))()( )( sin ufHufHvgHe i αααφ φ −−−+ − i.e. { } { } { } { }( )[ )()()()( (s))((s) )(2 cot 2 2 stfHstfHtgHthHe si −−= ααααφ { } { }( )])()()()(cos. stfHstfHe i −++ − ααφ φ { } { } { }( ))()()()()()( sin stfHstfHstgHe i αααφ φ −−−+ − (3.1.1) 3.2 Convolution of various combinations of even and odd functions: Next we consider different cases of convolution for even and odd functions. Case I: If the function f and g both are odd functions, i.e. )()( tftf −=− and )()( tgtg −=− then { } { } ( ) { } { } ).()()()( sin1)()()( cot 2 2 stgHstfHeesgfHsthH sii ααφφαα φ − +=∗= Case II: If the function f and g both are even function i.e. )()( tftf =− and )()( tgtg =− then { } { } { } { } )()().()(..cos)()()( cot 2 2 stgHstfHeesgfHsthH sii ααφφαα φ −−=∗= { } { } ).()().()(.cos cot 2 2 stgHstfHe si αα φφ φ         +− = Case III: If f is even, g is odd then { } { } { } { } ).()().()(.cos)( )()( cot 2 2 stgHstfHesthHsgfH si αα φφ αα φ         +− ==∗ Case IV: If f is odd, g is even then { } { } ( ) { } { } ).()()( )( sin1)()()( cot 2 2 stfHstgHeesgfHsthH sii ααφφαα φ −−−=∗= P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 167 Case V: If f is even function and g any function then, { } { } { } { } ).( )().( )(.cos)()( )( cot 2 2 stgHstfHesgfHsthH si αα φφ αα φ         +− =∗= Case VI: If f is odd function and g any function then, { } { } { } )( )()()()( cot 2 2 stfHesgfHsthH si αφαα − =∗= { } { }( ).)()( sin)()( stgHestgH i −− − αφα φ Case VII: If f is any function and g even function then, { } { } )()( )(2 cot 2 2 sgfHsthHe si ∗= ααφ { } { } ( ) { } ( )[ ].1)sin(cos)()()sin(cos1)()( )()( −+−+−+= −− φφφφ φαφαα ii estfHestfHstgH Case VIII: If f is any function and g is odd function then, { } { } { } ( )[ )cos(sin1)()( )( )()()(2 cot 2 2 φφφαααφ ++= −i si estfHstgHsthHe { } ( )].)cos(sin1 )( )( φφφα −+−− −iestfH 4. Modulation Theorem for fractional Hartley transform 4.1 If { } )()( stfH α is fractional Hartley transform of )(tf then { } { { } )sin()( 2 1)( cos)( cos2sin 4 2 φαφφα ustfHeesuttfH isu ui += −− { } }.)sin()(cos φαφ ustfHeisu −+ Proof: Using the definition of fractional Hartley transform { } [ ]).sin(csc).cos(csc 2 cot1)( cos)( cot 2 cot 2 22 stiesteeisuttfH i tisi φφ π φ φφφα − − = ∫ ∞ ∞− ,cos)(. utdttf P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 168 by solving, we get { } { { } )sin()( 2 1)( cos)( cos2sin 4 2 φαφφα ustfHeesuttfH isu ui += −− { } }.)sin()(cos φαφ ustfHeisu −+ (4.1.1) 4.2 If { } )()( stfH α is fractional Hartley transform of )(tf , then { } [ ({ { } { })(sin)sin()(cot 2 1)( sin)( cos2sin 4 2 tfHustfHieesuttfH isu ui −−+= −− ααφφα φφφ )] { } { }( )[ ]})sin()(sin)sin()(cot)sin( cos φφφφφ ααφ ustfHustfHieus isu −−−−−+ Proof: Using the definition of fractional Hartley transform { } [ ] dtuttfstiesteeiuttfH i tisi sin)(.).sin(csc).cos(csc 2 cot1(s) sin)( cot 2 cot 2 22 φφ π φ φφφα − − = ∫ ∞ ∞− by solving, we get { } [ ({ { } { } )])sin()(sin)sin()(cot 2 1(s) sin)( cos2sin 4 2 φφφφ ααφφα ustfHustfHieeuttfH isu ui +−−+= −− { } { }( )[ ]}.)sin()(sin)sin()(cotcos φφφφ ααφ ustfHustfHieisu −−−−− (4.2.1) 4.3 If { } )()( stfH α is fractional Hartley transform of )(tf then { } { { } φαφφα φφφ cotcot2sin 4 )cot1()sin()(.)cot1( 2 1(s) )( 2 isuisu uiiut eustfHeeetfH +++−= −− { } { } { }( )}.)sin()()sin()(sin)sin()( φφφφ ααα ustfHustfHiustfH −−−+−−− Proof: Using (4.1.1) and (4.2.1) 4.4 Parseval’s identity for fractional Hartley transform: Statement: If the fractional Hartley transform of )(tf and )(tg be { } )()( stfH α and { } )()( stgH α respectively then P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 169 { } { } { } { } { } { } { } { }∫∫ ∫∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∗ −+−− −−+= dsstgHstfHidsstgHstfHi dsstgHstfHdsstgHstfHdttgtf )()().()(sin2)()().()(sin2 )()().()( 2 sin)()().()( 2 cos)().()1 22 αααα αααα φφ φφ and { }( ) { }( ) ,)()( 2 sin)()( 2 cos)()2 22222 dsstfHdsstfHdttf ∫∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− −+= αα φφ where )(* tg is a complex conjugate of )(tg . Proof: The Parseval’s relation for the fractional Fourier transform is as follows, ∫∫ ∞ ∞− ∞ ∞− = dssGsFdttgtf )()()()( ** αα . (4.4.1) Now using the relation between fractional Fourier transform and fractional Hartley transform is as follows. { } { } { }[ ])()()1()()()1( 2 1(s) )( stfHestfHetfF ii −−++= −− αφαφ α ie { } { } { } { } { }( ) { } { }( )         −−− −−+−+ = )()()()(sin )()()()(cos)()()()( 2 1(s) )( stfHstfHi stfHstfHstfHstfH tfF αα αααα α φ φ and { } { } { } { } { }( ) { } { }( )         −−+ −−+−+ =• )()()()(sin )()()()(cos)()()()( 2 1(s) )( stgHstgHi stgHstgHstgHstgH tgG αα αααα α φ φ . Where )(sG • α is a complex conjugate of )(sGα , hence equation (4.4.1) becomes { } { } { } { } { } { } { } { } .)()().()(sin2)()().()(sin2 )()().()( 2 sin)()().()( 2 cos)().( 22 ∫∫ ∫∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∗ −+−− −−+= dsstgHstfHidsstgHstfHi dsstgHstfHdsstgHstfHdttgtf αααα αααα φφ φφ In particular if gf = then, P. Sontakke, A. Gudadhe / Eur. J. Pure Appl. Math, 2 (2009) 170 { }( ) { }( ) .)()( 2 sin)()( 2 cos)( 22222 dsstfHdsstfHdttf ∫∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− −+= αα φφ 5. Conclusion We have proved convolution theorem, modulation theorem and Parseval’s identity for fractional Hartley transform. Convolution of two functions for fractional Hartley transform may be used in filter design. References [1] A. I. Zayed: " Hand book of Generalized function and functional Analysis". Publisher CRC Press, 1996. [2] A. I. Zayed: " Convolution and product theorem for the fractional fourier transform" IEEE Signal Processing Letters, Vol. 5, No.4, April 1998. [3] P. K. Sontakke, A. S. Gudadhe: " Generalized fractional Hartley transform" Vidarbha Journal of Science, Vol. II, No.1, 2007. [4] Pei-Soo-Chang Jian-Jiun Ding: " Fractional cosin, sine and Hartley transform", IEEE, Trans. on Signal Processing, Vol. 50, No. 7, July 2002. [5] L.B.Almeida: "Product and Convolution theorem for the fractional Fourier transform" IEEE Ttrans. Signal Processing Letters, Vol. 4, P. 15-17, 1997. [6] R. N. Bracewell: " The Fourier transform and its applications". Mc. Grew-Hill 2003. P. K. Sontakke*1 and A. S. Gudadhe2 AMS subject code: 46F12 and 44