I Jourl of Mh Alyi, Vol, 008, o 0, 977-986 Alyiciy d Oprio Trform o Grlizd Frciol rly Trform *P K So d A S Guddh * VPM Collg of Egirig d Tchology, Amrvi-44460 (MS), Idi Gov Vidrbh Iiu of cic d umii, Amrvi-444604 (MS), Idi Abrc I hi ppr w hv dicud h lyiciy horm d ivrio formul for h grlizd frciol rly rform d uig h w hv provd uiqu horm Alo w hv dicud frciol rly rform of lcd fucio d obid oprio rform formul for hi rform Kyword: Frciol Fourir rform, rly rform, Tig fucio pc, Grlizd fucio Iroducio: Now dy, frciol igrl rform ply impor rol i igl procig, img rcorucio, pr rcogiio, ccoic igl procig [8], [9] Fourir lyi i o of h mo frquly ud ool i igl procig d my ohr ciific dicipli Bid h Fourir rform for im frqucy rprio of igl, Wigr diribuio, h mbiguiy fucio, h hor im fourir rform d h pcrogrm r of ud g i pch procig rdr, quum phyic I mhmic lirur grlizio of h Fourir rform o h frciol Fourir rform i udr Nmi [0] iroducd h cocp of Fourir rform of frciol ordr, which dpd o coiuou prmr Th grlizio of ordiry fourir rform d i propri wr dicud i Criolro l [3] Zyd [] Drgom [4] c Frciol fourir rform i furhr grlizd o h igrl wih rpc o w mur dρ d w grlizd igrl rform w obid by Zyd [] Bhol d chudhry [] hd xdd frciol fourir rform o h diribuio of compc uppor Th frciol Fourir rform wih corrpod o h clicl Fourir rform d frcio l Fourir rform wih 0 corrpod o h idiy opror I [5] ohr igrl rform of Fourir cl, h i coi rform, i
978 P K So d A S Guddh rform d rly rform, r lo grlizd o h corrpodig frciol igrl rform d udid by diffr mhmici Th rly rform i rciv lriv d covi rl rplcm for h wll ow complx Fourir rform rly rform i gig grr imporc i vrl pplicio I [6] Brcwll ugg h h rly rform i f or fr h h Fourir rform d c b ud rplcm for h Fourir rform Uig h igvlu fucio, ud i frciol Fourir rform, diffr igrl rform i Fourir cl r grlizd o frciol rform by Pi [5] rly rform i lo grlizd o frciol rly rform by him hd how h for ll o giv igr m, m ( ) i h ig fucio of h rly rform d hd giv h formul for frciol rly rform, whr K K d, π [( i ) c(cc φ ) + ( + i ) c( ccφ ) ] I hi ppr fir w dfi grlizd frciol rly rform i cio d prov i lyiciy Th fucio f () i h rcovrd from by m of h ivrio formul i cio 3 Uig ivrio formul w hv provd h frciol rly rformbl grlizd fucio hvig h m rip of dfiiio d h m rform mu b idicl, which i md uiqu horm i cio 3 Frciol rly rform of om lcd fucio d oprio rform formul r obid i cio 4 d cio 5 rpcivly Lly cocluio i giv i cio 6 Alyiciy Of Frciol rly Trform : ' Th Grlizd Frciol rly Trform o E : R d S d 0 L S { R,, > 0 L, if icoφ i coφ i coφ K [( i ) c(ccφ ) + ( + i ) c( ccφ ) ], π whr π φ, h K E( R ) γ K up D K < if { < < r E ( R ) i h ig fucio pc ' Th grlizd frciol rly rform of E ( R ) i dfid,
Alyiciy d oprio rform 979, K, () ' whr E ( R ) i h dul pc of h ig fucio pc Alyiciy Thorm : ' Thorm : L ( ) f E R d i frciol rly rform i dfid () Th i lyic o Supp S { : R,, > 0 D R if h f h [ ](, D K Proof : L (, ) R, w fir prov h : i diffribl d [ { f ( )]( f ( ), K W prov h rul for, h grl rul follow by iducio For om 0, choo wo cocric circl C d C wih cr d rdii r d r rpcivly, uch h 0 < Δ < r 0 < r < r < L Δ b icrm, ifyig Coidr, ( + Δ ) Δ ( ) f ( ), K ) f ( ), ΨΔ ( ), ) Δ whr, ) [ K, ) K ] K ) ΨΔ For y fixd ( + Δ R, {( ccφ K + i coφ K ) D i coφ i coφ, φ d K C, φ, whr, C i coφ π K [i(ccφ ) + i co(ccφ )]
980 P K So d A S Guddh Sic for y fixd R fixd igr d rgig from 0 o, D K ) i lyic iid d o C, w hv by Cuchy igrl formul, Δ M D i (, ) ΨΔ ( ) π z Δ z ( )( ) C dz, whr,, z, ) ( + Bu for ll z C d rricd o compc ub of R, 0, M D K i boudd by co K Thrfor w hv, K DΨΔ ( ) Δ ( r r) r ub of ' E Thu Δ 0, D ΨΔ ( ) d o zro uiformly o h compc R hrfor i follow h () ΨΔ covrg i E( R ) o zro Sic w coclud h () lo d o zro hrfor ( i S Bu hi i ru for ll, c ( diffribl wih rpc o i lyic d D [ ](, D K 3 Ivr Ad Uiqu Thorm : 3 Ivr Of Frciol rly Trform: Grlizd frciol rly rform dfid i () c b wri icoφ i coφ i coφ ( i ) c(ccφ ) + ( + i ) c( ccφ ) d π ( i π ) i co φ f ( ) (cc φ + ( + i ) f ( ) (cc φ,
Alyiciy d oprio rform 98 Pu (ccφ ) v v viφ ccφ Tig rly ivr of boh id which i m rly rform, iv i i coφ φcoφ ( vi φ ) cv d ( i ) f ( ) + ( + i ) f ( ) i ( ) co φ i ( ) co φ If f () i v h f ( ) h, w g i coφ g ( v) c v dv, iv i φco φ φ whr g( v) v i ) i coφ g( v) c v dv, wh f () i v Puig g (v) d olvig, w g d K(, ) ccφ i coφ i coφ whr K c(ccφ ) icoφ Now if f () i odd h f ( ) g( v) c v dv ( i i i( φ + coφ ) ), i coφ g( v) c v dv Agi puig g (v) d olvig, w g + ( + i ) i coφ ( )
98 P K So d A S Guddh d K(, ) ccφ i coφ i( φ ) i coφ whr, K c(ccφ ) i coφ π 3 Uiqu Thorm : Thorm : If d { g( ) r frciol rly rform of f () d g () rpcivly for o d upp f S : S { : R, d upp g S : S { : R, ( { g( ) h f g i h of quliy of D ' (I ) if { ) Proof : By ivrio horm, f g lim N π N N K [ { g( ) ]d Thu f g i D ' (I ) 4 Frciol rly Trform Of Slcd Fucio : Frciol rly Trform of lcd fucio r buld follow
Alyiciy d oprio rform 983 TABLE Sr No Sigl { Frciol rly rform ( (φ i) i φ δ ( ) { δ ( ) i coφ π i + co φ [ co(ccφ ) i i(ccφ ) ] 3 δ ( ) { δ ( ) i coφ π 4 i { ( + i ( ) ( ) ) φ φ i i( cφ ) i 5 co { ( + co ( ) ( ) ) φ φ i co( cφ ) i 5 Oprio Trform Formul : I hi cio w prov om oprio rform formul for frciol rly rform, for which followig wo lmm c b ily provd 5 Lmm : Frciol rly rform giv i cio, c lo b xprd i coφ i coφ i coφ π 5 Lmm : { f ( ) icoφ i coφ i coφ π [ co(ccφ ) i i(ccφ ) ] [ co(ccφ ) + i i(ccφ ) ] d d
984 P K So d A S Guddh 53 Lmm : icoφ i coφ i i coφ φ [ i(ccφ ) + i co(ccφ ) ] d π i coφ iφ { f ( ) 54 Formul : If FrT h d ( i coφ [ { ] { f ( ) d Proof : Sic i coφ i coφ i coφ π co φ d i Cφ d Solvig w g ico [ co(ccφ ) i i(ccφ ) ] d ' [ co(cc φ ) i i(cc φ ) ] d [ { ] { f ( ) φ 55 Formul: If FrT { h + i φ d { f ( ) + i φ { d ' ( icoφ { { f ( Proof : { ) Q [ ] ) { 56 Formul: + c) (c+ c ) i coφ { co(ccφ c) + iφ i(ccφ c) f ( ) { Proof : w ow h, + c) ( + c) i coφ i coφ i coφ π (c+ c ) d + i φ { f ( ) + i φ d (c+ c ) i coφ i coφ { i(cc φ c) { co(ccφ( + c) ) i i(ccφ( + c) ) d (c+ c ) i coφ i coφ { co(ccφ c) i coφ { i(ccφ c) + iφ { i(ccφ c) f ( ) d 57 Formul : i coφ { f ( Proof : Coidr, d d d d d i coφ π ) { co(ccφ ) i i(ccφ ) d
Alyiciy d oprio rform 985 i coφ iφ iφ iφ i {coφ + coφ { f ( ) i coφ { f ( ) { f ( ) + i coφ 58 Formul : L b fixd rl umbrth mppig f ( ) i coiuou lir mppig o S o S d { f ( ) Whr i rl umbr i coφ i coφ [ co(ccφ ) i i(ccφ ) ] { Proof : By h dfiiio of frciol rly rform, { f ( ), i coφ i coφ i coφ [ co(ccφ ) i i(ccφ ) ] π Puig x d olvig w g { f ( ) f ( ) d i coφ i coφ [ co(ccφ ) i i(ccφ ) ] { 59 Formul: If FrT f ( ) ( h i i( ) co φ ( ) co φ Proof : By h dfiiio of frciol rly rform, i( ) coφ f ( ) ( i coφ i coφ i coφ [ co(ccφ ) i i(ccφ ) ] i( ) coφ Puig i π T d olvig w g f ( ) ( i ( ) co φ ( ) co φ f ( ) d
986 P K So d A S Guddh 6 Cocluio : Th grlizd frciol rly rform i dvlopd i hi ppr Th oprio rform formul provd i hi ppr c b ud, wh hi rform i ud o olv ordiry or pril diffril quio REFERENCES Ahmd I Zyd, A Covoluio d produc horm for h rciol Fourir rform, IEEE Sigl procig lr, Vol 5, No 4, April 998 B N Bhol d Chudhry M S, Frciol Fourir rform of Diribuio of compc uppor Bull cl Mh Soc, 94 (5), P 349-358, 00 3 Criolro,l, Mulipliciy of frciol Fourir rform d hir rliohip IEEE Tr o igl proc Vol48, No, J 000, P 7-4 4 D Drgom Frciol Fourir rld fucio, Opic Commuicio, Vol 8, P 9-98, July 996 5 Pi Soo-Chg, Ji-Jiu Dig, Frciol coi, i d rly rform, IEEE, Vol 50, No 7, July 00 6 R N Brcwll, Th rly Trform, Oxford U K Oxford Uiv Pr, 986 7 R Scilr, Ergiv S d Ciiz N, Th u of h rly rform i gophyicl pplicio Gophyic, Vol 55, No, (Nov 990), P 488-495 8 Ti Aliv d Bi Mri J, O Frciol Fourir rform mom, IEEE Sigl procig Lr, Vol 7, No, Nov 000 9 Ti Aliv d Bi Mri J, Wigr diribuio d frciol Fourir rform for -dimiol ymmric bm, JOSA A, Vol 7, No, Dc 000, p 39-33 0 Vicor Nmi, Th frciol ordr Fourir rform d i Applicio o quum mchic, J I Mh Appic, (998), 5, 4-65 Rcivd: Fbrury 4, 008