Generalized Half Linear Canonical Transform And Its Properties

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2 Gnrlz Hl Lnr Cnoncl Trnorm An I Propr A S Guh # A V Joh* # Gov Vrh Inu o Scnc n Humn, Amrv M S * Shnkrll Khnlwl Collg, Akol M S Arc: A gnrlzon o h Frconl Fourr rnorm FRFT, h lnr cnoncl rnorm LCT h n u n vrl r, nclung opcl nly n gnl procng For prccl purpo hl lnr cnoncl rnorm CT mor uul Hnc n h ppr w hv prov om mporn rul g nvron horm, lnry, rnon, rvv propry, Prvl Iny or hl lnr cnoncl rnorm, lo rlon wn CT n Lplc rnorm LT cu Kywor: Lnr cnoncl rnorm, Frconl Fourr Trnorm Inroucon: Th o h rconl powr o Fourr opror ppr n mhmcl lrur rly n 93 I h n rcovr n qunum mchnc y Nm [5] n 98 H h gvn ymc mho or h vlopmn o rconl ngrl rnorm y mn o Egn vlu In h p c, FRFT h rc much non o h gnl procng communy h gnrlzon o FT Th rlvn hory h n vlop nclung uncrny prncpl, mplng hory, convoluon horm Numr o ohr ngrl rnorm lo hv n xn n rconl omn or xmpl, Aky [] h u rconl Mlln rnorm, Tyw, Guh [9] cu Frconl Hnkl rnorm, Frconl Hlr rnorm h n vlop y Zy [], Sonkk, Guh [8] u numr o propry o rconl Hrly rnorm, Joh, Guh [] work on gnrlz cnoncl n rnorm c Bhol n Chouhry [3] h vlop rconl Fourr rnorm gnrlz uncon Th rconl rnorm oun numr o pplcon n gnl procng, mg procng, qunum mchnc c Furhr gnrlzon o rconl Fourr rnorm known lnr cnoncl rnorm w nrouc y Mohnky [4] n 97 P, Dng [6,7] h u gn vlu pc Lnr cnoncl rnorm hr prmr lnr ngrl rnorm whch h vrl pcl c rconl Fourr rnorm, Frnl rnorm, Chrp rnorm c Lnr cnoncl rnorm n, [ LCT ] / /, c or =, or =, wh c =, whr,, c, n r rl prmr npnn on n Th ppr mphz on nng hl lnr cnoncl rnorm, rvng nvron horm, hn om propr o h hl lnr cnoncl rnorm r cu, lo rlon wn Hl Lnr Cnoncl Trnorm n Lplc rnorm n nlly concluon r gvn IJERTVIS699 wwwjrorg 7

3 Tng Funcon Spc E : An nnly rnl complx vlu uncon on R n long o E R n, or ch compc, I whr :,, } S S R n n n or k R, up k D, k I E No h pc E compl n Frch pc, l E no h ul pc o E Hl Lnr Cnoncl Trnorm: Dnon: n Th Hl Lnr Cnoncl Trnorm E R cn n y, CT } = <, K, > whr, K, n Hnc h gnrlz hl lnr cnoncl rnorm o E R cn n y, CT Snc h rng o ngron or h hl lnr cnoncl rnorm ju *, + n no or -, ung hl lnr cnoncl rnorm mor convnn hn ung h cnoncl rnorm o l wh h vn uncon Invron or Gnrlz Hl Lnr Cnoncl Trnorm: Any rnorm u o olv rnl quon, only nvr o h rnorm vll W on nvr o hl lnr cnoncl rnorm Invron horm or hl lnr cnoncl rnorm : I CT hl lnr cnoncl rnorm o gvn y, CT IJERTVIS699 wwwjrorg 7

4 CT hn, Proo: Th hl lnr cnoncl rnorm o gvn y CT F whr, CT F F C g whr, C g C F n g => C g By ung nvr ormul g C F IJERTVIS699 wwwjrorg 7

5 F F F F CT 3 Propr or Hl Lnr cnoncl Trnorm: 33 Lnr Propry or hl lnr cnoncl rnorm: rnorm o I CT }, CT g} no gnrlz hl lnr cnoncl,g n P,P r conn hn, CT [ P P g ]} P CT } P CTg } Proo: Th proo mpl n hnc om 33 Drnon propry o hl lnr cnoncl rnorm: I CT } no gnrlz hl lnr cnoncl rnorm o hn, ' } CT CT CT [ ] Proo: W hv, CT ' ]} ' C ' IJERTVIS699 wwwjrorg 73

6 whr, C C CT ' } ' C } ' C CT ' } CT C C ' } [ ] CT CT CT 333 Drvv propry o hl lnr cnoncl rnorm: I CT } no gnrlz hl lnr cnoncl rnorm, hn, ]} [ } } CT CT CT Proo: W hv, CT } CT } } ]} [ } CT CT CT 74 wwwjrorg IJERTVIS699

7 CT } CT[ ]} CT } 4 Prvl Iny or hl lnr cnoncl rnorm: I n g r h nvron hl lnr cnoncl rnorm o F n G rpcvly, hn g F G n Proo: By non o CT, CT g g F Ung h nvron ormul o CT g G Tkng complx conjug w g, g G g G Chngng h orr o ngron, w g, g G g G F g F G Hnc prov Pung g n quon 4, w g IJERTVIS699 wwwjrorg 75

8 F 4 Rlon wn hl lnr cnoncl rnorm n Lplc rnorm: 44 Rlon wn hl lnr cnoncl rnorm n Lplc rnorm: I c, hn CT chng no Lplc rnorm Proo: By non o CT, For CT c, w g, CT CT CT L[ ] Th h Lplc rnorm Tl or hl lnr cnoncl rnorm SN F HS [ P ] HCCT } Pg P CT } P CTg } HCST IJERTVIS699 wwwjrorg 76

9 Concluon: In h ppr, r nroucon o h gnrlz hl lnr cnoncl rnorm gvn n Invron horm prov Lnry, rnon, rvv propry, Prvl Iny, r on or hl lnr cnoncl rnorm whch wll uul n olvng rnl quon occurrng n gnl procng n mny ohr rnch o ngnrng Rrnc: [] Aky O n Brl, 998: Frconl Mlln Trnormon: An xnon o rconl rquncy concp or cl, 8h IEEE, Dg Sgn Proc Workhop, Bryc Cnyn, Uh [] Joh AV n Guh A S : Gnrlz Cnoncl Sn Trnorm, Sc Rv Chm Commun: 3, [3] Bhol BN n Chouhry MS : Frconl Fourr rnorm o ruon o compc uppor, Bull Cl Mh Soc, Vol 94, No5, [4] Mohnky, M97: Lnr cnoncl rnorm n hr unry rprnon, Jour Mh, Phy,Vol, No 8, P [5] Nm V 98: Th rconl orr Fourr rnorm n pplcon o qunum mchnc, Jour In Mh App, Vol 5, 4-65 [6] P n Dng, : Egnuncon o Lnr Cnoncl Trnorm Vol 5, No [7] P n Dng, : Frconl con, n n Hrly Trnorm, IEEE Trn On Sgn Proc Vol 5, No7, [8] Sonkk, Guh 9: Convoluon n Rylgh Thorm For Gnrlz Frconl Hrly Trnorm, EJPAM Vol, No, 6-7 [9] Tyw, Guh : On Gnrlz rconl Hnkl rnorm, Inr Journl o mhmcl Anly, Vol 6, No 8 [] Zy A I,Aug 998: Hlr Trnorm Aoc wh h Frconl Fourr Trnorm, IEEE gnl procng Lr, Vol5, No8 IJERTVIS699 wwwjrorg 77

Analytical Study of a Special Case of Complex Canonical Transform

Analytical Study of a Special Case of Complex Canonical Transform lobl Jornl o Mhmcl Scncs: hory n Prccl Volm, Nmbr 3 00, pp 6--70 Inrnonl Rsrch Pblcon Hos hp://wwwrphoscom Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm PR Dshmkh n AS h Pro Rm Mgh Ins o chnology & Rsrch,