Why is the Linear Canonical Transform so little known?

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1 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 Wh is th Linar Canonical Transorm so littl known? Adrian Strn Dpartmnt o Elctro Optical Enginring, Bn Gurion Univrsit o th Ngv, Br-Shva 8405, Isral, Phon: , strn@bgu.ac.il Abstract. Th linar canonical transorm (LCT, is th nam o a paramtrizd continuum o transorms which includ, as particular cass, th most widl usd linar transorms and oprators in nginring and phsics such as th Fourir transorm, ractional Fourir transorm (FRFT, Frsnl transorm (FRST, tim scaling, chirping, and othrs. Thror th LCT provids a uniid ramwork or studing th bhavior o man practical transorms and sstm rsponss in optics and nginring in gnral. From th sstm-nginring point o viw th LCT provids a powrul tool or dsign and analsis o th charactristics o optical sstms. Dspit this act onl w authors tak advantag o th powrul and gnral LCT thor or analsis and dsign o optical sstms. In this papr w rviw som important proprtis about th continuous LCT and w prsnt som nw rsults rgarding th discrtization and computation o th LCT. Kwords: linar canonical transorm, ractional Fourir transorm, Frsnl transorm, Laplac transorm, sampling, Wignr distribution, tim-rqunc rprsntation, phas spac. PACS: 0.30.Uu, 0.30.Nw, 4.40.L, 4.5.Bs. INTRODUCTION Th titl o this papr is borrowd rom R. [] qustioning wh is th Frsnl transorm so littl known. Hr w rpat th qustion or a mor gnral and mor widl usd transorm: th linar canonical transorm (LCT. This papr tris to convinc that LCT dsrvs spcial considration and should b bttr and widl known. Th main motivation is th act that LCT provids a gnral mathmatical tool with vr broad applicabilit in man ilds o scinc and nginring. Although it is not vr much known, its spcial cass ar widl usd in various ilds, otn undr dirnt nams. Thror, undrstanding th LCT ma hlp to gain mor insights on its spcial cass and to carr ovr knowldg gaind rom on subct to othrs. Th LCT is a our-paramtr (a,b,c,d class o linar intgral transorm []-[5]. It provids a canonical ormalism or th rspons o a vr larg class o phsical sstms. LCTs hav bn rinvntd or rconsidrd b man authors undr man dirnt nams at dirnt tims in dirnt contts [3], a act that implis univrsal importanc o th transorm. Th LCT is also known as th Gnralizd Frsnl Transorm [6] or ABCD transorm [7], Collins ormula [8] gnralizd Hugsn intgrals [9] and oshinsk and Qusn intgrals [], and is a spcial cas o th Spcial Ain Fourir Transorm [0].

2 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 Th LCT with paramtrs {a,b,c,d}o a unction (, is dind : = O ( ( = ( C, d, C ( { } ( (, = πb ( a + d b b a b whr and dt(=ad-bc=. For th cas o b=0 th LCT is dind as c d th limit o ( with b 0, ilding [4]: ( = O { ( }( = d ( d, b = 0. ( Plas not that sinc dt(= onl thr transorm paramtrs ar r. Thror th LCT can b dind also as a thr paramtr transorm [3]. Howvr hr w will adopt th our paramtr dinition sinc th matri ormalism is particularl hlpul in dining proprtis o LCT (s Sc. and or undrstanding th opration o th LCT on th phas spac (s Sc. 3. It is as to vri that th LCT with paramtrs {a,b,c,d}={cosθ,sinθ,-sinθ, cosθ} rducs to FRFT [3],[5] which, in th spciic cas θ=π/, bcoms th Fourir transorm. With paramtrs {a,b,c,d}={,b,0,} th LCT rducs to th Frsnl transorm [3],[5]. ultiplication b Gaussian or chirp unction is obtaind with an {a,b,c,d}={,0,c,}. Scaling oprator can b viwd as a spcial cas o th LCT with {a,b,c,d}={d -,0,0,d}. In this papr w rstrict ourslvs to th class o LCT with ral paramtrs {a,b,c,d}. In such cass th LCT oprator is unitar in L (R. With th pric o giving up on unitarit, othr important transorms can b ound as spcial cass o LCT. On such ampl is th Laplac transorm obtaind with {a,b,c,d}={0,,,0}. Th LCT was ound usul in man ilds in nginring. Th LCT dscribs th rspons o an quadratic-phas sstm [] and o cascad combinations o such sstms. In optics an sstm that is implmntd using an arbitrar numbr o thin lnss and propagation through r spac undr Frnl approimation, or through sctions o gradd-ind mdia, blong to th class o quadratic-phas sstms. Th propagation o ilds in irst ordr optics can b dscribd icintl b th LCT. Th LCT can also b applid in th thor o chirp lasr pulss []. In Radar thor, LCT can b usd in dscribing puls comprssion and chirp-componnt dtction. Th LCT can b usd in imag procssing or pattrn rcognition, imag dblurring, and watrmarking. In acoustics, LCT can b usd to dscrib bat-chirp tp signals and propagation. In communication thor LCT, was usd to dsign multichannl communication sstms and ma b usd to cop with multi-path problms. Th outlin o this papr is as ollows. In sction w summariz bril som important proprtis o th continuous paramtr (tim/spac LCT (CPLCT. In sction 3 w show th intrprtation o th LCT in oint paramtr spac (.g. phas spac or tim-rqunc spac. In sction 4 th sampling thorm or LCT is prsntd and in sction 5 th discrt LCT is drivd. Th matrial is ssntiall cd 0 (

3 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 tutorial. Howvr, th prsntation is original, and it includs a numbr o nw rsults rgarding th discrtization o th LCT (sctions 4 and 5.. SOE PROPERTIES OF THE CONTINUOUS PARAETER LCT In this sction w summariz som important proprtis o th CPLCT. Th LCT dind in (-( obs th additivit proprt: O O = O. (3 From (3 w s that th invrs LCT (ILCT is obtaind b an LCT with d b =, so that th intgration krnl in ( satisis C (, = C (,. c a Following this proprt, togthr with th dinition (, it is as to vri that i is ral thn th invrs transorm krnl is th Hrmitian conugat o th original transorm krnl: * C (, = C (,. (4 Equation (4 implis that th LCT with ral paramtrs is unitar. Associatd with th unitarit proprt is th gnralizd Parsval s rlation: ( g * * ( d = ( g( d (5 whr ( and g ( ar th LCTs o two arbitrar unctions ( and g(. In th spciic cas ( = g( (5 rducs to Parsval s rlation (nrg consrvation: =, whr dnots th rgular norm o ordr. ( ( Th LCT obs th associativ law; i.., ( C C C ( C C C =. This implis 3 3 that a cascad o quadratic phas sstms can b arbitrar rarrangd. Som usul proprtis o th LCT ar summarizd in Tabl. 3. GEOETRIC INTERPRETATION Usul insight on th opration o th LCT can b gaind b amining its ct on th Wignr distribution. Th Wignr distribution o ( is dind b [3],[5],[]: ξ ξ ( ω = + ξω W, dξ π. (6 It can b shown [3],[5], [3] that th Wignr distribution o ( is rlatd to that o ( b: W ( W [ d b c a ], ω = ω, + ω. (7 Th maning o (7 is that th LCT prorms a homognous linar mapping in th Wignr domain: a b =. (8 ω c d ω

4 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 TABLE. Proprtis o LCT. Th sign * in proprt 8 dnots convolution and in proprt 9 dnots corrlation. ( ( (- ( ( ( 3 ( / k ' a b / k k ( / k ' = ck d 4 ( 0 c 0 ac 0 ( a 0 5 ω0 ( dω 0 0 dbω ( bω 6 n n ( d d + b ( d 7 n n n d ( d n c a ( d d 8 { g} ( c a ( a g( 9 { g} ( ( d c( a + ( + a g * ( d 0 Eampls o th linar mapping (8 prormd b som spcial cas o LCT ar dpictd in Fig.. Th mapping (8 prsrvs th ara o th support in Wignr domain impling th consrvations o dgrs o rdom undr LCT, which is consistnt with th act that th LCT is invrtibl. Finall it is notd that th abov dscribd gomtric ct o th LCT in th (,ω domain holds not onl on Wignr distribution but on almost an Cohn class rprsntation [7]. 4. SAPLING OF THE LCT In this and th ollowing sction w considr onl LCT with b 0. This is th mor intrsting cas. As can b sn in quation ( th LCT with b=0 basicall scals th paramtr ais. Sampling is cntral in almost an domain bcaus it provids th link btwn th continuous phsical signals and th discrt tim domain. Sampling o LCTd signals was invstigatd in [4] and [5]. In [4] is dmonstratd that i a unction ( has a compact support such that (=0 or >, thn its LCT ( can b actl L

5 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 rconstructd rom its sampld vrsion T ( at points n =nt, n Z b th ormula: π ( = O rct( O { T } ( ( ( (9 ω L i th ollowing sampling condition is ulilld. π b T. (0 L On ma vri that or LCT with paramtrs {0,,-,0} implmnting a FT, condition (0 rducs to th Nquist sampling critrion and (9 prorms a rconstruction b appling a low-pass iltr in th rciprocal domain o (. Altrnativl, ( can b rconstructd using th intrpolation ormula [5]: [ ( ] T n d sin π d ( nt b b ( = ( nt. ( n Z π ( n T It can b shown that prviousl dvlopd sampling thorms or FT (Shannon sampling thorm, FRST and FRFT ar spcial cass o th LCT dscribd abov [4],[5]. ω ω (a (b ω ω (c (d Fig. Eampls o th ct o th LCT on th support o WD: (a FRST, (b FRFT, (d FT.

6 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun DISCRETIZATION OF THE LCT Lt us considr th sampld LCT signal ( b an impuls comb with intrvals T : ( = T ( k = δ obtaind b a multiplication o ( ( kt B using th wll known Poisson ormula or Dirac s comb ω kω π δ ( nt =,whr ω =, togthr with proprt 5 in Tabl, it n= π k = T is as to show that th ILCT o T ( givs [4] O p T ( = O n= ω π anω B dnoting p ( { T ( }( = T ( C ( kt, ( n= ( / ab P kω k= ( ω = π ( nω ( + nbω. n= anω π b =, (3 can b writtn as: T = ( / ab a a a mp ( mp P b b = ( + mp = b π b m= T m= ( nω O { ( }( + nbω a b ( mp = ( mp (4 W can s that th ILCT o T ( ilds a chirp-priodical signal b orming modulatd shits rplicas o th original signal (. Each rplica is multiplid b a constant and linar phas trm dpnding on th ordr o th rplica m, P and on th ratio a/b. Th rlation btwn T ( and its ILCT p ( is illustratd in Figs. (c and (d. Not that i ( has a compact support such that (=0 or > and i P ( = or P /. Thror, i L P, which is T quivalnt to th sampling condition (0, thn ( can b prcisl rconstructd rom th zro ordr rplica o p (. This obsrvation lads to th rconstruction ormula (9. L thn ( p L ( (3.

7 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 ( A ( Ar{ (} /T (a P ( P (b A T ( Arg{ ( } P T T Arg{ ( } (c (d T ( P A/T P ( T P Arg{ ( } ( ( P T ( P P T /T P T ( T T (g (h Fig.. (a Th unction ( and its ILCT (b, (c th ILCT o th sampld LCT (d, ( th sampld ( and its ILCT, ( sampld and priodicall rplicatd ( and its LCT (h

8 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 From similar considrations lading to (4 w ind that i w sampl th unction ( at intrvals T : ( T = ( δ nt n= T ( ( (5 thn th LCT o th sampld vrsion o is dual to (4, i..: d d ( lp b b P ( = ( nt C (, nt = ( lp. (6 n= T l= π b whr P =. Th rlation btwn th pair T ( T and P ( is illustratd in Figurs ( and (. Thus w s that sampling in domain causs chirp- priodicit in domain. For numrical calculation o LCT pairs w nd to considr sampling both and domains. Combining th rsults in Eqs. (4 and (6 togthr with th rsults in sction 4, w s that a signal sampld at rat /T and chirp-priodic with a priod P is a CPLCT priodic with a priod P dind on discrt valus o with intrval T, whr π b π b T = and T =. (7 P P This rlation is illustratd in Figs. (g and (h. Not that th numbr o sampls in P P P ach rplica in th domain [Fig. (g] is N = =, which is th sam as th T π b P P P numbr o sampls in ach rplica in domain; N = = = N. For utur us T π b w din N = N = N = N +. Th sampling- (chirp priodical rplication proprt shown in Fig. is rminiscnt o th sampling-priodical rplication o th Fourir pair. Thror, it is natural to din th discrt linar canonical transorm (DLCT pair in analog with th dinition o th discrt Fourir transorm: N i, T, T N T, T N [ a( nt nktt + d ( kt ] b [ k] = [] n, (8 T πb n= N and N i T, T N, T, T N [ d ( nt nktt + a( kt ] b [] n = [] k. (9 T πb n= N From Eqs. (4,(6 and Fig. w s that i (=0 or > L and L <P, and i its CPLCT ( is ε-concntratd in th rang /, i.., onl a small raction ε o its nrg is out this rang, thn ( P ( = or > L and T p

9 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 p ( ( or P /. This implis that i ( has compact support and i T th aliasing in domain is ngligibl, thn th DLCT pair dind b (8 and (9 is a good approimation o th o th CPLCT pair at discrt valus, that is:, T, T N kt k, (0 ( [ ] T, T N ( nt [ n]. ( Thus, or compact signals and ngligibl aliasing in th rciprocal LCT domain, (0 and ( ma b usd or numrical calculation o th LCT dind in (. As a inal not w point out that ast algorithms ma b usd to calculat th DLCT dind in Eqs. (8 and (9. On such wa is via th ast Fourir transorm [3]. Th othr is b appling a ast algorithm or transorms with quadratic krnls such as dvlopd in [6]. Although th ast LCT algorithm in [6] is dvlopd or a slightl dirnt DLCT krnl, it can b asil adaptd to th DLCT dind hr. 6. CONCLUSIONS In this papr w point out th usulnss o th LCT and giv a sktch o its thor. W hav summarizd important proprtis o th CPLCT and prsntd nw rsults rgarding th discrstization o th LCT. W impl that mor broad awarnss o th LCT and its proprtis could b o grat bnit, as it provids a uniid mathmatical tool o man commonl usd tools. As such, it givs a mor gnral prspctiv on man spciic mathmatical tools and a bttr insight to thm. orovr, it hlps to carr ovr knowldg rom on ild to othr and avoids rdiscovring th sam thoris in dirnt ilds ust bcaus dirnt trminolog is usd. For ampl, thoris o Frsnl transorm studid or optical ild propagation, thoris dvlopd or quadratur-phas iltring usd in signal procssing and or puls comprssion, thoris dvlopd or chirp puls propagation in radar, in optical communication thor and or cho bat propagation, and thor o th imaginar Gauss-Wirstrass transorm studid in mathmatics, phsics and hat thor, wr dvlopd sparatl although th ma b uniid as an LCT with paramtrs (,b,0,. So, atr pointing out th bnits o LCT w rturn to th qustion in th titl: wh is th LCT still so littl known? On possibl rason is that th LCT can b prssd in trms o th Fourir transorm; it can b writtn as suitabl iltring oprator on th Fourir transorm o som suitabl modiication o th signal. But isn t th prrnc o such a point o viw biasd rom our gnral tndnc to analz automaticall vr problm through Fourir glasss, with which ar w usd to? Also, on ma argu that prssing th LCT in trms o oprations on Fourir transorm is in concordanc with th gnral approach with which complicatd things ar undrstood b dcomposing thm into lmntar ons. Indd this approach is gnrall usul. But somtims th opposit approach ma b mor advantagous; i.., idntiing simplr things (.g. Fourir transorm, Frsnl transorm tc. as a subclass o som mor complicatd on (.g. LCT. For instanc a chair ma b bttr dind as a tp o urnitur usd to sit on rathr than as a crtain combination o wood pics, scrws and nails.

10 AIP Conrnc Proc., 5 th Int. Workshop on Inormation Optics, Vol. 860, pp. 5-34, Jun. 006 REFERENCES. F. Gori, Wh is th Frsnl transorm so littl known? in Currnt Trnds in Optics, J. C. Daint, d. London, Acadmic, 994, pp oshinsk and C. Qusn, J. ath. Phs., ( 8, ( H.. Ozaktas,. A. Kuta, and Z. Zalvsk, Th Fractional Fourir Transorm With Applications in Optics and Signal Procssing. Nw York: Wil ( K. R. Wol, Intgral transorms in scinc and nginring, NY, Plnum Prss, 979, chps. 9,0. 5. S.C. Pi, and J. J. Ding, IEEE Trans. on Signal Proc, 49 (8, ( D. F. V. Jams and G. S. Agarwal, Opt. Comm.., 6, 07 ( L.. Brnardo, Opt. Eng., 35,(3, ( S. A. Collins, J. Opt. Soc. Am. A, 60, ( Nazarath and J. Shamir, First-ordr optics a canonical oprator rprsntation: losslss sstms, J. Opt. Soc. Am. 7, pp , S. Ab, J.T. Shridan, Opt. Ltt., 9, ( J. Bastiaans, J. Opt, Soc, Am. 69, (979.. E. Sigman, Lasrs, Caliornia, ill Vall, Hnnll and J. T. Shridan, J. Opt, Soc, Am. A.,, 97-97, ( A. Strn, Signal Procssing (publication o EURASIP, 86 (7, 4-45 ( A. Strn, Sampling o compact signals in th ost linar canonical domain submittd to publication 6.. Hnnll and J. T. Shridan, J. Opt, Soc, Am. A.,, (005.

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals