Fourier Eigenfunctions, Uncertainty Gabor Principle And Isoresolution Wavelets

Transcription

1 XX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES-SBT 0, DE OUTUBRO DE 00, RIO DE JANEIRO, RJ Fourir Eigucios, Ucraiy Gabor Pricipl Ad Isorsoluio Wavls L.R. Soars, H.M. d Olivira, R.J.S. Cira ad R.M. Campllo d Souza Absrac Shap-ivaria sigals udr Fourir rasorm ar ivsigad ladig o a class o igucios or h Fourir opraor. Th classical ucraiy Gabor-Hisbrg pricipl is rvisid ad h cocp o isorsoluio i joi im-rqucy aalysis is iroducd. I is sho ha ay Fourir igucio achiv isorsoluio. I is sho ha a isorsoluio avl ca b drivd rom ach ko avl amily by a suiabl scalig. Idx Trms Gabor-Hisbrg iqualiy, Fourir igucios, im-rqucy aalysis, isorsoluio avls. I. PRELIMINARIES Th Fourir rasorm is o irprd as a liar opraor F. A irsig problm i his ramork is o id ou h so calld igucios i h laguag o opraors -]. L V b a vcor spac quippd ih a liar rasorm, T:V V, v T(v). Udr h liar rasorm T, igucios ar soluios o T{v}λv, hich corrspods hr o F{()} λ.() or som L (R), λ a scalar. Thy ar a qui rmarkabl class o ucios, hich prsrvs h shap udr Fourir rasorm: Boh h sigal ad is spcrum (im ad rqucy rprsaio) hav h sam shap. I joi im-rqucy rprsaio 4, 5] his aur ca rprs a vry good balac b h o domais. I is ll ko ha h Gaussia puls is a sigal hos shap is prsrvd udr h Fourir opraor:. () This ca asily b drivd by riig j d F( ). Drivig his quaio ad usig igral by pars, id ou F'()-.F(). Th soluio o h dirial quaio F'()+.F()0 udr h iiial codiio F(0) is F( ). I ollos promply ha λ. L.R. Soars, H.M. d Olivira, R.J.S. Cira ad R.M. Campllo d Souza, Dparm o Elcroics ad Sysms, Fdral Uivrsiy o Prambuco, Rci-PE,Brazil, lusoars@up.br, hmo@up.br, rjsc@.up.br, ricardo@up.br. This papr is ddicad o Dr. Max Grk (i mmoriam). Th qusio is: Ar hr ohr igucios? This mar is addrssd i h x scio. I is orhhil o bar i mid ha som rsuls i his papr ar dlibraly o ova, sd ov. II. SHAPE-INVARIANT SIGNALS: EIGENFUNCTIONS OF THE FOURIER OPERATOR L E (rspcivly O) do h ucioal ha xracs h v (rspcivly odd) par o a sigal. Proposiio. L () F() b a arbirary Fourir rasorm pair. Th h sigal h( ( )] E F( )] ) : E + is ivaria udr Fourir rasorm. Furhrmor, { h ( )} h( ) F. I ollos rom h diiio o h(.) ha ( ) + ( )] + F( ) + F( )] h( ). () Takig h Fourir rasorm, F( ) + F( )] + ( ) ( )] H( ) + () ad h proo ollos. Corollary. Each v ucio () F() iducs a Fourir ivaria h ( ) ( ) + F( ). For isac, h sigals h ( ) + +, h ( ) hav spcra ih similar shap. A xra rmarkabl xampl is sc h sc h. (4) Proposiio. L () F() b a arbirary rasorm pair. Th h sigal h( ) : O ( )] O F( )]

2 XX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES-SBT 0, DE OUTUBRO DE 00, RIO DE JANEIRO, RJ is ivaria udr Fourir rasorm. Furhrmor, F Similar o proposiio. { h ( )} h( ). Corollary. Each odd ucio () F() iducs a Fourir ivaria h( ) ( ) F( ). L us o ocus o a paricular ad impora class o Fourir ivaria, hich gras a orhogoal ad compl s. To bgi ih, l us do by E g a class o igucios o h Fourir opraor did accordig o h olloig: Proposiio. A sigal () is i E g i h sigal saisis h dirial quaio ''()- ()κ(), or som scalar κ C. ( ) ( ) λ ( ) (hypohsis) Th propris o im ad rqucy diriaio or F giv: ''() (j) λ(), (-j) () λ''(). (5) Addig boh mmbrs, driv ''()- () λ''() - ()]. (6) Thus, h sigal ''()- () has also is shap prsrvd, providd ha isl prsrvs is shap. Thror, ''()- () E g, ha is, ar lookig or sigals such ha ''()- ()κ(), sic hy hav idical igvalus. ( ) Th sigal () saisis h dirial quaio ''()- ()κ(), k C. (hypohsis) By akig F, (j) F()+F''()κλF(), (7) so ha F''()- F()κλF(), i.., is spcrum also obys a similar dirial quaio. Thror, ad F hav idical shap, sic hy ar soluios o h sam dirial quaio. Th ky quaio or shap-ivaria sigal is hus ''()- ()κ(). L us ry soluios o h orm ( ) N.B. Subracig: ''()+ () -λ''() + ()]. p( ).. (8) Thror, ] " p( ). p( ). κ p( ).. (9) Ar simpl algbraic maipulaios, driv + ] p( ). p" ( ) p ( ) κ 0 (0) A sadard dirial quaio o h abov orm is 6] p " ( ) p ( ) + p( ) 0, igr. () Thus, or a suiabl choic κ ( + ) (igvalus), h soluios p() ar xacly Hrmi polyomials () 6], hich orm a compl orhogoal sysm. p( ) H ( ). () (H 0 (), H (), H ()-+4, H ()-+8, H 4 () , c.) Proposiio 4. Possibl igvalus o h Fourir rasorm ar h our roos o h ui (±,±j) ims. L us do by F () h opraor corrspodig o ira ims h opraor F. L ' Ω b h Fourir domai or h ira Fourir rasorm. Obsrv ha ( E g ) F () {()}(-') ad F (4) {()}4 (Ω). () Bu F () {()}λ (') ad F (4) {()} λ 4 (Ω). (4) From () ad (4) i ollos ha W coclud ha { } ( ) : H ( ). λ C has ordr 4. 0 ψ ar shapivaria udr Fourir opraor associad o ) λ ( j. Thror, H ( ). ( j ) H ( ).. (5) Aohr irpraio ca b drivd vokig Rodrigus' ormula 6]: d H ( ) ( ).. (6) d Th d-ordr dirial quaio hold by ivaria sigals is y''+(+-x )y0. (7) Th abov dirial quaio is xacly h clbrad Schrödigr quaio or h harmoic oscillaor 7].

3 XX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES-SBT 0, DE OUTUBRO DE 00, RIO DE JANEIRO, RJ III. CONSEQUENCES ON THE TIME-FREQUENCY PLANE L us o ivsiga crai cosqucs o igucios o h Fourir opraor o h im-rqucy pla 4, 8]. L () b a ii rgy sigal E, ih a rasorm F(). Th im ad rqucy moms o ar did by: ad : E : E ( ) F ( ( )d ) ( )d ( ) d, F ( ) F( )d + ( )F( )d F( ) d. By aalogy o Probabiliy Thory, h rm () /E dos a "im-domai" rgy dsiy, hr E is a ormalisig acor so as o mak h hol igral o h dsiy b qual o o. I is cusomary o dal ih h rgy spcral dsiy G()F(), hos igral ovr a rqucy bad givs h rgy co o h sigal ihi such a bad. L us suppos i h squl, ihou loss o graliy, ha E (rgy ormalisd sigals). Th "civ duraio" (rspcivly "civ rqucy idh") o a sigal () (rspcivly F()) is origially did via: : : ( ) ( ) r.m.s. duraio, (8a) r.m.s. badidh, (8b) ad corrspod o h sadard dviaio (i.., spradig masurs). Hovr, ohr commo ad much hadir diiios ar A. Rvisiig h Gabor Pricipl By applyig argums rom quaum mchaics 7], Gabor 9, 0] drivd a ucraiy rlaio oadays calld Gabor-Hisbrg pricipl or sigals:. /, provig ha im ad rqucy cao b xacly masurd (simulaously). Th Gabor-Hisbrg ucraiy pricipl sas a lor boud o h produc. (or. ).. (0) Proposiio 5. Th Gabor lor boud is oly achivd by h irs ivaria sigal (igucios o F opraor). Skch o h proo: From (0), h boud is achivd i '()k.(). This codiio ca b irprd as: 'Drivaiv i im domai' 'drivaiv i rqucy domai'. Thror ''()k()+.'()]k()+k () () or ''()- (k) () k(). () Ad so ''()-k(+k )()0. Th oly soluios o E g corrspod o k±, i.., "+(- )0 or "-(+ )0. Proposiio 6. Ay ral sigal () F() such ha, ', F, F' L (R) has ii rsoluios. Applyig h Parsval-Plachrl Thorm 6], i ollos ha ad ( )d F' ( F( ) ) ' ( )] d. d, d j ( )] j ( )] d jf( )] jf( )] () d (4) Clarly, : ( ) : ( ). (9),... ( ) ( ) Thror F' ( ) d <, (5) F( ) d ' ( ( ) d <. (6) ) d

4 XX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES-SBT 0, DE OUTUBRO DE 00, RIO DE JANEIRO, RJ is hror giv by h squar roo o h raio b h rgy o h sigal drivaiv ad h rgy o sigal isl. Thus, h rsoluio or h Fourir ivaria sigal sch(.) giv by q(4) is sic ha (7) sc h ( ζ ) dζ, gh ( ζ ).sc h ( ζ ) dζ, +. ζ sc h ( ζ )dζ. (8) Proposiio 7. 9]. Tim-rqucy ucraiy o Fourir Eigucios { ψ ( )} rach quaizd valus o h Gabor-Hisbrg lor boud, i.... (+),. (+). (7a) (7b) Tha is hy Gabor ucios ar rlva i som problms (.g. ]). IV. THE CONCEPT OF ISORESOLUTION WAVELET Th cocp o isorsoluio aalysis is iroducd i his scio. Accordig o h Gabor pricipl, i o icrass rsoluio i o domai, h rsoluio mus dcras i h ohr domai so as o guara h lor boud giv by (0). Wh aalysig sigals i joi im-rqucy pla, rquly, hr is o grouds o assur a br rsoluio i a domai ha i h ohr domai. As a irsig propry, ay Fourir igucio achivs isorsoluio as i ca b s by: Proposiio 8. Fourir-ivaria sigals prorm a isorsoluio, ha is,. Supposig ha E g, h F()λ (). Thror F( ) F ( )d F( ) d ad h proo ollos. λ λ ( )d ( )d This is a irsig propry or sigallig o h joi im-rqucy pla. ψ ) H ( )xp( ) xp( j + φ ) ( 0 0. I is suggsd hr h chagig o h im-rqucy rsoluio by a propr scalig ha allos or idical rsoluio i boh domais. Proposiio 9. I ψ() has civ duraio ad civ badidh, h h scald vrsio ψ ( ) achivs isorsoluio. Scald vrsios ψ(a), a 0, hav rsoluios /a ad a.,, so a ca b approprialy chos. Th ssial ida o isorsoluio ca b placd i h avl srucur. Normally, h basic avl o a amily b ψ holds h admissibiliy codiio bu o a a dos o achiv isorsoluio. W propos hr o rdi h basic avl o a amily so as o achiv isorsoluio. Tak as a modl h sadard Mxica ha avl, Mha ψ ( ), did by ( ) 4 4. (8) Th isorsoluio Mxica ha avl ca b oud applyig proposiio 9: 7 Mha 7 ψ (9) 5 5 For ay isorsoluio avl, h scalig by a> or a< corrspods o ubalac rsoluio i a dir ay. Tabl displays boh im ad rqucy rsoluio or ko coiuous avls: Gaussia drivaivs, Mxica ha, Morl, rqucy B-Spli, Shao ad Haar ]. Gaus is a ivaria avl hror i achivs isorsoluio, i accordac o proposiio 8. I is valuabl o mio ha compac suppor avls (i im or rqucy) cao aai isorsoluio, sic o sigal ca simulaously b im ad rqucy limid ]. TABLE. RESOLUTION OF FEW STANDARD CONTINUOUS WAVELETS. Wavl am Tim rsoluio Frqucy rsoluio Isorsoluio acor Gaus Mxica ha Morl bsp Shao Haar 0. -

5 XX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES-SBT 0, DE OUTUBRO DE 00, RIO DE JANEIRO, RJ V. PERSPECTIVES AND CLOSING REMARKS Eigucios o h Fourir opraor r ivsigad ad h Gabor pricipl as rvisid diig h cocp o isorsoluio, i., a sigal ih h sam im ad rqucy rsoluio. Th { ψ ( )} ucios (s q(5)) ur up as a vry appalig choic or dsigig rprsaios such as avls. I is im o ry idig avls sarig ih h quaio (). Sic hy ar soluios o a av quaio (d ordr dirial quaio), our approach (Mahiu 4], Lgdr 5], Chbyshv 6]) ca b usul o cosruc avls: Th Quaum Wavls, or Gabor-Schrödigr avls. Th cosrucio o avls basd o hs compl, orhogoal, domai shap-ivaria sysm is currly big ivsigad. Th ida is o adap h cocp o isorsoluio i orhogoal mulirsoluio aalysis 7, 8]. 4] M.M.S. Lira, H.M. d Olivira, R.J.S. Cira, Ellipic-Cylidr Wavls: Th Mahiu Wavls, IEEE Sigal Procss. Lrs, accpd, 00. To appar. 5] M.M.S. Lira, H.M. d Olivira, M.A. Carvalho Jr, R.M.C. d Souza, N Orhogoal Compac Suppor Wavls Drivd From Lgdr Polyomials: Sphrical Harmoic Wavls, 7h WSEAS I. Co. o SYSTEMS 00. 6] R.J. d Sobral Cira, L.R. Soars ad H.M. d Olivira, Filr Baks ad Wavls basd o Chbyshv Polyomials, 7h WSEAS I. Co. o SYSTEMS, 00. 7] H.M. d Olivira, L.R. Soars, T.H. Falk, A Family o Wavls ad a N Orhogoal Mulirsoluio Aalysis Basd o h Nyquis Cririo, Rv. da Soc. Bras. Tlcomm., Brazil, Ju, 00, o appar. 8] N. Hss-Nils ad M.V. Wickrhausr, Wavls ad Tim- Frqucy Aalysis, Proc. o h IEEE, vol.84,.4, April, pp.5-540, 996. ACKNOWLEDGEMENTS Th auhors hak Mr. M. Müllr or som moivaig discussio. REFERENCES ] I.N. Hrsi, Topics i Algbra, Blaisdll Pub. Co., Mass., 964. ] I.S. Sokoliko, R.M. Rdhr, Mahmaics o Physics ad Modr Egirig, d Ed., Tosho: McGra-Hill Kogakusha, 966. ] S-C. Pi ad J-J. Dig, Eigucios o Liar Caoical Trasorm, IEEE Tras. o Sigal Procssig, v.50,., Ja., pp.- 6, 00. 4] A. Coh, Tim-Frqucy Aalysis, Pric-Hall Sigal Procssig Sris, ] S. Qia ad D. Ch, Udrsadig Joi Tim-Frqucy Aalysis, IEEE Sigal Proc. Mag., March, pp ] M. Abramoiz ad I. Sgu (Eds.), Hadbook o Mahmaical Fucios, N York: Dovr, ] A. Bisr, Cocps o Modr Physics, McGra-Hill Sris i Fudamal Physics, ] P.M. Olivira ad V. Barroso, Ucraiy i h Tim-Frqucy Pla, Proc. o h Th IEEE Workshop o Saisical Sigal ad Array Procssig, Pocoo Maor, PA, USA, pp.607-6, ] D. Gabor, Thory o Commuicaios, J. IEE (Lodrs), vol.9, pp , ] D. Gabor, Commuicaio Thory ad Physics, IEEE Tras. Io. Thory, vol. IT-, pp , Fb. 95. ] K. Okajima, Th Gabor Fucio Exrac h Maximum Iormaio rom Ipu Local Sigals, Nural Norks, v., pp.45-49, 998. ] M. Misii, M. Misii, G. Opphim, J-M. Poggi, Wavl Toolbox, Th Mah Works, 00. ] J.M. Wozcra ad I.M. Jacobs, Pricipls o Commuicaio Egirig, N York: Wily, 967.