Computational Intelligence and Application of Frame Theory in Communication Systems

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1 America Jural Eieeri ad Applied Scieces Oriial Research Paper Cmputatial Itelliece ad Applicati Frame Thery i Cmmuicati Systems Rajupillai, K., S. Palaiammal ad 3 K. Bmmuraju Departmet Mathematics, Gvermet Cllee Techly (Autmus, Cimbatre, Tamil Nadu, Idia Pressr ad Head, Departmet Sciece ad Humaities, Sri Krisha Cllee Techly (Autmus, Cimbatre, Tamil Nadu, Idia 3 Deparrtmet ECE, Gvermet Cllee Eieeri, Sriraam, Tamil Nadu, Idia Article histry Received: Revised: Accepted: Crrespdi Authr: K. Rajupillai Departmet Mathematics, Gvermet Cllee Techly (Autmus, Cimbatre, Tamil Nadu, Idia rajupillaimathsct@mail.cm Abstract: I this study, we have t discuss the reducti ise due t peak amplitude ad pwer rati i multi carrier mdulati scheme like Orthal Frequecy Divisi Multiplei (OFDM prcess usi Frame thery. The rame peratr the rame which is psitive, sel adjit, ivertible ad it cmmutes with sythesis peratr. I X is dual rame i H with rame peratr S ad aalysis peratr T, the T is quasi rmal peratr. I Y is dual rame r X, T is quasi uitary peratr. I T ad Q are pseud iverse ad sum its iverse with its ad jit multiplicati is rame peratr, the X is Bessel s sequece ad dual rame. Keywrds: Hilbert Space, Baach Alebra, Frame, Dual Frame, Quasi Nrmal Operatr Itrducti Frames were rmally deied i Hilbert spaces by Dui ad Schaer (95 t deal with harmic Furier series. Ater a cuple years, rames were bruht t lie Daubechies et al. (986, i the ctet Pailess rthal epasis ad Peter G. Casazza ad their rame thery research cetre discussed (Casazza ad Christese, 997; Casazza, 000; Casazza ad Christese, 997; Obeidat et al., 009. Frames are eeralizatis rthrmal basis. The liear idepedece prperty r a basis which allws each elemet i the space t be writte as a liear cmbiati ad this is very restrictive r practical prblems. A rames allws each elemet i the space t be writte as a liear cmbiati the elemets i the rames, here liear idepedece betwee the rames elemet is t required. This act plays imprtat rle i sial prcessi, imae prcessi, cdi thery ad sampli thery. Prelimiaries ad Ntatis Let H be Hilbert space ad L(H be a set all liear buded peratrs H. We ca deie the llwi peratrs: { } T: I H, Ta a, r a a I Is called sythesis peratr r pre rame peratr ad the ad jit peratr is ive that: { } T H I T :,, Is called the aalysis peratr. The cmpsiti peratr T with its adjit T is deted by: S T T e., S: H H, S, r H is called the rame peratr. Every rame at least has a dual.a dual rame which is t the caical dual rame is called a alterate dual rame. Bere i t deiiti Stable ad ustable, let us deie buded ad ubuded sial r rames. I the sial is buded, the its maitude is always be iite e., m, therwise ubuded. A system is said t be ustable i the utput the system is ubuded r buded iput. A system is called Stable i the utput system is buded r every buded iput r BIBO stable. We bei with rame deiitis. Let H be separable Hilbert space with the ier prduct, liear i the irst etry ad all ide sets are assumed t be cutable. 05 Rajupillai, K., S. Palaiammal ad K. Bmmuraju. This pe access article is distributed uder a Creative Cmms Attributi (CC-BY 3.0 licese.

2 Rajupillai, K. et al. / America Jural Eieeri ad Applied Scieces 05, 8 (4: DOI: /ajeassp Deiiti 3. Let H be separable Hilbert space ad a sequece { } H is called a rdiary rames. I there eist cstats A, B >0, such that:,, A B r all H Deiiti 3. Let H be separable Hilbert space. A sequece { } H is called a Bessel Sequece. I here eists cstat B >0, such that, B, r H. Deiiti 3.3 Let H be Hilbert space, the: A sequece { } i, r H A sequece { } r { } S Therem 4. is called a dual rame r { } is called a caical dual rame i, r H Let H be separable Hilbert space ad the rame peratr S the dual rame r { } is psitive, sel adjit, ivertible ad cmmutes with sythesis peratr, the: T is quasi rmal peratr i Hilbert space H I { } is dual rame r { } Pr uitary peratr We kw that (Gavruta, 0:, the T is quasi { } i T: l H, Ta a, r a a l Is called sythesis peratr r pre rame peratr ad the ad jit peratr is ive that: { } T H l T :,, Is called the aalysis peratr. The cmpsiti peratr T with its adjit T it deted by S TT, e., S : H H, S, r H is called the rame peratr. Sice T is buded liear peratr: TS T, (, T, w, (, TS T, Frm (. ad (., we have TS ST: e. TTT T TT Therere T is quasi rmal peratr.. Sice { } is dual rame r { } Frm (.: Frm (.: S we have:, TS TS I, ST TS I TS ST I (. (. Therere T is quasi uitary peratr i Hilbert space H. Therem 4. Let H be separable Hilbert space ad let P be a rthal prjecti, { } i ad ly i { } H is a dual rame r H H is a dual rame r P(H, with lwer rame bud A ad upper rame bud B ad S q is rame peratr. Pr Sice P is a rthal prjecti i Hilbert space H ({ } { } { } P P Hece the pr. is a dual rame. 634

3 Rajupillai, K. et al. / America Jural Eieeri ad Applied Scieces 05, 8 (4: DOI: /ajeassp Prpsiti 4.3 Let H be separable Hilbert space ad S q is quasi rmal rame peratr which is buded liear, the T(TT ad (TTT have the same zer Eie value. Pr Fr: Nw: H,0 TH suchthat T T T λ adt TT µ λ µ, λ µ, λ, µ, T T T, T TT, 0 We have: ( (, T, TS, T,, Sice{ } Ad:, T,, c, TS, K, is rame i H: TS, K, KB TS, K, KB Sice S q is quasi rmal rame peratr. Therere λ µ, Hece the pr. Therem 4.4 Let H be separable Hilbert space ad TH is reductive quasi similar t quasi rmal peratr ad S is rame peratr the rame { } rmal peratr. Pr Fr each H, lim m ( m H. The T is quasi ad S T which is reductive quasi similar t quasi rmal peratr. Nw: ( m ( S ( S S ( S S S ( T TT m m T TT, lim T TT, T TT, ( e T TT T TT, 0 ettt m m m m m T TT Therere T is quasi rmal peratr. Nw: There eist cstats B KB<ad A KA> 0, we have: Ad: Therere: TS, B TS, A A TS, B where, lwer rame bud A ad upper rame bud Bwith rame peratr S q TS Lemma 4.5 Let H be Baach alebra ad let H be the set all ivertible elemets H, r H ad h H'; with h < /, the: Ad: Therem h H ( 3 + h + + h h Let H be separable Hilbert space. I T ad Q are pseud iverses ad (T + Q (T + Q is rame 635

4 Rajupillai, K. et al. / America Jural Eieeri ad Applied Scieces 05, 8 (4: DOI: /ajeassp peratr, the { } i rame i Hilbert space H. Pr is Bessel s sequece ad dual Let T ad Q be pseud iverse i H (Di, 003: ( ( + + T Q T Q, ( ( T + Q T + Q, ( T Q, ( T Q ( T + Q TQ, ( T + Q TQ + + T + Q Q + Q TQ, T + Q Q + Q TQ (( T + Q + Q + Q TQ, (( T + Q + Q + Q TQ (( T Q Q Q TQ (( T Q +Q + Q TQ By lemma 4.5: 3 Q T B e T + Q ( T + Q, B e T + Q T + Q, e, Therere { } Prpsiti 4.7 is dual rame the rame { }. Let H be Hilbert space ad T ad T be sel ad jit peratrs i H, the: S T T 0 is sel adji peratr I T ad T are shit ivariat peratrs i H I the sequece (a ah, the the rame peratr S is stable Pr T ad T are sel ad jit peratrs i Hilbert space: { } { } :,, T l H T a a r all a a l T : H l, T,, r all H Fr every l, there is ad T. H such that T ( Sice T ad T are sel ad jit peratrs i H. We et S T T which is eative ad sel ad jit i H. T prve S is shit ivariat. Sice T ad T are shit ivariat peratrs i H: ( ( k ( k ( k T ad T k k k k T T k k k T T S Therere S is shit ivariat. Sice the sequece (a a i the Hilbert space which is buded (BIBO. Therere the system is stable. Therem 4.8 Let { } { } M ad N be subspaces Hilbert space H which is rthrmal basis ad i there eists aalysis peratr T, Frames with juk {R σ }H such that R σ τ + σ, r all τm, σn, the TR σ τ. This is recstructi riial irmati (Areijamaal ad Zekaee, 03. Cclusi We cclude that the mai prblem cmmuicati systems is ise, which is elimiated by rame thery peratr i the mdes liear, Shit ivariat ad rthal. The rthrmal Frames i Hilbert space used r reduce ise t received riial data. Frames play a imprtat rle t ly the theretic but als may applicatis i Eieeri ad Techly. Ackwledmet We wuld like t thak Pressrs Peter G. Casazza ad Lara Gavruta r brii t ur atteti their recet wrks rame thery. We wish t thak Pressrs S. Palaiammal, Sri Krisha Cllee Techly, Cimbatre ad K. Parthasarathi, Ramaujam Istitute advaced Study i Mathematics, Uiversity Madras r his several suestis. We urther thak the aymus reeree r very valuable suestis which imprve the paper. 636

5 Rajupillai, K. et al. / America Jural Eieeri ad Applied Scieces 05, 8 (4: DOI: /ajeassp Authr s Ctributi K. Rajupillai: Mauscript described ad writi wrk. S. Palaiammal: Mauscript crrecti ad uid K. Bmmuraju: Discussi r techical cmmuicat Ethics Tday ise is biest prblem i cmmuicati system. The aalysis rthrmal peratr i quatizati errr r ise is similar t the aalysis quatizati errr r ise due t A/D prcess. Recstructi riial irmati which is elimiated ise. Reereces Areijamaal, A.A. ad E. Zekaee, 03. Sial prcessi by alterate dual Gabr rames. Applied Cmput. Hrm. Aal., 35: DOI: 0.06/j.acha Casazza, P.G. ad O. Christese, 997. Apprimati the rame ceiciets usi iite dimesial methds. J. Electr. Ima., 6: DOI: 0.7/ Casazza, P.G. ad O. Christese, 997. Perturbati peratrs ad applicatis t rame thery. J. Furier Aal. Applic.,.3: DOI: 0.007/BF Casazza, P.G., 000. The art rame thery. Taiwaese J. Math., 4: 9-0. Daubechies, I., A. Grssma ad Y. Meyer, 986. Pailess rthal Epasis. J. Math. Phys., 7: DOI: 0.063/ Di, J., 003. New perturbati results pseudiverses liear peratrs i Baach spaces. Liear Alebra Applic., 36: DOI: 0.06/S ( Dui, R.J. ad A.C. Schaeer, 95. A class harmic urier series. Tras. Am. Math. Sc., 7: DOI: 0.090/S Gavruta, L., 0. Frames r peratrs. Applied Cmput. Harm. Aal., 3: DOI: 0.06/j.acha Obeidat, S., S. Samarah, P.G. Casazza ad J.C. Tremai, 009. Sums Hilbert space rames. J. Math. Aal. Applic., 35: DOI: 0.06/j.jmaa