# AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION

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1 AN ALGEBRAIC APPROACH TO -BAN WAELETS CONSTRUCTION Toy L Qy S Pewe Ho Ntol Lotoy o e Peeto Pe Uety Be 8 P. R. C Att T e eet le o to ott - otool welet e. A yte of ott eto ote fo - otool flte te olto e o S Sl le eooto eeloe to ele to oe ele welet e of e let. Alo te oety of oet tete to o welet otto oe w oe ote wy to ote -el - flte. Key Wo Welet tfo - welet le o oet. Itoto It well ow tt - otool welet ffe fo eee ott oto otl yet le-e otool welet o ot et. Botool welet ltwelet - welet e ee ltete fo oe feeo te. T e eote to - welet otto. Te e of - le e flte ee ette []. Sl wo w le [] t e o te-o ott oly fo. I [] [] flte of K oet o tte K- el fte let e otte. Sl flte e ote e o te tooelto o l flte oto. Te t eteo y olye eooto o tte-e teto e to ott te - welet flte to ot ely etee y te l flte le te - e. Botool - flte e ee lft ee []. Follow te e eete [] fo - otool welet t e le o ooe to ott - welet y ol te ott eto. I ott wt Foe-e eto fo o few elete flte o le o e te elt fol wt oe fee ete to oe ele welet e o w we f ote oe fo ef lto. Te et of t e oe follow. Seto ef eew of le o o - otool welet otto. Seto toe oe o owlee of - welet. O le eto eete Seto to ott - otool welet e. I Seto oto of two oet e ee tete to o otto eto wt te et elt e te well-ow ee l flte. Te olo e e Seto.. Reew of te le o fo - otool welet eote te et of ll tee ϕ ϕ e two l fto tfy te ole l eto ϕ ϕ ϕ ϕ Te oeo welet fto e ϕ ϕ wee. Ully we te low- - oeffet f I to te le e oety ee te otool oto oe o ϕ ϕ e elet to δ δ wee δ we δ otewe. Ge ete l { } wt fte eey te ete otool welet tfo

2 eooe { } to { } { } te ee tfo eott { }. Teefoe te efet eotto PR oto δ w oe to e elet to δ Coeetly yte of te ott eto of otool welet e ee δ Coe te - t e ele let we e te ott eto yte : ; If we te fee ete te te olto Fo. we et te - t ee otool flte. I l e we et te 9- t ee otool flte.. Bo tel o - welet e fe ote tee. A fly of loe e e tele fto e L lle lteolto of L f te follow oto ol: f f oly f f fo ll ; U ee L {} I ; { } ; otool of fo oe. Te fto lle - l fto tfe te efeet eto wee { } tfe. It ey to ee tt > < R δ Aetly te otool of } { / W e te otool oleet of : W It e oe tt tee et L tt / } { eoe te otool of W. > < R te L te low - oeffet we e: δ oeoe te otool oety ee tt R ξ π ξ ˆ δ δ Ge ete l { } wt fte eey te eooto eotto fol e efe H. Fo tee K we y H K oet o H K-el f tee et olyol Q tt Q H K Fo fe ote tee K < te efto of oet K elet to y of follow tteet: ˆ π... ˆ π... e π Wt e to H we ow tt H t let oet.. Cotto of - welet wt otly ote let L

3 L e fe ote tee. Fo lty we oe L ftly wle eel e. tete lte. oe eely we oe { }. e : L I oe to etl te yte of ott eto o { } efet eotto oto e eee tte Teoe. Teoe. Pefet eotto oto of { } fo tfo e δ Poof: [ / ] fo we e te To ee efet eotto follow eto ol e tfe: δ w e elet to. Teoe. Te ott eto et fo { } [ ] A B [ ] A B AB A A B B AA BB I wee A B e two te A B. Tt A B It e ely oe wt eto. Te otto of - flte { } ee to ole te ott eto et. Be o Sl-le-eooto S we e Teoe. Soe ol t wee o otool.e. el ty t U [ α α α α ] wee α te -t ol eto ote otool t [ β β β β ] wee 8 β α β te ft eleet of β let I te A U B U e te olto of eto. Poof: Ay t A e wtte S fo A U wee U e ot otool t U U I I U toe of U ol t wt o-ete ol ete. U [ α L α ] [ β L β ] L te we e A U ' α β 9 wee α β e -eol ol eto. Teefoe t ey to ee feely lly ee oe f we ee te oeo ol U lteoly. A U B U. Clely f tfy I te A B e te olto of.. Fo I we et. Wt e to we e. Teefoe o fo I. Wt oe elt oto e folte [ ] U [ ] U [ ] we ot [ ] U [ ]. Te we e β α w ow tt te ft ow of etee y U. Note tt te ft ow of ee ole to e β α [ α α α α α α α α α α α α ] α α α α α α α α α α α α Te eto to e te flte otte. Ftly et o fo I. Seoly ooe oe otool

4 t U [ α α α α ]. Flly et te ft ow of to eet 8 wt ote tee ow etee y St otoolto oe we e A U B U to e te olto of te eto. Oe eto : oe teoe e ll te olto fo? Te we ye Teoe followe. Teoe. Fo te A B te S A U B U f A B e te olto of te eto yte te I otool te U U tfy Q U U wee A eote te of A otool te tfy P wee. Poof: Wt e to A B O we et A B A B. Fo A A B B I we et A B A A B B. Atlly we e A A A y S. Teefoe A B B. A U B U ot S fo te we te te l te L L L L >. Fo I A A B B we et. Set P we e P P P o P P I. Hee P P P P So. Slly we e P P I to ole tt. T. < Fo O A B U U we e U U O Set Q U U. U otto of t lo. Q Q Q Q Q Q Q It ole tt Q o. Fo A B O we e P O. Slly we et. Bee t ole tt e P P Alo fo P P I we et wt P P I y. Alo we ee to ee te elto etwee U U. Fo P we et P P P P It ole tt te ft ol of e otool to te lt ol of. Slly te ft ol of U e otool to te lt ol of U. Fol 9 ow tt A oly elte to te ft ol of U B oly elte to te lt ol of U. Teefoe ow to ooe te lt ol of U e o ffeee. Ptlly we ol ooe U U wle P Q I.. e : L fo y I t e te ott eto yte tll t wt eo. Alo te olto oe e l. Te le fto otol te eo of te ott eto et olto. So we oe te ffeee lte.. e : L> I t e te ott eto et ot of L te eote { A A L A L }. Fo ele we L te ott eto e [ ] A A A A [ ] A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A I wle e to e foe y ft teeto of two eee of { A A L A }. To Teoe ot l fo t e we tll et tl olto y te wy ee Teoe. Tt te e U e eloye te S fo fo ll te { A A L A }. A U. It ey to ee tt tfe f

5 I It ole tt te ol ete of e o e t oe o te ol to o A. I l e 8 ee to eet. Te ltto of t eto oe A to e we L o oe eel olto e ele to ott loe flte we L.. oet fo L. Cott fol fo l flte Sl flte { } o { } fo lty oet ee: Fol lwy te e y l flte t let oet. Fol ft-t f te ft oto eoe te: Fo L fol elet to Te we e If te fee ete we et. Sl flte otto U A U B efe Teoe I wee o fo 8 we e Te l flte.e. te ft ol of A B e wtte wt : 8 Ce : If ttte 8 to we et [ ] [ ] [ ] 9 Te we e Bee δ we e Te t ole tt te

6 We e lfe eto yte ol : 8 To ole t we f fo oltow e ow : Tle. Te we ote l flte { } Fo fo olto of owee elt of { } e te e te ft ol of Tle eet fo te ffeee te eee oe. Ce : α β. Fo 9 we et α β α β α β Sl flte e ote fo te lt two ol of U: α β We ote te lt two ol of U to ot l flte. Howee t ee tt te l flte ote e te e te ft ol of Tle. Ce : Sl to e we et te 8 Sl flte { } e ote follow: I le e o ew l flte e fo fo t e.. Colo T e e le o to ott - otool welet e w ele to ot ele welet e fo eleto. Alo we e ote wy to ote -el - flte. O fte wo to ote loe flte to tete te le-e oety f ole to f ette flte fo welet-e e o. Refeee [] A. K. So P. P. yt T. Q. Nye Le e ty flte : teoy ftoto e IEEE T. Sl Poe [] O. Al H. Cl e of effet - oe wt le-e efet-eotto oete IEEE T. Sl Poe ol. No. Jly 99. [] P. N. Helle R welet wt N oet SIA Jol of t Al. Al [] N. B X. R. Q.Y. S Cotto of otly ote - welet Ale Cottol Ho Aly []. Lo Botool -B Flte Cotto U te Lft See Nel Alot ol [] Q. Y. S Botool welet teoy tee fo e o Po. SPIE 998 ol Tle : olto of e Tle : flte fo L wt oet

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### Physics 232 Exam II Mar. 28, 2005

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