A FAMILY OF WAVELETS AND A NEW ORTHOGONAL MULTIRESOLUTION ANALYSIS BASED ON THE NYQUIST CRITERION

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1 A FAMILY OF WAVELETS AND A NEW ORTHOGONAL MULTIRESOLUTION ANALYSIS BASED ON THE NYQUIST CRITERION HM de Oliveira Member IEEE LR Soares TH Falk Suden M IEEE CODEC - Communicaions Research Group Deparameno de Elerônica e Sisemas - CTG- UFPE Laboraório Digial de Sisemas de Poência LDSP- UFPE CP Recife - PE Brazil hmo@npdufpebr lusoares@npdufpebr iagofalk@gocom ABSTRACT- A generalisaion of he Shannon complex wavele is inroduced which is relaed o raised cosine filers This approach is used o derive a new family of orhogonal complex waveles based on he Nyquis crierion for Inersymbolic Inerference ISI eliminaion An orhogonal Muliresoluion Analysis MRA is presened showing ha he roll-off parameer should be kep below /3 The pass-band behaviour of he Wavele Fourier specrum is examined The lef and righ roll-off regions are asymmeric; neverheless he Q-consan analysis philosophy is mainained Finally a generalisaion of he square roo raised cosine waveles is proposed Key-words- Muliresoluion Analysis Waveles Nyquis Crierion Inersymbolic Inerference ISI INTRODUCTION Wavele analysis has maured rapidly over he pas years and has been proved o be valuable for boh scieniss and engineers [] Wavele ransforms have recenly gained numerous applicaions hroughou an amazing number of areas [3 4] Anoher srongly relaed ool is he Muliresoluion analysis MRA Since is inroducion in 989 [5] MRA represenaion has emerged as a very aracive approach in signal processing providing a local emphasis of feaures of imporance o a signal [ 6 7] The purpose of his paper is wofold: firs o inroduce a new family of waveles and hen o provide a new and complee orhogonal muliresoluion analysis We adop he symbol := o denoe equals by definiion As usual Sinc:=sin/ and Sa:=sin/ The gae funcion of lengh T is denoed by Waveles are denoed T by ψ and scaling funcions by φ The paper is organised as follows Secion generalises he Sinc MRA A new orhogonal MRA based on he raised cosine is inroduced in secion 3 A new family of orhogonal wavele is also given Furher generalisaions are carried ou in secion 4 Finally Secion 5 presens he conclusions A GENERALISED SHANNON WAVELET RAISED-COSINE WAVELET The scaling funcion for he Shannon MRA or Sinc MRA is given by he sample funcion: Sha φ = Sinc A naive and ineresing generalisaion of he complex Shannon wavele can be done by using specral properies of he raised-cosine filer [8] The mos used filer in Digial Communicaion Sysems he raised cosine specrum wih a roll-off facor was conceived o eliminae he Inersymbolic Inerference ISI Is ransfer funcion is given by w < = cos w w < 4 w The "raised cosine" frequency characerisic herefore consiss of a fla specrum porion followed by a rolloff porion wih a sinusoidal forma Such specral shape is very ofen used in he design of base-band digial sysems I is derived from he pulse shaping design crierion ha would yield zero ISI he so-called Nyquis Crierion Noe ha is a real and nonnegaive funcion [8] and in addiion w l = l Z Furhermore he following normalisaion condiion holds: dw = We propose here he replacemen of he Shannon scaling funcion on he frequency domain by a raised cosine wih parameer Fig We assume hen ha = In he ime domain his corresponds exacly o he impulse response of a Nyquis raisedcosine filer Figure Fourier Specrum of he raised cosine scaling funcion The generalised Shannon scaling funcion is herefore: GSha cos φ = Sinc 3 In he paricular case = he scaling funcion simplifies o he classical Shannon scaling funcion As a consequence of he Nyquis crierion he scaling funcion presens zero crossing poins on he

2 unidimensional grid of inegers n=±±±3 This scaling funcion φ defines a non-orhogonal MRA Figure shows he scaling funcion corresponding o a generalised Shannon MRA for a few values of φ GSha Figure Scaling funcion for he raised cosine wavele generalised Shannon scaling funcion 3 MULTIRESOLUTION ANALYSIS BASED ON NYQUIST FILTERS A very simple way o build an orhogonal MRA via he raised cosine specrum [8] can be accomplished by invoking Meyer's cenral condiion [6]: w n = 4 Comparing eqn o eqn4 we choose Φ w = ie a square roo of he raised cosine specrum Le hen w < = cos w w < w 5 Clearly w n = so he square roo of n he raised cosine shape allows an orhogonal MRA The scaling funcion φ is ploed in he specral domain Fig 3 Figure 3 Specral Characerisic of he "de Oliveira" orhogonal MRA The cosine pulse funcion PCOS defined below plays an imporan role on he raised cosine MRA Definiion The cosine pulse funcion of parameers θ w and B is defined by w w PCOS w; θ w B : = cos w θ B θ w B R <B<w I corresponds o a cosine pulse in he frequency domain wih frequency and phase θ wih duraion B rad/s cenred a w rad/s Some ineresing paricular cases include: The Gae funcion: w = PCOS w; B B A Gae shifed by w w w = PCOS w; w B B 3 An infinie cosine pulse: cos w θ = PCOS w; θ B Denoing he inverse Fourier ransform of PCOS by pcos ; θ w B : = I PCOS w; θ w B he following resul can be proved Proposiion Given θ w and B parameers of a PCOS he inverse specrum pcos is given by: pcos ; θ w B = j w { [ ] [ ]} w θ j w w e Sa B e θ Sa B B 6 Proof Follows applying he convoluion propery for he following couple of ransform pairs: θ jθ [ δ e δ e ] cos w θ B e jw Sa j and B w w B I is ineresing o check some paricular cases: w; B PCOS w; B pcos B w Sa B B pcos B PCOS w; B ; [ δ δ ] cos w which follows from he propery of he sequence lim Sa = δ 7 ε ε ε Propery Time shif: A shif T in ime is equivalen o he following change of parameers: pcos T; θ w B = pcos ; T θ w B In order o find ou he scaling funcion of he new orhogonal MRA inroduced in his secion le us ake he inverse Fourier ransform of The specrum can be rewrien as a sum of conribuions from hree differen secions a cenral fla secion and wo cosine-shaped ends:

3 w = cos w θ B w cos w θ B wih parameers B := θ : = I follows from Definiion ha = PCOS w; B w B 8 : = and PCOS w; PCOS w; and herefore φ deo = pcos ; B pcos ; pcos ; Afer a somewha edious algebraic manipulaion we derive deo φ = Sinc[ ] { cos sin } 9 A skech of he above MRA scaling funcion is shown in figure 4 assuming a few roll-off values Sha lim Clearly φ = φ The low-pass H filer of he MRA can be found by using he so-called wo-scale relaionship for he scaling funcion [7]: w w w = H How should H be chosen o make eqn hold? Iniially le us skech he specrum of and w/ as shown in figure 5 The main idea is o no allow overlapping beween he roll-off porions of hese specra Imposing ha > i follows ha </3 remember ha << This is a simplifying hypohesis I is quie usual he use of small roll-off facors in Digial Communicaion Sysems w/ φ deo Figure 5 Draf of and w/ Figure 4 "de Oliveira" Scaling Funcion for an Orhogonal MRA = and /3 The scaling funcion φ can be expressed in a more elegan and compac represenaion wih he help of he following special funcions: Definiion Special funcions; ν is a real number Hν : = νsinc ν ν and ν ν ν Μ : cos sin ν = ν ν ν ν ν ν I follows ha Sha φ = H φ [ ] { } = H Μ w I is suggesed o assume ha H = Subsiuing his ransfer funcion ino he refinemen equaion eqn resuls in w = The above equaion is acually an ideniy for w > Ino he region w < i can be seen ha w = under he consrain </3 The orhogonal "de Oliveira" wavele can be found by he following procedure [7]: jw / * w w Ψ = e H Insering he shape of he filer H in he above equaion i follows ha: jw / w Ψ = e w 3 In order o evaluae he specrum of he moher wavele we plo boh w and w again under he consrain </3 Fig 6 In his case < and < /33 3

4 Defining a shaping pulse w S deo : = w Φ 4 he wavele specified by eqn can be rewrien as jw / Ψ = e S The erm e -jw/ accouns for he wave while he erm S accouns for le a w / furhermore ha making he wavele reduces o he complex Shannon wavele I is quie apparen from figure 7 he band-pass behaviour of he wavele Ψ Observe ha he lef and righ roll-off is no exacly symmerical Insead despie heir similar shape hey occur a differen scales a ypical behaviour of waveles S deo b w S deo /33 /33 c Figure 6 Skech of he scaling funcion specrum: a scaled version Φ w c simulaneous plo of a and b Φ w b ranslaed version From he figure 6 i follows by inspecion ha: if w < - w if - w < if w < S deo = w if w < if w 5 Insering he square roo raised cosine forma of resuls in: if w < - cos w if - w < S deo = if w < cos w if w < 8 if w 6 The complex "de Oliveira" wavele is given by jw / Ψ = e S and is modulo deo deo Ψ = S w is depiced in figure 7 Observe Figure 7 Modulo of he "de Oliveira" Wavele frequency domain The ime domain represenaion of he wavele can be derived by aking he inverse Fourier ransform: ψ = I Ψ Denoing by s S he corresponding ransform pair i follows ha ψ = s The shaping pulse can be rewrien as: S deo = PCOS w; 3 PCOS w; 3 PCOS w; Finally applying he inverse ransform we have s deo = pcos ; 7 3 pcos ; 3 pcos ; The pcos signal is a complex signal when here are no symmeries in PCOS The real and imaginary pars of he pcos funcion can be handled separaely according o pcos θ w B = rpcos j ipcos where ; : = Re pcos ; θ w B and = Im pcos ; w B rpcos ipcos : θ 4

5 Aiming o invesigae he wavele behaviour we propose o separae he Real and Imaginary pars of s deo inroducing new funcions rpc and ipc R eψ = Re s Im{ ψ } = Im s 8 Proposiion Le w : = w B and : = w B ; θ : = B w θ and : = B w θ be auxiliary parameers Then Re ψ rpc i i sen cos i { } i { = i icos sen } i ipc i i sen sen i { } i { = i icos cos } Proof Follows from rigonomery ideniies A his poin an alernaive noaion ; ; rpc = rpc θ and ipc = ipc θ can be inroduced o explici he dependence on hese new parameers Handling apar he real and imaginary pars of s deo we arrive a Re s = rpc ; rpc ; rpc ; 9 Applying now proposiion afer many algebraic manipulaions: s = { Η Η Μ Μ } R e = Re ψ = Re s / and The analysis of he imaginary par can be done in a similar way Definiion 3 special funcions; ν is a real number H ν cos ν : = ν ν and ν ν ν ν Μν ν The imaginary par of he wavele can be found by I m s deo = : = = i [ ] { sin ν } ν ν ν ν cos { Η Η Μ Μ } and Im = Im s / ψ The real par as well as he imaginary par of he complex wavele ψ are ploed in figure 8 for = and /3 Im ψ deo a b Figure 8 Wavele ψ : a real par and b imaginary par 4 FURTHER GENERALISATIONS Generally he approach presened in he las secion is no resriced o raised cosine filers Algorihm of MRA Consrucion Le be a real band-limied funcion = w> which saisfies he vesigial side band symmery condiion ie { w } = for w < Then he scaling funcion Φ w = defines an orhogonal MRA Proposiion 3 If w; is a Nyquis filer of roll-off and λ is an arbirary probabiliy densiy funcion << hen he scaling funcion Φ w = λ w; d defines an orhogonal MRA Proof I is enough o show ha w n = Given an ineger n hen w n = λ w n; d Taking he 5

6 square of boh members and adding equaions for each ineger n w n = λ w n; d Since w; is a Nyquis filer P w n; = and he proof follows The mos ineresing case of such generalisaion corresponds o a "weighing" of square-roo-raisecosine filer Corollary Generalised raised-cosine MRA If w; is he raised cosine specrum wih a coninuous roll-off parameer ie w < w; : = cos w w < 4 w and λ is an arbirary probabiliy densiy funcion defined over he inerval << he scaling funcion Φ w = λ w; d defines an orhogonal MRA 5 CONCLUSIONS This paper inroduced a new family of complex orhogonal waveles which was derived from he classical Nyquis crierion for ISI eliminaion in Digial Communicaion Sysems Properies of boh he scaling funcion and he moher wavele were invesigaed This wavele family can be used o perform an orhogonal Muliresoluion Analysis A new funcion ermed PCOS was inroduced which is offered as a powerful ool in maers ha concern raised cosines An algorihm for he consrucion of MRA based on vesigial side band filers was presened A generalisaion of he square roo raised cosine wavele was also proposed yielding a broad class of orhogonal waveles and MRA ACKNOWLEDGMENTS The firs auhor hanks Professor R Campello de Souza for his consrucive criicism REFERENCES [] CK Chui An Inroducion o Waveles San Diego: Academic Press 99 [] I Daubechies The Wavele Transform Time- Frequency Localizaion of a Signal and Analysis IEEE Trans Info Theory IT 36 Sep 999 pp96-5 [3] Waveles and heir Applicaions in Compuer Graphics Alain Fournier Ed Siggraph 94 Course Noes 995 fpcsucbca/pub/local/bobl/wvl [4] J Gomes L Velho e S Goldensein Waveles: Teoria Sofware e Aplicações Rio de Janeiro IMPA pp6 987 [5] S Malla A Theory for Muliresoluion Signal Decomposiion: The Wavele Represenaion IEEE Trans Paern Analysis and Machine Inelligence Vol n7 July 989 pp [6] Y Meyer Ondelees e opéraeurs Hermann Paris 99 [7] DB Percival e AT Walden Wavele Mehods for Time Series Analysis Cambridge Press pp 594 [8] KS Shanmugam Digial and Analog Communicaion Sysems Singapore: John Wiley 985 6