On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets *

Transcription

1 Applied Mathematics,, 3, Published Olie July ( O the Zeos of Daubechies Othogoal ad Biothogoal Wavelets * Jalal Kaam Faculty of Sciece ad Geeal Studies, Alfaisal Uivesity, Riyadh, KSA jkaam@alfaisal.edu Received Febuay 3, ; evised Apil 7, ; accepted May 4, ABSTRACT I the last decade, Daubechies wavelets have bee successfully used i may sigal pocessig paadigms. The costuctio of these wavelets via two chael pefect ecostuctio filte bak equies the idetificatio of ecessay coditios that the coefficiets of the filtes ad the oots of biomial polyomials associated with them should exhibit. I this pape, othogoal ad Biothogoal Daubechies families of wavelets ae cosideed ad thei filtes ae deived. I paticula, the Biothogoal wavelets Bio3.5, Bio3.9 ad Bio6.8 ae examied ad the zeos distibutio of thei polyomials associated filtes ae located. We also examie the locatios of these zeos of the filtes associated with the two othogoal wavelets db6 ad db8. Keywods: Othogoal; Biothogoal Wavelets; Biomial Polyomials. Itoductio The Daubechies wavelets costuctio, equies the fidig of a scalig fuctio t ad a wavelet fuctio t []. This costuctio is best descibed via a two-chael pefect ecostuctio filte bak [,] ad depeds o the distibutio of the zeos of some polyomials i the plae. The liteatue povides may theoems descibig geometic locatios of the oots of cetai polyomials [3,4]. Idetifyig ecessay coditios fo the coefficiets of the filtes associated with the costuctio is vital fo othogoality, Biothogoality ad alias cacellatio. The distibutio of the zeos of the biomial polyomial elated to the costuctio of these wavelets wee poved to eside iside the uit cicle [5], ad bette limits fo these oots based o a geealizatio of the Kakeya-Eestom Theoem wee deived i [6] whee it was show that if y is a oot of a biomial poly- omial of degee p, the: y. p A subclass of polyomials is deived fom this costuctio pocess by cosideig the atios of cosecutive biomial polyomials coefficiets essetials to these costuctios. We showed mathematically that the oots of this class of polyomials eside iside the uit cicle. I Sectio 3, the costuctio of Daubechies othogoal wavelets is detailed. I Sectio 4, the ew class of polyomials is i- * Dedicatio: This pape is dedicated to my pofessos ad supevisos Kal Dilche ad William Phillips. toduced ad the distibutio of thei zeos is examied ad mathematically show to eside iside the uit cicle. A example is the peseted illustatig the locatio fo the zeos of the deived polyomial of the othogoal mothe wavelet db6. The case of db8 is the examied i Sectio 5 ad esults ae obtaied. Sectio 6 descibe the coclusio of this wok.. Related Woks A two-chael filte bak has a low-pass ad a high-pass filte i the decompositio (aalysis) phase ad aothelow-pass ad a high-pass filte i the ecostuctio (sythesis) phase. Let H ad G deote the low-pass filte coefficiets ad the high-pass filte coefficiets espectively of the aalysis phase, the give the coefficiets of H, it is show i [,7] that the coefficiets of the filtes H, G ad G that lead to othogoality ca easily be deived fom the coefficiets of H. Theefoe, to costuct a Daubechies othogoal wavelet, all we eed to do is to fid the coefficiets of the filtes H associated with it. The distibutio of the zeos of a family of polyomials havig thei coefficiets as the atios of those of the biomial polyomials is cosideed i Sectio 4 ad poved to eside iside the uit cicle. Simila discussios about Daubechies Biothogoal wavelets family ae also icluded alog with the costuctios of Bio3.5, Bio3.9 ad Bio6.8. I the othogoal case, the scalig ad wavelet fuctios ae deived fom the coefficiets of the Copyight SciRes.

2 J. KAR 779 filtes H, H, G ad G. They must satisfy espectively the followig two equatios []: l t h ktk k l () W t g k tk () k Biothogoal filte baks poduce Biothogoal wavelets. This calls fo a ew scalig fuctio t ad a ew wavelet fuctio wt. Hee, oe eeds the follow- ig coditios: Hz H z ad Gz H z [8]. The wavelet filtes fo aalysis baks ae deived [] fom the scalig filtes usig the two elatios: h g (3) g h (4) The aalysis scalig ad wavelet equatios thus become: N t h kt k N (5) wt g k tk (6) whee ad h g ae the evese of the oigial filtes h ad g espectively. The costuctio of t, wt, t ad wt stats with imposig the Biothogoality coditios o the filtes. The lowpass aalysis coeffi ciets h k ae the poduct of a double shift Biothogoal to the lowpass sythesis coefficiets h k : h k h k (7) g k g k (8) Ad the highpass filte is Biothogoal to the lowpass filte: h k g k ad g k h k Figue shows the fequecy esposes of the decompositio ad ecostuctio filtes ad, the decompositio ad ecostuctio scalig ad wavelet fuctios of the Biothogoal (Bio3.9) [9]. Figue shows this Biothogol wavelet zeos distibutio of its decompositio ad ecostuctio filtes. This wavelet possesses the popeties of beig smooth with a liea phase ad shot legth filtes. Also, Table displays the coefficiets of the low-passes ad high-passes filtes of Bio3.9. Figue 3 ad Figue 4 show the zeos distibutios fo the filtes associated with Bio6.8 ad Bio3.5 espectively. (9) Figue. Bio3.9 impulse espose fo the decompositio ad ecostuctio filtes. Copyight SciRes.

3 78 J. KAR Figue. Bio3.9 zeos distibutio of the decompositio ad ecostuctio filtes. Figue 3. Bio6.8 zeos distibutio of the decompositio ad ecostuctio filtes. Copyight SciRes.

4 J. KAR 78 Figue 4. Bio3.5 zeos distibutio of the decompositio ad ecostuctio filte. Table. Filte coefficiets of the biothogoal wavelet Bio3.9. H H H H Costuctio of Daubechies Othogoal Wavelets Wavelets such as db6 ad db8 have played a vey essetial ole i a vaiety of speech ecogitio ad compessio paadigms itoduced last decade [,]. The low-pass ad high-pass filtes i the decompositio phase of a two chael filte bak ae depicted i Figue 5 ad two moe filtes of the ecostuctio phase ae displayed i Figue 6. h ad g Let deote the low-pass filte coefficiets ad the high-pass filte coefficiets espectively i the aalysis phase. To obtai pefect ecostuctio, these two filtes must satisfy the followig coditios [,]: ) Fo the low-pass filte h : h h h h k k ) Fo the high-pass filte g : whee g g g g k k k is the Diac delta fuctio defied by: () () Copyight SciRes.

5 78 J. KAR Figue 5. Aalysis phase filtes of a two-chaels filte bak. Figue 6. Sythesis phase filtes of a two-chaels filte bak. k if k o othewise. Give the coefficiets of h, it is show i [,7] that the coefficiets of the filtes h, g ad g that lead to othogoality ca easily be deived fom the coefficiets of h. Theefoe, to costuct a Daubechies othogoal wavelet, all we eed to do, is to fid the coefficiets of the filte h associated with it. The costuctio of the filte bak amouts to []: ) Desig a poduct low-pass filte satisfyig: P z P z z () P l ) Facto P i HH, the fid G ad G. Ad ca be educed eve futhe by defiig: l l Pz z P z ad substitutig Pz by zp z. Hece, the pefect ecostuctio coditio becomes [8]: PzPz (3) Pz is a half bad filte [] with all which implies that of its the coefficiets zeos except the costat tem. Futhemoe, the odd powes cacel whe we add P z to Pz. The desig of the low-pass ad the high-pass filtes of the sythesis ad aalysis filte baks of a Daubechies othogoal wavelet, cosides the followig two popeties []: ) These wavelets filtes must be othogoal. ) Ad must have maximum flatess at w ad w π i thei fequecy esposes. The low-pass filtes will have p zeos at π, ad have a total of p coefficiets, (legth of the filtes). This filte bak is othogoal ad the poduct filtes P z ad P z have a legth of 4p. The costuctio of Daubechies othogoal wavelets begis by choosig the umbe of zeos p at π. The zeos the filtes associated with the db6 ae depicted i Figue 7. Hee, we also eed to choose the biomial polyomial B p y associated with it which has a degee of p. The coefficiets of these polyomials ca be foud ecusively fo p by usig the followig equatio: pi bi bi (4) 4 pi Fo a give value p, the coefficiets of Bp y ae i a ascedig ode [5]. To get the oots of B p y, oe scales b by 4 ad to facilitate the umeical calculatios, oe uses the vaiable 4y istead of y. The atio of ay two cosecutive coefficiets is: b p k! p! k! k k (5) b p! k! pk! k Which i its simplest fom ca be expessed as: i i, i,,,, p i p (6) This equatio will be used i Sectio 5 to costuct the family of polyomials with coefficiets equal to the atios of this polyomial cosecutive coefficiets. Now to compute the p zeos of P z othe tha, we ote that accodig to [,9] the fequecy espose of the half-bad filte P w is give by: p Pw y By whee cosw y o y w cos. Figue 7. Zeos of B(y) fo db6. Copyight SciRes.

6 J. KAR 783 O the uit cicle we have: z z w cos y. Also, off the uit cicle we use the same elatio betwee z ad y. Reaagig these tems leads to: zz yz (7) Now let x y with ad ad u x zz xz z, this implies that: z x u z x u the, B y. ae the two oots of P z fo each oot y of Note that x u x u. That is, we have p oots ad thei iveses, amely: ad x y Figue 8. Zeos of P(z) fo db6 ad the fequecy esposes of the filtes P, H ad H. ad u x z xu, x u. The distibutio of these zeos i the plae is show i Figue 8. Fom Pz, P z is the deived ad all is left is to factoize P z. Daubechies did the followig factoizatio foud i []: whee Q p z Pz Qp z (8) z is a polyomial of degee p. p The Costuctio of db6 Fo p = 6, the db6 wavelet is obtaied. Figue 8 shows the locatio i the complex plae fo the zeos of B6 y associated with the Daubechies db6 othogoal wavelet. The fequecy esposes of the aalysis lowpass filte H z ad sythesis lowpass filte H z of this wavelet ae depicted i Figue 9. Theefoe completig the costuctio of the scalig fuctio alog with the mothe wavelet. The decompositio ad ecostuctio fuctios fo the mothe wavelet db6 ae plotted i Figue 9, while Figue shows the impulse espose of the fou filtes associated with it. h, h,, h Now, give the coefficiets of the low-pass filte h, it is show i [,7] that the coefficiets of the filtes g that h, g ad Figue 9. The db6 aalysis ad sythesis scalig ad wavelet fuctios. lead to othogoality ca be deived fom the coefficiets of h as follows: Fist, the coefficiets of the high-pass filte g of the aalysis bak ae obtaied fom those of the low-pass filte h by the alteatig flip. This ca be epeseted by thee opeatios o the coefficiets of h. ) Revese the ode; ) Alteate the sigs; 3) Shift by a odd umbe l. This takes the low-pass filte coefficiets ito a othogoal high-pass filte [] which is epeseted i the Copyight SciRes.

7 784 J. KAR followig equatio: Figue. Impulse espose fo the ecostuctio ad decompositio filtes of db6. g h l (9) The, the coefficiets of the high-pass filte g of the sythesis bak ae obtaied by the evese of the coefficiets of the high-pass filte g of the aalysis bak. They ca be geeated by the followig equatio: g h l () l (3) k W t g k t k 4. Zeos of Ratio Coefficiets Polyomials Now we coside the class of polyomials with coefficiets those of the atios obtaied i Equatio (4). A optimal limit of these zeos i the complex plae was peseted i []. Othe theoems ad alteative appoach to povig this distibutio ca be foud i [3]. Now the atios ca be expessed as follows: The coefficiets of the low-pass filte g of the sythesis bak ae the alteatig flip of the coefficiets of g. They ca be geeated by the followig equa- Qp z z z tio: p p h 3 3 p p gl () z z. 4 p p whee l is the legth of the low-pass filte h []. The scalig ad wavelet fuctios ae the deived We ote that these coefficiets ae i a ascedig fom the coefficiets of these filtes. The scalig fuctio ode whee ak ak. satisfies the equatio []: Theoem I: The oots of the polyomials Qp lie iside the uit disk fo all p. l t h kt k () Poof: Fo a, coside k p whee, h k is the evese of h k ad the wavelet p zq z z apz fuctio is the deived fom the scalig fuctio by the equatio: whee Copyight SciRes.

8 J. KAR 785 ad Fo p k k k k. z a a a z z, we have: p p p k p k k z a z a a z. k z Now fo z, z p p z a a k k a k a a a p a a a p z Replacig z with, we get: z Hece, if z p p z a a a z fo p p.. z. a a ap z (i.e. z ), the: a z Q z z a z p p p p a z z a z a a a z p p p p z z y z. (6) The coefficiets of B8 y ae displayed i Table. If oe sets x y ad u x the, z = x + u ad z x u ae the two oots of Pz fo each oot y of B 8 y. Note that x u x u. That is, we have 7 oots ad thei iveses, The p lot of these zeos i the plae ae show i Figue. Also, the values of x, y ad u ae listed i Table 3. Fom the defiitio of Pz, P z is obtaied ad all is left ow is to facto ize P z. Wee Q 4 z is a polyomial of degee 4 ad cho se to satisfy Equatio 6. It is equal to B4 y. The diffeet factoizatios of P z ito H z H z lead to diffeet mothe wavelets. Choosig H z to have its seve zeos iside the uit cicle ad H z to have its seve zeos outside the uit cicle leads to the Daubechies othogoal mothe wavelet db8. The scalig ad wavelet fuctios of oe of the Daubechies wavelets family membe called db8 ae show i Figue 3. The same figue also shows the impulse espose of the fou filtes associated with that wavelet ad Table 4 displays The coefficiets of the of db8 filtes. The Distibutios of Zeos fo db6 Fo the Daubechies wavelet db6 this polyomial is: 3 4 Q4 z68z9z 3z 4z with maximum module of.935. The oots of this polyomial ae depicted below i Figue ad obseved to eside all i the uit cicle. 5. The Costuctio of db8 The scalig ad wavelet fuctios ae the deived fom the coefficiets of these filtes. They satisfy espectively the equatios []: l t h k tk (4) k l W t g k t k k (5) To compute the 4 zeos of P z othe tha, ote that o ad off the uit cicle we have the followig elatio betwee z ad y: z z y. This implies that the equatio: Figue. Zeos of Q 4 (z) fo Daubechies db6 othogoal wavelet. Table. The coefficiets of the polyomial B 8 (y). b(). b(8).95 b(7).489 b(6).7734 b(5).89 b(4).875 b(3). b().5 Copyight SciRes.

9 786 J. KAR Table 3. The oots of Q 4 (z) iside the uit cicle z = x u ad outside of it z = x + u. u x y z z Ta ble 4. The coefficiets o f the of db8 filtes. H H To cay oizatio we o P z has 6 oots at z ad 4 othe oots whic h occu i pais (z ad z ). This meas that we have 7 oots iside the uit cicle ad the othe 7 oots outside the uit cicle. The oots i side the cicle ae the oots fo the filte H z comig fom the equatio: z x u ad the oes outside it ae fo filte H z obtaied fom the equatio: z x u. These oots whe factoized lead to the coefficiets of these two filtes ad they ae show i Table 3 fo Q4 z. The coefficiets of the high-pass filtes G z ad G z ae simply the deived fom the low-pass filte coefficiets by the alteatig sig popety. The scalig ad wavelet fuctios of oe of the out the fact te that Figue. Zeos of P(z) fo db8 ad the fequecy esposes of the filtes: P, H ad H. Daubechies wavelets family membe called db8 ae show i Figue 3. The same figue also shows the impulse espose of the fou filtes associated with that wavelet. The geeal chaacteistics of this wavelet iclude compact suppot fo which exact ecostuctio ae possible with FIR filtes. Its associated scalig filte is a miimum-phase filte. This wavelet is a membe of the othogoal set of wavelets that ae usually deoted by: db N N epesets the ode of the ecostuctio ad decompositio wavelet. Thei coespodig filte legth is N. 6. Coclusio I this pape we costuct Daubechies othogoal wavelets via the two chael pefect ecostuctio filte bak. The cases of db6 ad db8 ae examied whee we deived the coefficiets of the filtes associated with these wavelets ad the oots of the biomial polyomials that made this costuctio possible. The locatios of the zeos of the polyomials ivolved i this costuctio wee foud ad thei locatios wee discussed. The distibutio of the zeos of a family of polyomials havig thei coefficiets as the atios of those of the biomial polyomials was examied ad wee poved to eside iside the uit cicle. Simila discussios about the Daubechies Biothogoal wavelets family ae icluded alog with the costuctios of Bio3.5, Bio3.9 ad Bio Ackowledgemets The autho would like to thak Alfaisal Uivesity ad its Office of Reseach fo secuig the time, eviomet ad fuds to complete this eseach poject. This wok Copyight SciRes.

10 J. KAR 787 Figue 3. db8 Wavelet ad scalig fuctios. The impulse espose fo the ecostuctio ad decompositio filtes of the wavelet db8. was also suppoted by the Alfaisal Uivesity Stat-Up F ud (No. 449). REFERENCES sig, IEEE Tasactios o Sigal Pocessig, Vol. 4, No. 9, 99, pp doi:.9/78.57 [8] M. Vetteli ad J. Kovacevic, Wavelets ad Subad Codig, Petice Hall, Eglewood Cliffs, 995. [9] M. Misiti, Y. Misiti, G. Oppeheim ad J. Poggi, Matlab [] [] G. Stag, ad T. Nguye, Wavelets ad Filte Baks, Wellesley-Cambidge Pess, Wellesley, 996. I. Daubechies, Te Lectues o Wavelets, SI, Wavelet Tool Box, 997. [] J. Kaam, A Compehesive Appoach fo Speech Related Multimedia Applicatios, WSEAS Tasactios o Philadelphia, 99. Sigal Pocessig, Vol. 6, No.,, pp. -. [3] S. Kakeya, O the Limits of the Roots of a Algebaic [] J. Kaam, Radial Basis Fuctios With Wavelet Packets Equatio with Positive Coefficiets, Tohoku Mathe- Fo Recogizig Aabic Speech, The 9th WSEAS Itematical Joual, Vol., 9, pp [4] J. Kaam, O the Kakeya-Eestom Theoem, MSc. Thesis, Dalhousie Uivesity, Halifax, 995. atioal Cofeece o Cicuits, Systems, Electoics, Cotol ad Sigal Pocessig, Athes, Decembe, pp [5] J. Kaam, Coectig Daubechies Wavelets with the Kakeya-Eestom Theoem, Iteatioal Joual of Applied Mathematics, Vol. 4, No., 3, pp [6] J. Kaam, O the Roots of Daubechies Polyomials, Iteatioal Joual of Applied Mathematics, Vol., No. 8, 7, pp [7] M. Vetteli, Wavelets ad Filte Baks: Theoy ad De- [] J. Kaam, O the Distibutio of Zeos fo Daubechies Othogoal Wavelets ad Associated Polyomials, 5th WSEAS Iteatioal Cofeece o Applied Mathematics, Athes, 9-3 Decembe, pp. -5. [3] C. A. Muesa, Compaative Methods fo the Polyomial Isolatio, Poceedigs of the 3th WSEAS Iteatioal Cofeece o Computes, 9, pp Copyight SciRes.