Introduction to Wavelets and Their Applications

Transcription

1 Itroductio to Wavelet ad Their Applicatio a T. Bialaiewicz Uiverity of Colorado at Dever ad Health Sciece Ceter Wavelet are a powerful tool for tudyig time-frequecy behavior of fiite-eergy igal. Advatage of the wavelet decompoitio over Fourier traform preet themelve whe traformig a igal that ha time varyig ad/or igularity characteritic. Multireolutio ad wavelet theory ha recetly foud applicatio i a remarkable rage of diciplie uch a data compreio, igal proceig, image aalyi, tatitic, ad modelig of oliear dyamic procee. Excellet preetatio of the ubect i give by Mallat []. Note: It i ot the itetio of thi hort paper to give a broad itroductio to the ubect or a log lit of referece. CONTINUOUS WAVELET TRANSFORM The cotiuou wavelet traform Wf(,b) of a fiite-eergy igal f(t) i defied a it calar product with the wavelet ψ b (t). I other word, the wavelet traform repreet the correlatio of the igal f(t) ad the wavelet ψ b (t). I additio, due to the Pareval theorem, thi relatiohip ca alo be tated i the frequecy domai. The followig equatio give both repreetatio: b dt fˆ( ) ψˆ ( ) b Wf (, b) ψ e d () π where the wavelet / + ψ ( ( t b)), R, b R () b ad ψ(t), kow a the mother wavelet, i a fuctio, which atifie the followig equatio: dt. (3) () t ψ t/t Fig. Morlet wavelet

2 Oe uch fuctio, which we ued i our reearch, i the Morlet wavelet, how i Figure. It ca be ee from equatio () that the parameter ad b i equatio () ca be iterpreted a the cale ad the tralatio. The mother wavelet cale. The icreae of i equivalet to tretchig the mother wavelet or movig it frequecy cotet toward lower frequecie. O the cotrary, the decreae of caue the compreio of the mother wavelet or movig it frequecy cotet toward higher frequecie. Varyig b for the fixed cale, we are lidig a wavelet with a fixed badwidth B ad with ome fixed ceter frequecy alog the aalyzed igal. For a give poitio b of the aalyzig wavelet, we calculate the value of the wavelet traform. It tell u, to what extat the frequecy cotet of the aalyzed igal, i the eighborhood of b, i cloe to the frequecy cotet of the wavelet at the give cale. To make the decribed procedure ueful, we have to etablih the relatiohip betwee the cale ad the frequecy cotet, at thi cale, of a particular aalyzig wavelet. Thi i accomplihed by defiig two frequecy characteritic of the mother wavelet []: the ceter frequecy ψˆ ( ) π d, (4) ad the badwidth B σ, cetered aroud, with σ ( ˆ ) ψ ( ) d. (5) π For a wavelet of the cale, the ceter frequecy ad the badwidth are, repectively, defied by the followig equatio: B max mi, B, (6) with max + B /, mi B /. Note that by thi covetio B B ad. It i very importat to ote that both the ceter frequecy ad the badwidth have differet value for every wavelet. I other word, it i importat to chooe the mother wavelet with high frequecy reolutio or arrow badwidth. The icreae of the cale improve the frequecy reolutio at lower frequecie. I practice, for both cotiuou-time ad dicrete-time wavelet igal aalyi, we have to ue the dicrete-time repreetatio of the aalyzed igal. Thi igal i ampled with ome frequecy T, where T i the amplig period. I thi cae, the badwidth of the ampled igal i limited by the Nyquit frequecy T. The frequecy variatio of the cotiuou-time igal betwee ad T correpod to the repreetative poit movig betwee thee value i the complex plae alog imagiary axi. However, the correpodig frequecy variatio of the dicrete igal hould be repreeted i the complex plae z. The mappig betwee imagiary axi of the plae, for which i ad the z plae i determied by the followig equatio: it z e or z T, z, (7) which mea that whe the frequecy of the cotiuou-time igal varie i the iterval [, T ], the agle or the frequecy of the dicrete-time igal, repreeted i radia uit, varie i the iterval [,π]. Thi i utified by the followig calculatio:

3 T π z π T /. (8) T I other word, the frequecy of T rad/ of the cotiuou-time igal i repreeted by the frequecy of π radia of the dicrete-time igal. Thi utifie the ue of the relative frequecy of the dicrete-time igal, which i obtaied by dividig, by π, every frequecy from the iterval [,π]. The, the relative frequecy /π correpod to the highet frequecy of the cotiuou-time igal preerved i the ampled igal, i.e., the frequecy T rad/ or the frequecy f T Hz, with T πf T. It i coveiet to repreet the magitude of the Fourier traform of the wavelet with differet cale a a fuctio of the relative frequecy. The, we obtai the real frequecy value multiplyig the relative frequecy by the amplig frequecy repreeted i radia per ecod or i Hertz. The magitude of the Fourier traform of the Morlet wavelet for everal cale (,,3,4,5) i repreeted i Figure. The Morlet mother wavelet i give by the followig equatio: t / e co(5t). (9) ψˆ ( ) /π Fig. Fourier traform of Morlet wavelet for differet cale For Morlet wavelet 5 ad B.7 (radia uit). Uig thee two parameter, we ca determie ad B for ay value of. To covert frequecy radia to radia per ecod, we have to multiply the ratio by the Nyquit frequecy T rad/. For π example, ettig 5, we obtai for Morlet wavelet the ceter frequecy / 5 ( T ) B / 5 T rad/ ad the badwidth B (.34 5 T ) T 5 rad/. I other π π π π word, for a cotat cale we ca vary the frequecy rage of aalyi uig differet amplig frequecie T. It i clear from Figure that the magitude of the Fourier traform of the Morlet wavelet for i almot etirely poitioed to the right of /π

4 or to the right of the Nyquit frequecy. I other word, it i almot completely ucorrelated with the dicrete repreetatio of the origial igal or the ueful badwidth of the Morlet mother wavelet i almot etirely above the half of the amplig frequecy. It i apparet from thi figure that the frequecy reolutio i much better for the lower relative frequecie (or for the lower cale). Thi immediately offer olutio to the problem of poor reolutio at higher frequecie, i.e., the aalyzed igal ca be overampled at uch rate that lower cale ad, i particular, cale equal oe will ot cotai ay ueful frequecie. I other word, the bad of the lower cale wavelet (or wavelet with frequecy reolutio too mall for the performed aalyi) will be outide the eetial badwidth of the aalyzed igal. Aume that a igal ha a compoet of frequecy ad that mi max () or mi max. () It mea that the igal compoet of frequecy will ifluece all wavelet coefficiet Wf(b,) with withi thi iequality boud ad ay b R. DISCRETE WAVELET TRANSFORM We hall dicu ow the dicrete wavelet traform. A a reult of the quatizatio of the parameter ad b, we obtai the followig equatio, which correpod to (): m / m ψ t) a ψ ( a t b. () m ( )( m, ) Z I particular, we ca chooe a ad b ad obtai the followig dyadic orthoormal wavelet bai of the L (R) pace: m/ m m ψm () t ψ( ( t )). (3) Note that for a fixed value of m we have a cotat cale m. Uig ψ m a a baic wavelet at the level m, we ca rewrite (3) a m ψm ψm ( t ), (4) which mea that m i the time tep of hiftig the wavelet ψ m. I the dicrete wavelet aalyi of the igal f(t) L, we determie the ier product of the igal f(t) ad the equece of fuctio ψ m. Thee ier product are kow a wavelet coefficiet. The dicrete wavelet expaio of the igal f, which i the ivere dicrete wavelet traform, ca be expreed a f, ψ m ψ m d m[ ] ψ m, (5) m, where the wavelet coefficiet d m[ ] f, ψ m repreet the commo feature of the igal f ad the wavelet ψ m. The parameter m ad allow to acce the particular igal feature. The parameter i ued to localize the requeted time itat ad the parameter m allow to chooe the cale level or frequecy rage, i which we wih to examie the igal frequecy pectrum. Let u ow cocetrate o the iterpretatio of (5). For a fixed value of m, the right-had ide of thi equatio (or the ier um i (5)) repreet m

5 a orthogoal proectio of f oto a ubpace W m L (R). Deotig thi proectio by f, we ca repreet (5) a PW m f m m PW f + P f. (6) m m W m + We ca treat the ecod um a a orthogoal proectio V L (R). We ca rewrite (6) a m PV f of f oto certai ubpace f PW m f + PV f. m (7) Thi lead to the followig relatio: V V + W +, Z, (8) which mea that V + i the orthogoal complemet of W + i V. If the et { ϕ( t ); Z} i a orthoormal bai i V, the we call ϕ the calig fuctio ad we defie a orthoormal dyadic bae i V a / ϕ ϕ( t ), Z (9) The we ca rewrite (7) a d [ ] + a [ ] ϕ, () Z where a [ ] f, ϕ ad i kow a the approximatio coefficiet of the igal f at the level ad at the time itat. The level et the coaret cale, at which f i decompoed. Thi i the celebrated multireolutio repreetatio of the fuctio f. We ca approximate the igal f by it proectio oto ome ubpace V l. With thi aumptio, we obtai Z d [ ] + a [ ] ϕ. () l Z By thi approximatio, we aume that the frequecy cotet of the igal i frequecy bad correpodig to the cale level <l i eglegible. Equatio () i a approximate multireolutio repreetatio of the igal f(t). To make thi multireolutio repreetatio complete, let u ote that L ( R ) V K Vl K V K V { }. () I each tep of the proce of the multireolutio aalyi, we obtai the detail of a igal, which repreet the proectio of it previou approximatio oto a ubpace W with the ubcript icreaed by ad the ew approximatio, which repreet the proectio of the previou approximatio oto the ubpace V with the ubcript icreaed by oe. Fially, whe we top the proce, the igal repreetatio coit of a equece of detail ad the lat approximatio. We are aumig that the wavelet ytem i orthoormal, i.e., we have Z ϕ, ϕ k ψ, ψ k,, ϕ, ψ k k k ψ, ψ ik,, where where, i. (3) The wavelet fuctio ψ(t) ad the calig fuctio ϕ(t) are both recurively defied a

6 N g( ) ψ (t ), (4) N ϕ h( ) ϕ (t ), (5) i which g() ad h() are equece of N umber repreetig FIR filter, high-pa ad low-pa, repectively. Thee filter are related by g( ) ( ) h( N ). (6) The fat wavelet traform algorithm, kow a Mallat algorithm, i implemeted with thee filter []. SCALOGRAM AND COSCALOGRAM The calogram illutrate the eergy evolutio with time i a igle quare-itegrable igal by providig a time-frequecy map of it quared wavelet coefficiet. Replacig the quared wavelet coefficiet value of a igal with the product of wavelet coefficiet of two differet igal, oe obtai a repreetatio of the correlatio betwee the igal, called a cocalogram. It ha the advatage of revealig time-varyig correlatio a a fuctio of frequecy ad, i fact, it repreet cro-eergy of thee igal i their commo time-frequecy domai of iteret. Repreetig wavelet atom ψ b (t) i frequecy domai, we ca ay that mot of their eergy i i the iterval of the legth of B B / cetered aroud / or i time domai aroud b R i the iterval of the legth σ t with the tadard deviatio σ t defied by the followig itegral: σ t ψ ( t dt. (7) t ) The Morlet wavelet tadard deviatioσ t Note that i time-frequecy plai we have characterized a mother wavelet by three parameter:, σ (or B ), ad σ t. Thi defie, for every wavelet ψ b (t), the Heieberg box [] i the time-frequecy plae, cetered at ( b, ) with the dimeio of σ t alog the time axi ad σ / alog the frequecy axi. It area σ t σ i idepedet of frequecy or cale ad it ide repreet local time ad frequecy reolutio. It i illutrated i Figure 3. Note that for a fixed cale (or frequecy) the dimeio of the Heieberg boxe i time directio i cotat but (for a fixed time) decreae a frequecy icreae or i proportioal to cale. The calogram of a igal f(t), deoted by P f b, ), i defied a W ( PW f ( b, ) Wf ( b, ) Wf ( b, ). (8) The magitude quare of a wavelet coefficiet i (8) i replaced by the quare of a wavelet coefficiet of real-valued igal. It repreet a local time-frequecy eergy deity, which meaure the eergy of f i the Heieberg box of each wavelet ψ b (t) cetered at ( b, ). Thi decriptio of

7 igal eergy (i time-cale domai) facilitate idetificatio of time-varyig eergy flux, pectral evolutio, ad traiet burt ot readily viible i time or frequecy domai repreetatio [-5]. σ t σ ( ) ψˆ π b π ( ) ψˆ b σ t σ b b Fig. 3 Heieberg boxe i time-frequecy domai ψ ψ b b t We will how ow that calogram, defied for the CWT, actually repreet a local time-frequecy eergy deity of a igal f. The eergy of a igal f ca be repreeted by the itegral of it quare or it itataeou power with repect to time. Thi eergy, uig CWT expaio of f, ad aumig that we repreet f by a fiite umber of it wavelet traform value i a bouded time-cale rectagle (or a time-frequecy rectagle related to it through a mother wavelet ued) cotaiig mot of the igal eergy, ca be expreed a follow: dt Wf (, b ) ψ dt, (9) where, b ) pair repreet cale-time poit i a bouded rectagle, at which wavelet ( traform W f(,b ) i calculated by covolvig f with ψ ( t) ψ ( ( t b )). Applyig to (9) orthoormality property of the wavelet bai, we obtai Wf (, b ) dt Thi ca be repreeted a f dt Wf (, b ) Wf (, b ) + Wf (, b ) Wf ( /, b ) where the firt um i the igal eergy i the highet frequecy ubbad (correpodig to the lowet calig level ) ad the lat um repreet the igal eergy i the lowet frequecy ubbad (correpodig to the highet calig level of iteret ). Sice (3) i, (3) (3)

8 the expreio for the igal eergy i the time iterval of iteret with amplig poit, the quare of wavelet coefficiet Wf (, b ) repreet local eergy deity of a igal i it time-frequecy domai or it calogram. Let u coider two igal f (t) ad f (t) ad let u repreet their cro-eergy uig their CWT expaio. Proper calig of wavelet coefficiet reflect their differet frequecy cotet. Uig the ame procedure a for repreetig eergy of a igle igal ad the orthoormality property of the wavelet bai, we obtai the followig croeergy repreetatio of two igal: Wf (, b ) Wf (, b ) dt Thi ca be rewritte a f dt Wf (, b Wf (, b ) Wf (, b ) Wf ( ) Wf, b (, b ) Wf ( ) +, b ), Wf ( where the firt um i the cro-eergy of two igal i the highet frequecy ubbad (correpodig to the lowet calig level ) ad the lat um i the cro-eergy of the two igal i the lowet frequecy ubbad (correpodig to the highet calig level of iteret ). Sice (33) repreet the cro-eergy of the two igal i the time iterval of iteret with amplig poit, the product of the wavelet coefficiet Wf (, b ) Wf(, b ) of thee igal repreet local cro-eergy deity i their timefrequecy domai or their cocalogram., b ) Wf (, b ) (3) (33) REFERENCES [] S. Mallat, A Wavelet Tour of Sigal Proceig, d ed., Academic Pre, 999. []. T. Bialaiewicz, D. Gozález,. Balcell ad. Gago, Wavelet Aalyi of the Effectivee of Coducted EMI Reductio i Power Coverter, i Proc. IECON 5, 5. [3] N.D. Kelley, R.M Ogood,.T. Bialaiewicz ad A. akubowki, Uig Wavelet Aalyi to Ae Turbulece/Rotor Iteractio, Wid Eergy, vol.3, pp. - 34, No.3,. [4] N. D. Kelley, B.. okma,. T. Bialaiewicz, G. N. Scott ad L. S. Redmod, The Impact of Coheret Turbulece o Wid Turbie Aeroelatic Repoe ad It Simulatio, i Proc. AWEA Widpower 5, Dever, 5. [5] K. Gurley ad A. Kareem, Applicatio of Wavelet Traform i Wid, Earthquake, ad Ocea Egieerig, Egieerig Structure, vol., pp , No.,, 999.