The Fourier-Like and Hartley-Like Wavelet Analysis Based on Hilbert Transforms

Transcription

1 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP Th Fourir-Lik Hrtly-Lik Wvlt lyi B o Hilbrt Trfor Luci R Sor, Hélio Olivir Rto J Sobrl itr btrct I cotiuou-ti wvlt lyi, ot wvlt prt o ki of ytry B o th Fourir Hrtly trfor krl, w wvlt ultirolutio lyi i propo Thi pproch i b o pir of orthogol wvlt fuctio i th Fourir-Lik Hrtly-Lik wvlt lyi Hilbrt trfor lyi o th wvlt thory i lo iclu Ix Tr Wvlt ig, Hilbrt trfor, Fourir trfor krl, Hrtly trfor krl, Fourir-Lik wvlt, lytic wvlt, Hrtly-Lik wvlt, otiuouti, Wvlt trfor T I INTRODUTION h i of coprig Fourir lyi with wvlt copoitio i th trtig poit for itroucig lyi b o coupl of orthogol wvlt fuctio, o with v ytry othr with o ytry Thi pproch i hr prt th Fourir-Lik Hrtly-Lik wvlt lyi I th tr Fourir lyi, igl x(t) i iultouly lyz by v o (i qurtur) fuctio, big rprt by: x(t) c tr + coi tr + i tr () I th tr wvlt ultirolutio lyi (WR) [], x(t) y b rprt by: x(t) ϕ tr + ψ tr, () whr ϕ tr ccout for th lyi of x(t) with clig fuctio ϕ(t), ψ tr rprt tho o riv fro cl vrio of othr wvlt fuctio ψ(t) oprig th WR th Fourir lyi qutio, th clig cofficit, ϕ tr, ply rol tht corrpo to th c tr of th Fourir ri; th wvlt cofficit, ψ tr, c b viw th hroic copot of th Fourir ri, ic th hroic r cl vrio of th ifiit Fourir krl Luci R Sor, Hélio Olivir Rto J Sobrl itr r with th Dprtt of Elctroic Syt, Frl Uivrity of Prbuco, Rcif-PE, Brzil, E-il: luor@ufpbr, ho@ufpbr, rjc@ufpbr Thi work w prtilly upport by PES NPq I cotiuou-ti wvlt lyi, ot vilbl wvlt fuctio prt o ki of ytry [] Thu, wh v wvlt i u to lyz ytric igl, it o prt y ot b proprly lyz Thrfor, it i xpct tht th igl lyi y b iprov by icluig w tr o th WR Hc, it turl to ocit to ch wvlt with v ytry othr o of o ytry, vic-vr Th Hilbrt trfor c turlly b ivok to riv th i qurtur vrio of ytric (or ti-ytric) wvlt Hc, x(t) y b rprt o w cotiuouti WR by: x(t) ϕ tr + ψ tr + orthogol of ψ tr () ccorigly, w wvlt fuctio tht look lik th Fourir Hrtly trfor krl r ivok So, by logy to Fourir Hrtly trfor, th coi i krl i rplc by ψ Hilbrt trfor of ψ i thi w cocpt of wvlt lyi I orr to llow furthr ivtigtio o th Fourir-Lik Hrtly-Lik wvlt lyi, brif rviw of th Hilbrt trfor i prt, wll th rult of pplyig o wvlt proprti to th Hilbrt trfor of wvlt II THE HILBERT TRNSFOR Th Hilbrt trfor of fuctio g(t) i fi by []: Hb { g( t) ( t) + g p v t, () π x t whr pv i th uchy pricipl vlu of th itgrl ftr chg of vribl, th Hilbrt trfor c b writt covolutio: Hb { g( t) g( t) π t By tkig th Fourir trfor of (), w hv: F { Hb{ g( t) j ( ) F{ g( t) () g, (6) whr F i th Fourir trfor oprtor g() i th igu fuctio, which i fi by:

2 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP +, > g( ), (), < It follow fro (6)-() tht th Hilbrt trfor of fuctio ipo ull t -π/ ph ly o th frqucy rpo of tht fuctio Othr itrtig proprti of th Hilbrt trfor r []: fuctio it Hilbrt trfor r orthogol ovr th ifiit itrvl; Th Hilbrt trfor of rl fuctio i rl fuctio; Th Hilbrt trfor of v fuctio i o fuctio, vic-vr I th frwork of wvlt, th Hilbrt trfor of ytric (or ti-ytric) rl wvlt i rl tiytric (or ytric) fuctio Howvr, it i cry to vrify whthr th rultig fuctio i lo wvlt fuctio or ot III THE HILBERT TRNSFOR ON THE WVELET NLYSIS fuctio ψ(t) i othr-wvlt, if oly if, (i) ψ(t) i i th pc of fiit rgy fuctio L (R), (ii) ψ(t) tifi th iibility coitio [] So proprti r xplor i th followig propoitio i viw of pplyig Hilbrt trfor to wvlt oir Ψ() th Fourir trfor of ψ(t), Hb{ψ(t) th Hilbrt trfor of ψ(t), E[ψ(t)] th rgy of ψ(t), [ψ(t)] th iibility cofficit of ψ(t) Propoitio : If ψ(t) i rl wvlt, th Hb{ψ(t) i lo rl wvlt with rgy iibility cofficit of it grtig wvlt, ψ(t) Proof: If ψ(t) i rl wvlt, th ψ(t) blog to L (R) tifi th iibility coitio Ivokig Prvl thor, th rgy iibility cofficit of Hb{ψ(t) r giv by: [ { ( ) + E Hb ψ t j g( ) Ψ( ), π ( ) ( ) [ { ( ) + j g Ψ Hb ψ t ipl ipultio yil to [ { ( ) + E Hb ψ t Ψ( ) π, [ Hb{ ψ ( t) ( ) pplyig Prvl' thor o th right-i of th rgy qutio, it i trightforwr to coclu tht Hb{ψ(t) i lo i L (R) orovr, ψ(t) Hb{ψ(t) hv th rgy ψ(t) L (R) Ψ ( ), it proptly follow tht Hb{ψ(t) lo tifi th iibility coitio: [ Hb{ ψ ( < + Furthror, ψ(t) Hb{ψ(t) hv iibility cofficit Propoitio : Lt ψ(t) b wvlt with N vihig ot, th Hb{ψ(t) h t lt N vihig ot Proof: Th th ot of ψ(t) i fi by []: [ ( )] + t t ψ ( t) t ot, th [ ψ ( t) ] ψ ψ(t) h N vihig, fro to N- I th frqucy oi, th ot of ψ(t) r xpr by []: [ ψ ( t) ] ( Ψ ) ( ) ( π j), whr th uprcript () ot th th rivtiv of Ψ() Hc, th th ot of Hb{ψ(t) i giv by [ Hb{ ψ ( t) ( ( ) ( )) ( j g Ψ ) ( π j) oqutly, it follow tht Hb{ψ(t) h t lt N vihig ot I viw of Propoitio, it follow tht Hb{ψ(t) i wvlt with rgy, iibility cofficit t lt ubr of ull ot th it grtig wvlt, ψ(t) Figur how fw cotiuou-ti rl wvlt thir corrpoig Hilbrt trfor () (c) () orlt it Hilbrt Trfor Ti Ti xic Ht it Hilbrt Trfor Gu- it Hilbrt Trfor Ti () (f) (b) yr it Hilbrt Trfor Ti Gu- it Hilbrt Trfor Ti Gu- it Hilbrt Trfor Ti Figur : otiuou-ti wvlt thir Hilbrt trfor: () orlt; (b) yr; (c) xic Ht; () Gui-; () Gui-; (f) Gui-

3 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP IV THE FOURIER KERNEL ON THE WVELET NLYSIS It i ti to fi w wvlt fuctio tht look lik th Fourir trfor krl, which c lyz both ytri of ytric igl Th Fourir trfor krl, or Fourir krl, i fi j t co t + j i t by: ( ) ( ) Evokig tht Hb{ co ( t) i( t) c lo b writt j t co( t) j Hb{ co( t), th th Fourir krl Thi iv obrvtio otivt th fiitio of Fourir- Lik wvlt b o rl wvlt it Hilbrt trfor Lt u fi th Fourir-Lik wvlt, Ft{ψ(t), by: Ft { ψ ( t) ( ψ ( t) j Hb{ ψ ( t) ) (8) Propoitio prov tht Ft{ψ(t) i lo wvlt tht th fctor i ipo o to gurt tht th Fourir krl hol th rgy iibility cofficit of it grtig wvlt itiolly, Propoitio how tht Ft{ψ(t) h vihig ot th it grtig wvlt, ψ(t) Propoitio : If ψ(t) i rl wvlt Hb{ψ(t) it Hilbrt trfor, th Ft{ψ(t) i lo rl wvlt with rgy iibility cofficit of it grtig wvlt, ψ(t) Proof: If ψ(t) i rl wvlt Hb{ψ(t) it Hilbrt trfor, th ψ(t) Hb{ψ(t) blog to L (R) tify th iibility coitio Th rgy iibility cofficit of Ft{ψ(t) r giv by: E [ Ft{ ψ ( t) [ Ft{ ψ ( t) ipl hlig giv ( t) j Hb{ ψ ( t) t, ( ) g( ) Ψ( ) ( t) + Hb{ ψ ( t) [ { ( ) E Ft ψ t t,, [ Ft{ ψ ( t) + Ψ( ) <, Fro Propoitio, it follow tht Ft{ψ(t) lo blog to L (R) it h th rgy ψ(t) Oc ψ(t) L (R), it follow tht Ft{ψ(t) lo h th Ψ ( ) iibility cofficit of ψ(t) Propoitio : Lt ψ(t) b wvlt with N vihig ot, th Ft{ψ(t) h lo N vihig ot Proof: Fro Propoitio, it follow tht th th ot of Ft{ψ(t) i giv by: [ Ft{ ψ ( t) ( ( ) ( ) ( )) ( Ψ g Ψ ) ( ) ( π j), which c lo b writt [ Ft{ ψ ( t) [ ψ ( t) ] + [ Hb{ ψ ( t) j Th, Ft{ψ(t) h lo N ull ot I th frqucy oi, Fourir-Lik wvlt r ull for > For <, thy hv th gitu rpo of th grtig wvlt ultipli by clr fctor Thi i typicl bhvior of lyticl igl lytic Wvlt B o th rult obti fro Fourir-Lik wvlt it i ctully ipl to fi lytic wvlt lytic fuctio {g(t) i coplx igl ig by rl fuctio g(t) it Hilbrt trfor Hb{g(t) [] I th frwork of wvlt, lytic wvlt, {ψ(t), c b fi by: { ψ ( t) ( ψ ( t) + j Hb{ ψ ( t) ) () lytic wvlt hv lo rgy, iibility cofficit ull ot th thir grtig wvlt, ψ(t) Th proof r iilr to tht of Propoitio I th frqucy oi, lytic wvlt r ull for < For >, thy hv th gitu rpo of th grtig wvlt ultipli by clr fctor B oputig th Wvlt lyi of ytricl Rl Sigl It y b xpct tht uig Fourir-Lik or lytic wvlt, v o prt of ytric rl igl c b bttr lyz, rpctivly, by v wvlt it Hilbrt trfor, i o wvlt I both c, it will b cry to prfor coplx wvlt lyi It i lo poibl to lyz both ytri of rl igl uig rl wvlt I tht c, th Hrtly krl houl b ivok V THE HRTLEY KERNEL ON THE WVELET NLYSIS Th Hrtly trfor krl, or Hrtly krl, i fi c t co t + i t by th coi i fuctio: ( ) ( ) ( ) Rcllig tht Hb{ i( t) co( t) lo b writt c( t) co( t) Hb{ co( t), th Hrtly krl c or

4 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP ( t) i( t) Hb{ i( t) c + Thi ipl rrk otivt th fiitio of Hrtly-Lik wvlt by tkig th u or th iffrc of giv rl wvlt it Hilbrt trfor Lt u fi th Hrtly krl of wvlt, or Hrtly-Lik wvlt, Ht{ψ(t), by: clr fctor itiolly, thy ipo ±π/-hift o th ph rpo of th grtig wvlt Figur how o cotiuou-ti wvlt thir corrpoig Hrtly krl uig th itio oprtor i () Ht { ψ ( t) ( ψ ( t) Hb{ ψ ( t) ) () () 8 6 orlt it Hrtly Krl (b) 8 6 yr it Hrtly Krl Propoitio prov tht Ht{ψ(t) i lo wvlt tht th fctor k th Hrtly krl it grtig wvlt to hv rgy iibility cofficit itiolly, Propoitio 6 how tht Ht{ψ(t) h ull ot th thir grtig wvlt, ψ(t) Propoitio : If ψ(t) i rl wvlt Hb{ψ(t) it Hilbrt trfor, th Ht{ψ(t) i lo rl wvlt with rgy iibility cofficit of it grtig wvlt, ψ(t) Proof: If ψ(t) i rl wvlt Hb{ψ(t) it Hilbrt trfor, th ψ(t) Hb{ψ(t) blog to L (R) hol th iibility coitio Th rgy iibility cofficit of Ht{ψ(t) r giv by: E ( t) ± Hb{ ψ ( t) t, ( ) ± j g( ) Ψ( ) ipl ipultio yil to E ( t) + Hb{ ψ ( t) t, ( ) Fro Propoitio, it follow tht Ht{ψ(t) lo ty o L (R) it h rgy of ψ(t) itiolly, Ht{ψ(t) h lo iibility cofficit th it grtig wvlt, ψ(t) Propoitio 6: Lt ψ(t) b wvlt with N vihig ot, th Ht{ψ(t) h lo N vihig ot Proof: Fro Propoitio, it follow tht th th ot of Ht{ψ(t) i giv by: ( ( ) ( ) ( )) ( Ψ ± j g Ψ ) ( ) ( π j), which c lo b writt [ ψ ( t) ] ± [ Hb{ ψ ( t) Th, Ht{ψ(t) h lo N ull ot I th frqucy oi, Hrtly-Lik wvlt hv th gitu rpo of th grtig wvlt ultipli by (c) () Ti xic Ht it Hrtly Krl Ti Gu- it Hrtly Krl Ti Ti Gu- it Hrtly Krl Ti Gu- it Hrtly Krl Ti Figur : otiuou-ti wvlt thir Hrtly krl: () orlt; (b) yr; (c) xic Ht; () Gui-; () Gui-; (f) Gui- VI SOE EXPLE SES OF SIGNL NLYSIS USING FOURIER-LIKE ND HRTLEY-LIKE WVELETS Th wvlt propo i thi ppr wr iult uig th TLB Wvlt Toolbox [] Str pl igl wr lyz to illutrt th bhvior of th propo wvlt oir th wvlt trfor,,b cofficit, giv by:, b + t b f ( t) ψ t, () whr (>) b r, rpctivly, rl cl trltio clr pplyig th Hrtly-Lik Wvlt lyi () (f) Figur how igl copo by two uitryplitu oil fuctio, Hz Hz, Figur how 8-lvl wvlt lyi (wvlt trfor) of tht igl uig th orlt wvlt, it Hilbrt trfor, it Hrtly krl

5 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP Figur : igl copo by two oil fuctio wvlt c rtriv or ifortio fro igl rgrl it ytry th ki of ytry of th lyzig igl i ot priori kow, th u of Hrtly-Lik wvlt o th cotiuou-ti wvlt trfor y chiv bttr rult Figur how th igl riv fro th t-lvl wvlt trfor of th ltr igl uig thi pproch So iprovt c b chiv uig o or ytricl wvlt to lyz o igl, th uig v wvlt 8 bolut Vlu of,b officit for orlt it Hilbrt Trfor it Hrtly Krl l c 6 u p lit ti (or pc) b () Figur : Th t-lvl wvlt trfor of cobi igl uig th orlt wvlt, it Hilbrt trfor it Hrtly krl bolut Vlu of,b officit for l c l c ti (or pc) b (b) bolut Vlu of,b officit for B pplyig th lytic (or Fourir-Lik) Wvlt lyi Hilbrt trfor h lry b utiliz i wvltb igl lyi [] I thi ctio w crry prliiry ivtigtio bout lytic wvlt lyi Figur 6 how tr frqucy brkow igl Figur prt -lvl wvlt lyi, by th lytic xic Ht wvlt Fro Figur, it c b obrv tht th xic Ht wvlt c itify th prc of both frquci wll th ti wh occur th frqucy chgig, which c b bttr ccurt t lowt cl vlu Th oulu of tht lytic wvlt lyi how tht th high frqucy igl c b viw t lowt cl vlu th low frqucy igl t highr cl vlu ti (or pc) b (c) Figur : 8-lvl wvlt lyi of ltr igl uig: () th orlt wvlt; (b) it Hilbrt trfor, (c) it Hrtly krl Th cl whr Hrtly-Lik orlt wvlt y prfor bttr igl lyi th th ipl orlt wvlt c b obrv through th cl ti chrt Fro Figur, igifict iffrc o th wvlt trfor t lvl c b, wh uig v, o or ytricl wvlt It follow tht, pit th ti hiftig ipll by th wvlt trfor, ytricl Figur 6: tr frqucy brkow igl

6 XXII SIPÓSIO BRSILEIRO DE TELEOUNIÇÕES - SBrT, -8 DE SETEBRO DE, PINS, SP l c l c Rl prt of,b for l c 6 8 ti (or pc) b oulu of,b for 6 8 ti (or pc) b l c Igiry prt of,b for 6 8 ti (or pc) b gl of,b for 6 8 ti (or pc) b Figur : -lvl wvlt lyi of tr frqucy brkow igl uig th lytic xic Ht wvlt xpl, Figur 8 how, rpctivly, th th th-lvl wvlt trfor, coirig wvlt riv fro th xic Ht wvlt It c b th low high frqucy igl, wh uig rl wvlt Figur how th orliz oulu of th th th-lvl wvlt lyi, wh uig th lytic, or Fourir-Lik, xic Ht wvlt Thi ftur how tht wh lyzig iiviully cl, th ti itrvl wh occur iffrt frqucy cott c b tit xic Ht it Hilbrt Trfor it Hrtly Krl it Fourir Krl Figur 8: Th cotiuou-ti wvlt trfor of tr frqucy brkow igl uig th xic Ht wvlt, it Hilbrt trfor, it Hrtly krl, it Fourir krl othr wvlt uig th cl prtr Th oulu i u for th Fourir-Lik wvlt lyi th cl prtr Th oulu i u for th Fourir-Lik wvlt lyi u p lit 8 6 Lvl Lvl 6 8 Figur : Norliz oulu of th th th-lvl wvlt cofficit of tr frqucy brkow wh uig th lytic, or Fourir-Lik, xic Ht wvlt VII ONLUSIONS Nw wvlt fuctio hv b itrouc tht ight iprov th cotiuou-ti igl lyi tiytricl wvlt c b ig throughout th Hilbrt trfor of ytricl wvlt, vic-vr Togthr, thy r cll Hilbrt trfor pir of wvlt Fourir- Lik Hrtly-Lik wvlt hv b riv fro th Fourir Hrtly trfor krl, which hv b writt, o th bi of Hilbrt trfor Th xpl c illutrt tht, pit th ti hiftig ipll by th wvlt trfor, ytricl wvlt c grp or ftur fro igl hvig o prticulr ytry itiolly, Fourir-Lik lytic wvlt hv pottil pplictio for iturbc frqucy tctio REFERENES [] K hui, Itrouctio to Wvlt, S Digo, : cic Pr, [] iiti, Y iiti, G Opphi, J- Poggi, Wvlt Toolbox Ur Gui, Nw York: Th thwork, Ic,, [] R N Brcwll, Th Fourir Trfor It pplictio, Nw York: cgrw-hill,, 8 [] V Opphi, R W Schfr, J Buck, Dicrt-Ti Sigl Procig, Nw Jry: Prtic Hll, 8 [] E Ozturk, Okucur, G tki, Wvfor Ecoig of Biry Sigl uig Wvlt it Hilbrt Trfor, Proc of th IEEE ISSP, vol, pp 6-6, Itbul, xic Ht it Hilbrt Trfor it Hrtly Krl it Fourir Krl Figur : Th cotiuou-ti wvlt trfor of tr frqucy brkow igl uig th xic Ht wvlt, it Hilbrt trfor, it Hrtly krl, it Fourir krl othr wvlt uig