Higher-Order Differential Energy Operators

Transcription

1 P Highr-rdr Diffrntial Enrgy prators Ptros Maragos and Alxandros Potaianos Instantanous signal oprators Abstract of intgr ordrs ar proposd to asur th cross nrgy btwn a signal and its drivativs. Ths highr-ordr diffrntial nrgy oprators contain as a spcial cas, for, th Tagr-Kaisr oprator. Whn applid to possibly odulatd) sinusoids thy yild svral nw nrgy asurnts usful for paratr stiation or AM FM dodulation. Applying th to sapld signals involvs rplacing drivativs with diffrncs which lads to svral usful discrt nrgy oprators dfind on an xtrly short window of sapls. I. HIGHER-RDER ENERGY MEASUREMENTS Instantanous diffrncs in th rlativ rat of chang btwn two signals can b asurd via thir Li brackt bcaus thn #"%'& )*)& +,".-/02145& 6-795& #" bcos th continuous-ti Tagr-Kaisr nrgy oprator [1], [2] <6=45&,?@BA C1 +& " 0-/. Dots dnot ti drivativs. Not th antisytry: #"%1: 6,". If 21'&, which has bn usd for tracking th nrgy of a sourc producing an oscillation [2], [1] and for signal and spch AM FM dodulation [4], [5]. In th gnral cas, if and rprsnt displacnts in so gnralizd otions, th quantity +& "B1D& )& EFGA has dinsions of nrgy pr unit ass), and hnc w ay viw it as a cross nrgy btwn and. This nrgy-lik quantity H& & IJBA was usd in [2], [] to analyz th output <2K of th nrgy oprator applid to a su of two signals. In our work w us th cross nrgy btwn a signal and its highr-ordr drivativs to dvlop highr-ordr nrgy asurnts. Spcifically, w dfin th LM.N -ordr diffrntial nrgy oprator DE) QP <R,S PUXW "61'& S P/T%V,W S 'LZ1\[Y]_^ ]Q`abaca as yilding th cross nrgy btwn a signal d<, and its fl_g^/,m.n drivativ or intgral), whr jjji P PYW k +., - Lonp^ S.,h ljjj +.,Y L21\[ qsr PYu+V,W T6t S <v0 v0'lowpx^ Subittd: Nov Accptd: April This rsarch work was supportd by th National Scinc Foundation undr Grant MIP Th authors ar with th School of Elctrical & Coputr Enginring, Gorgia Institut of Tchnology, Atlanta, GA 02, USA.

2 q & P 1 P IEEE SIGNAL PRCESSING LETTERS, VL.XXX, N.XXX, XXX dnots a signal drivativ for positiv ordr L or an intgral for L ngativ. f practical currnt intrst ar th DEs of positiv ordrs. Th scond-ordr DE, asuring th nrgy of a haronic oscillator producing a signal xp, givs to th na nrgy sinc it is idntical to th standard nrgy oprator. Th zroth-ordr oprator is. 1 o this lattr xprssion was rcognizd in [] as th ngativ of th nrgy of th signal intgral. Th first-ordr DE yilds zro for any signal. Two nw and usful nrgy asurnts ar givn by th third- and fourth-ordr DEs: <6='& GA E*0 S Not that as also obsrvd in []).=1 <.=1 <6='& 0 S <6 W *0 S Hnc th third-ordr DE is an nrgy vlocity oprator, whras th fourth-ordr DE has dinsions of nrgy acclration. In gnral, th highr-ordr oprators can b gnratd by lowr-ordr oprators with a 2-stp rcursion: Finally, not that QP <IKh1 P QP.=1.K QP/T%V QP <K < PUT & 6 P/T%V,W,S "5K & PUXW 6,S " a Whn th nrgy oprators xp ar applid to a sin wav, thy yild products of powrs of th aplitud and frquncy. Spcifically, th cosin +.,=1 hk rprsnting th rspons of an undapd haronic oscillator satisfis th otion quation A crats th nrgy rcursion with initial conditions 1: indx L yilds P ok 1 [. This P PUT P QP 14 h%k# " and V 1\[. Running this rcursiv quation in both forward and backward ordr hk "61 If th aplitud and/or frquncy abov nrgy quations ar approxiatly valid providd that uch with rspct to th carrir frquncy. Furthr, bcaus i k l [ V,u L21 ]_^ ] 0] abaca Xx^/ L21 []Q`]abaca of d<, ar slowly ti-varying, i.., if is an AM FM signal, thn th., and 1 <, do not vary too fast or too ar low-pass signals, th abov instantanous nrgy asurnts can b usd for robust stiation of instantanous aplitud and frquncy in odulatd sinusoids. An application of th fourth-ordr DE, in conjunction with th standard nrgy oprator hk : <6 <6 1 #" " 1. % <6 stiat th aplitud and frquncy of a possibly odulatd) sinusoid +.,=1, is to

3 ` u u IEEE SIGNAL PRCESSING LETTERS, VL.XXX, N.XXX, XXX 4 This is an nrgy sparation algorith, slightly diffrnt fro th on in [5], which can also b usd for AM FM dodulation. An application of th third-ordr DE is to stiat th nrgy dissipation rat in dapd oscillations. Naly, givn a dapd cosin, th daping factor can b found using and th nrgy oprator. Thus, if T +.,=1 r hk#, [, thn x1: < 1: ^. <6 ` II. DISCRETE-TIME PERATRS Applying th nrgy oprators to sapld signals rquirs rplacing drivativs with diffrncs. This lads to a varity of discrt nrgy oprators for ach ordr L bcaus thr ar any diffrnt ways of discrtizing drivativs. Th siplst approach is to first discrtiz th Li brackt by rplacing drivativs with ti shifts. Naly, rplacing continuous-ti signals d<, with squncs E1\+ of thir sapls, also dnotd as 6", and first-ordr drivativs d., & with backward diffrncs 6"614. 6", RI^ "< - convrts th continuous-ti oprator +," <, into th discrt-ti oprator < 6", 6"<= 6"b C ^ "@ 9 ^ " 6" whr w hncforth assu 41 ^. Using sytric diffrncs 6"d1. CKp^ " ^ "< - ` to rplac ti drivativs yilds a sytric discrt oprator qual to th avrag of 6"%1 K ^" aks idntical to th discrt Tagr-Kaisr nrgy oprator [1]. 6" R 6"* o ^ " K ^"61. 6"f ok ^" Gnralizing th abov rsult by using asurnts for a signal at two conscutiv sapls.) Using 6"61 2K L" in 6". For xapl, w dfin th LM.N -ordr discrt 1 nrgy oprator P. 6". 6"f EK L_ ^ "< L21 [0^ `5 ababa 1 6" KFL_ `/"0* 9g^"b KgL) ^ " lads us to dvlop discrt-ti highr-ordr nrgy For LE14^ w always gt zro sinc V p[. For LE1 ` w obtain th standard discrt nrgy oprator For L21, w obtain an asytric discrt nrgy vlocity oprator < 0= 5 u+v * whras L21 yilds a discrt nrgy acclration oprator: % < = * Iportant aspcts of ach xp ar th lngth of its corrsponding indx window and its ti alignnt a)sytry. Nxt w invstigat ths issus for L21. Sinc rquirs a 4-sapl oving window 1 For notational siplicity w us th sa sybol for both th continuous- and discrt-ti highr-ordr oprators nrgy oprator, sinc th input signal can rval this aspct of th oprator. u I*^ )K@`7", its output and th Tagr-Kaisr.

4 ` T V " " IEEE SIGNAL PRCESSING LETTERS, VL.XXX, N.XXX, XXX 5 at th window s cntr occurs at th continuous ti instant h14 xk [a. n sipl approach to liinat this ti isalignnt is to rplac < with its avrag ovr two conscutiv sapls and thus hav a sytric third-ordr nrgy oprator with a 5-sapl window /< = 9 `52Kg`/". < 0K Applying th oprators xp to discrt possibly dapd) cosins yilds discrt nrgy quations QP K# "61 uput. flig^/ which ar usful for paratr stiation in sinusoids. In addition, ths nrgy quations hold approxiatly whn th cosin has ti-varying aplitud and frquncy that do not vary too fast or too uch with rspct to th carrir, i.., whn th input is a sapld AM FM signal. This allows us to find discrt AM FM dodulation algoriths by cobining th abov nrgy quations of various ordrs. For xapl, by using, HP, and th undapd cosin nrgy quations K # "1 fl=9^/ " for LZ1 `5, a discrt algorith was proposd in [6] for instantanous frquncy tracking, which is closly rlatd to th discrt nrgy sparation algorith in [5]. W conclud by noting that, all th abov discrt highr-ordr nrgy oprators can b unifid as spcial cass P of a class of quadratic nrgy oprators, or thir wightd linar cobinations, whr P < 6"<=\ 6"b KgL"0* Co"b KgLQKo" for LZ1 [ ^ `5ababa, 1:^# `5 acaba. Siilar oprators hav )P also bn studid indpndntly by Kaisr [7]. Th class P V QPYu V V contains all th discrt highr-ordr nrgy oprators sinc.g., and. For LZ1 [ th oprators can also b viwd as spcial cass of th class of quadratic dtctors BK " o" proposd in []. Th gnral oprators In addition, ach P P P provid so intrsting nrgy quations: up K# "%1 KFL0 can b gnratd rcursivly fro oprators of lowr ordrs L6. III. ALTERNATIVE DISCRETIZATINS Instad of discrtizing th Li brackt and rplacing drivativs with ti shifts, an altrnativ approach to P discrtizing is to rplac ach om.n -ordr signal drivativ involvd in its xprssion with backward diffrnc T%V oprators 1 or sytric diffrncs 1. For L21p` th asytric diffrnc yilds a 1-sapl shiftd vrsion of th discrt nrgy oprator, whras th sytric diffrnc yilds a -point avrag of, as shown in [4]. For LZ1 using th diffrnc yilds anothr asytric discrt nrgy vlocity oprator < 1 ` <6" r T%V < s.

5 & P T IEEE SIGNAL PRCESSING LETTERS, VL.XXX, N.XXX, XXX 6 which is coputationally or coplicatd than. Th only slight advantag of ovr is that, whn th input is a discrt cosin 1 )K #, it givs as output - ` which for ^ is uch closr to zro than th output f` of. Rcall that th continuous-ti third-ordr nrgy of a cosin is zro. Using th diffrnc for L21 yilds anothr sytric discrt nrgy vlocity oprator. R. " r V 1 u+v T%V < +K < < s <, which yilds a zro whn th input is a discrt cosin consistnt with th continuous-ti rsult) but has a longr window than th sytric oprator, i.., it nds a 7-sapl window or coplicatd xprssions, using ithr th or th diffrncs. C 2K ". For L w gt vn P PYW A final approach w hav considrd for discrtizing th continuous oprators is to i) rplac drivativs PUT%V,W +S with diffrncs, ii) shift th diffrncs by any rquird nubr of sapls so that th two trs PYW ss & and 0S ar coputd at th sa ti location aftr discrtization. dd-ordr drivativs ar cntrd at ti instants g[a if L is vn, this is balancd by th othr odd drivativ in th product which is cntrd at )]@[a. This approach yilds th sa discrt oprator for LZ1 ` _P but crats altrnativ discrt oprators with intrsting proprtis for L `. Thus, for L21\`5 w obtain th altrnativ oprators )P : < < < 1 For xapl, to obtain < fro <R'& BA ) 0S backward diffrncs) and cntrd at ti )o[0a # GA and 0S ".0. s T%V < =1 < u+v < sj` < K. T%V.I@ udv. s, th odd ordr drivativs &, +S ar discrtizd using, whil th discrtization of A is cntrd at ti so that ar coputd at th sa ti instant). Not that yilds zro whn th input is a discrt cosin, as in th continuous-ti cas. Th abov discrt P xp oprators rquir a sall window and satisfy rcursiv forulas QPU QPUT of th sa typ as th rcursion <6=1. - 5& % satisfid by thir continuous countrparts with drivativs appd to diffrncs highr-ordr nrgis dpnds on th spcific application. ). This is thir ain advantag. In gnral, th bst typ of discrtization of REFERENCES [1] J. F. Kaisr, n a sipl algorith to calculat th nrgy of a signal, in Proc. IEEE ICASSP-90, Albuqurqu, NM, April 1990, pp [2] J. F. Kaisr, n Tagr s Enrgy Algorith and Its Gnralization to Continuous Signals, in Proc. 4th IEEE Digital Signal Procssing Workshop, Mohonk Nw Paltz), NY, Sp [] J. F. Kaisr, So Usful Proprtis of Tagr s Enrgy prators, Proc. IEEE ICASSP-9, Minnapolis, MN, April 199, pp. III: [4] P. Maragos, J. F. Kaisr, and T. F. Quatiri, n Aplitud and Frquncy Dodulation Using Enrgy prators, IEEE Trans. Signal Procssing, vol. 41, pp , April 199.

6 IEEE SIGNAL PRCESSING LETTERS, VL.XXX, N.XXX, XXX 7 [5] P. Maragos, J. F. Kaisr, and T. F. Quatiri, Enrgy Sparation in Signal Modulations with Application to Spch Analysis, IEEE Trans. Signal Procssing, vol. 41, pp , ct [6] L. Frtig and J. H. McClllan, Instantanous Frquncy Estiation Using Linar Prdiction With Coparisons to th DESAs, subittd to IEEE Signal Procssing Lttrs, [7] J. F. Kaisr, privat counication, Nov [] L. Atlas and J. Fang, Quadratic Dtctors for Gnral Nonlinar Analysis of Spch, Proc. IEEE ICASSP-92, San Francisco, CA, Mar. 1992, pp. II:9-12.