Parametric envelope in LPC speech coders

MELECON '85/ Volume 11: Digi al Signal P ocessing A. Luque, A.R. Figuei as Vidal, V. Cappellini (eds.l Else ie Science Publishe s B. V. (No h·Holland)/ C) IEEE 1985 .. ;· :-;.- . :. :: .! -{ ~ :·=· · .: ·: .·-- . ~ .. -. . .. ,.______ .. -· : -;: __ ~ : ~-~ -- ~- · . . .· :- . _ __ , .. . .- ·-.-. .. - - ~ - ~ - ;:- ·· ';";" . .. ...: .• _ : _._ ; ~ . ~~: . . - . : .·:..- ·· - 107 PARAMETRIC ENVELOPE IN LPC SPEECH CODERS A. lob eno, M.A. Iagunas, F. Vall e d 1. P ocesado de sei'lal en ca unicaciones E. T. S. I. Telecoounicaci6n Apdo 30002 08071 Ba celona Spain Abs ac : Du ing he las decade, many e o s ha e been de o ed o he ela i e impo ance o associa ed lmc ions like magni ude a l );:hase o Fou ie T ans o ms in image and signal ba lwid h educ ion. 'Ihe epo ed ' IO k deals wi h he impo ance o he eal en elope and ins an aneoJS equency in signal analysis/sin esis p oblems. In his pape au ho s show a me hod o pa ame ize he en elope and ins an aneaJS ~ equency o a eal signal. 'Ibis me hod is e y closed o spec al analysis me hods m he sense ha wi h an app opia e s udy, il e do ain and equency da ain can be analyced in a simila way. I. INTRODUCTION Signal p ocessing echniques a e cha ac- e ised in he equency domain by means o he magni ude and he phase o he asso- cia ed Fou ie ans o m o he signals unde analysis. I is clea ha he e a e no simila i ies be ween ime domain and equency domain signal ep esen a ions, in he sense ha one domain uses a eal signal and he o he wo signals. Many au ho s ha e epo ed some in e es ing esul s which deals wi h his opic and acing he ques ion o edundancy in he ep esen a ion o a eal signal in he equency domain -1- ,-2- • In some_ way we can ecognize ha ega dless o he p e iosly men ioned wo k in signal p ocessing, we a e e y amilia; wi h magni ude/phase ep esen a ion in he equency domain ins ead o eal and imagina y pa s o he complex Fou ie ans o m. In o he wo ds, no ma e edundancy, a designe can always ecognize be e , a low pass il e in a magni ude- -phase plo han om he cu es co es- ponding o hei eal and imagina y pa . The poin is, up o wha deg ee he eal pa o he anali yc signal, he gi en signal x( ), p o ides a be e ep esen- a ion. o he phenomena unde s udy. Follow1ng he equency domain expe ience, in a e y heu is ical way, he conclussion will be ha magni ude and ins an aneous phase a e a good in o ma ion suppo o ~he designe . I'n o he wo ds, en elope and 1ns an aneous equency, looks like, hey dese e he same impo ance in he ime domain ha magni ude and phase in he equency domain. In his pape , he au ho s will explo e he po en ial o such ep esen a ion and he di icul ies a ound handling i in CH 2185-7/85/0000-0107(8)/$01.00 © 1985 IEEE signal p ocessing p oblems. To be mo e concise, en elope and ins an aneous e- quency a c compu ed o simula ed signals using wo app oaches and he p oblems which a ise a e epo ed. Also, and dealing wi h a p oblem o bandwidh educ ion in a signal communica ion sys em, i is shown he ad an ages o well-known spec al es ima- ion p ocedu es in pa ame ic models o en elope and ins an aneous phase. The pape is o ganized as ollows: Sec ion II p esen s an in oduc ion o he gene al concep s o in e es abou anali yc signals. Sec ion III epo s he ela ion- ship be ween spec al es ima ~on p ocedu es and en elope smoo hing. A e Sec ion III, he nex one will show undamen al ques ions abou en elope and ins an aneous equency. Finally, some p elimina y esul s a e in oduced. II ANALITYC SIGNAL Any signal x( ) has an associa ed complex signal a ( ) being i s Fou ic ans o m wice hl ig h hand side o he o iginal X(w) • [ 2 X(w) ; w>O A (w)= X 0 ; w<O (1) Clea ly, a ( ) is a complex signal, and i s ime o mula ion includes as eal componen he o iginal da a signal x( ) and as imagina y componen he so-called Hilbe T ans o m o i . ax( ) = x( ) + j hx( ) whe e hx( )=(l/ { 00 (X( ' )/( - ' ))d ' ( 2) (3) This las o mula s eems om he de ini- ion o he analy ic signal in he equen- : ! . ... . . ·. . ....... · ~ .. --~-:.. .. · .. ,._ .... ___ .. . ... .. ........ ·-. .. . . 108 A. Mo eno e si. cy domain. Mo e conc e ly, de i ~d om he causal condi ion o A (w). I 1s easy ~o conclude h!"~' a ) ol~::·s he same ole 1n he ime domainx ha X(w) doe:;; .'.!! he e- quency domain. Howe e , he main conce n o ::his wo k a e no x( ) and h ( ), bu he al e na i e ep esen a ion s~own in (4), a ( ) ,. x( )+jh ( ) = e ( ) exp ~ x( ) (4) X X X being e ( ) and 41 ( ) he al eady en elop/ and ins a~ aneous phase. e e ed e 2 ( ) = x 2 ( ) +h 2 ( ) = I ax ( ) I 2 X X (S.a) ~ ( ) = an- 1 (h ( )/x( )) = Phase X X I is wo hwhile o men ion ha gi en x( ) he e is no an uniquiness · in inding unc ions e ( ) and ~ ( ). To check his, jus add s~me esidu~l only o he ima- gina y oa o (2). The uniquiness ? ex~ 2 cos( ( )) in ep esen 1ng x( ) l1e::; wl u he '*inimum phase condi ion o a ( ). Exac ly, he en elope/phase ep esen ~ ion is conec ed wi h he anali yci y o a ( ;) in he uppe hand side o he ( plane. ¥ ~us no poles can be inside he uppe hand s de o plane in o de o gua an ee he causa- li y cons ain o A (w) holds. Wi h espec he i~o ance o bo h pa a- me e s, no e ha he en elope p o ides in onna ion conce ning o he ime ene gy dis ibu ion and he ins an aneous equency wi h ze o-c ossing in o ma ion. III SPA METUODS IN ENVELOPE REPRESENTATION F om he de ini ion o en elope o a gi en analy ical signal i can be in e ed ha he squa e o he en elope can be iewed as a ime domain pe iodog am. In o he wo ds, i we know he Fou ie T ans o m o he analy ic signal Ax(w), we can ob ain he en elope in he same way as we ob ain he Pe iodog am o a gi en da a signal. Fou ie A (w) -------~ ax( ) -~1·1 2 -•en elope ~ . T ans o m Fcu ie 2 (6) x( ) ------• X(w) --•l'l -•Pe iodog am T ans o m F om he p e ious simila i y we can conclude ha he mos amilia p ocedu es applied in pa ame ic spec al es ima ic:m could also be appl~d o e A (w) o ob a n an es ima ion o e ( ). x Conc e ly, he m~ popula maximum en o- py echnique can be used o e a sampled e sion o he ·causal signal A (w). I A (w) is gi en as a da a egis e engh o W2 samples wi h index 1 (i.e. Ax(l); 1 = 0, N/2-1), hen a linea p edic o o ::oe icien s a (q) (q = l,Q) can be designed by minimizing he squa e e o : (1). Thus he linea p edic o is: "' Q A ( 1 ) = I: ( q) Ax ( 1-q) ( 7 ) X q=l The p~dic o esidual :(l) is de ined as A (1)-A (1) and he quan i y o be m nimiz~d is: N2 2 I: l :(lll (8) l=N1 . The e a e many well-known p ocedu es in he li e a u e o sol e (8) depending on he choice o N1 and N2• We selec he p ocedu e o co ~la ion, ega dless i is no ecommended when he signal unde analysis is de e minis ic in na u e, as i is he case due o he causal cha ac e o A (1). Anyway, he consequences de i ed ~om he use o Le inson algo i hms in he minimiza ion o (8) will be g ea e deg ee o smoo hing o low esolu ion, in e ms o spec al es ima ion, u - he esul ing pa ame ic en elope ep esen a ion. O he p ocedu es can be ca ied o e he p oblem p c iosly s a ed in (7) and (8) (sec o example-s~. As conce ns wi h his pape , i is no e y ele an he spec al es ima ion p ocedu e selec ed and he eade could change i acco dingly wi h he desi ed ea u es in he esul ing en elope es ima e. The en elope es ima e, one~ coe icien s a(.) ha e been ob ained, can be de i ed assuming he whi e cha ac e o he esidual sequence (1), Wi h his assump ion he squa e magni ude o he in e se disc e e Fou ie ans o m e(n) is cons an , so ha ; £ (l)=a(,) *A(,) X e (1)= 8 (,) • ax(n) ( 9) and because he whi e cha ac e o d 1) is assumed, (10) being k he a e age po~.Je o dl) (i.e. he minPmum o he objac i e in he design p ocess (8)) In summa y, 2 K0 = E I 1 d 1 >I 1 ;~(n) = K0 /l8 Cnll 2 whe e Q (ll.a) (ll.b) 8(n) = 1+! a(q) exp (j2nqn /N) (12) q=l The es ima e shown in (ll.b) ha e been applied succes ully o oice speech eco ds. In Fig 1 he eade can see such !.. ·.· ·. .. - ~ . . ... . . _. .· :: ·.:. ·. ·: .:. .. . := · ···· • :. ~ : .. ~ - . < :-~- < : . ... :- . :· . .. . ,. .: .. ·. Pa ame ic En elope in LPC Speech Code s 109 en elope es ima e compa ed wi h he ac ual ~n clope and he da a signal eco d •. The da a leng h was 32 ms and he. l1ne~ p edic ion o de 20 • I can be Vlewed 1n his plo ha he LP es ima e p o ides a smoo hed e sion o he ac ual en elope. The eplica ob ained wi h he es ima e esul s accu a e when he o de 0 is almos double han he numbe o pe iods included in he o iginal signal. (a) (b) (c) Fig.l. (a) Voiced speech signal, (b) en clope,(c) pa ame ic en elope. No e ha he co esponding analy ic signal o he so-called pa ame ic en elope is always a minimum phase signal in he sense ha a (~) has no ze os in he uppe hand side o he complex plane ~ • This p ope y gua an ees ha he loga i m o he es ima e en elope and he phase o a (n) a e a Hilbe T ans o m pai . X IV INSTA."lTANEOUS FREQUENCY FROM PARAMETRIC ENVELOPE. In he p eceding sec ion i is in oduced how en elope in o ma ion could be smoo hed by using well-known L.PC echniques. T~is poin is e y in e 'cs lng. as conce n~ w~ h da a a e educ ion o s1gnal ansm1ss1on pu poses: The ques ion which emains is ha , a non minimum phase signal ~eods also, o be eco e ed a he ece1 e , ins an aneous phase in o ma ion. To ealice how ins an aneous phase o equency can be ep esen ed by a ini e se o pa ame 2 s, i wi 11 be in e es ing o explo e how classical equency disc i- mina o s wo k o de ec ins an aneous equency. . Conside ing he linea sys em depl ed in Fig.'2, i he inpu signal is gi en wi h e ( ) almos cons an , he ou pu signal wh.J. be: . / h( ') e ( - ') cos <I> ( - ' )d '; X ~ X - e J ~ X( ) h( ')cos<l> x( - ') d ' -~ _ .. (13) : .. · .· ·.· . . :· -. -. . . · .. :: ·. . . ~' .. . . . · : ·:· :· ·:· ·-~· . -. - .. .. '1',. - > Fig 2. Ins an aneous equency . hough linea sys ems. Thus,assuming ha wo e ms o Taylo ' s se ies o he phase e m is adequa e o ep esen i in he con olu ion in eg al, < >x( - ') = < >x( ) - ' ~ x( ) he ou pu signal will be: (14) (15) ~ being H(.) he ans e esponse o he linea sys em (i.e. he Fou ie T ans o m o h( )), F om (15) i is easy, o conclude ha , being e ( ) almos cons an , he en elope o y( ) xis gi en om ans e . esponse and he ins an aneous equency ljlx o he inpu signal. This easonin'J d9 no p ;eclu- de he anali yci y o e ( ).H( ~ x)exp(J ~ xl' hus o a oid his as~ump ion he aoo e compu a ions will be ca ied o e he complex signal exp(j<l>x>· The men ioned complex signal is ob ained om he quo ien o he gi en signal a.x.( ) and i s ac ual en elope ob ained wi n a Hilbe ans o m. x ( ) = a ( )/ex( ) m x (16) No e ha x ( ) is no analy ic in gene al. Only n he case o ex( ) band limi ed o w and cos (cp ( ll no spec aly o e lapped iR his band, ~ ( ) will ha e an analy ic cha ac e . This cWse is no longe ue o many p ac ical signals x( ). Rega dless o he analy ic condi ion o x ( ), i i is a~plied o a linea sys em hT l, he ou ~u signal will be: • ym( ) .. H(<l>xl exp(j<Px) (17) As a consequenc~ we could use p ocedu es o smoo h E {I y I} in he same ashion we use hem in h~ p e ious s:lc ion. Fu he - mo e i H(.) is a de 'i a i c in he ime domain he magni ude o 1 Ym( ) 1 wil~ be ap oxima ely l$ ( ll • In summa y, as.ln a classic disc ilbina o , using a llnea sys em wi h cons an slope equency 'es- ponse we can ob ain he ins an aneous e- quency as he magni ude o a complex signal. . . ; .• .· . • - ', •- ' I ~ • ... · . ... .. . , . .. '•· . . · '· .. 110 A. Mo eno e al. No e ha o eco e exac ly he ins an- aneous ~equency i mus be posi i e. This p~oblem could be a oided by adequa e scaling and ca~ i~e modula ion. Anyway he designe mus gua an ee he basic assump- ion o no ab up phase changes in he ins an aneous phase (low ins an aneous equency) o de i e he p e ious esul s. To check ou he esul ing pe o mance o he p ocedu e, a 128 da a s ample eco d wi h ins an an eous equency e olu ion as is shown bellow, was used. [ 0.125 : n = 1, 32 i(n) linea : n = 33, 95 0.25 : n = 96,128 The linea sys em was implemen ed by a ans e unc ion ha a ies linea y whi h he equency. The ins an aneous equency wich esul s can be iewed in Fig. 3 oge he wi h he o iginal signal x(n). Fig 2. Top, o iginal signal. Bo om, ins an aneous ~equency de ec ed. I can be iewed some side e ec s due o he 1"indow e ec s in he de ec ed ins an aneous equency. I is cspec ed ha his dis o sion wich also appea s in he en elope will no appea in a con inous p ocessing o he signal unde analysis. Anyway, om he esul s ob ained by he au ho s he app oxima ion in he cen a zone, in a block p ocessing ashion, is accou a c enough o ep oduce he o iginal signal. CONCLUSIONS The 'main con ibu ion o his wo k esides in e ealing he in e es o hese unc- ions associa ed o any signal o be p o- cessed. No essen ial e o s in he pas ha e been de o ~d o such ep esen a ion o€ signals, jus in modul a ion p oblems some au ho s epo ed in e es ing esul s in such ield Cu en ly, ime- equency ep esen a ions exhibi a new look o he p oblem. We eally belie e ha his unc ions dese e mo e a en ion, because i ia well ecognized ha essen ial in o ma ion is in ol ed inside he en elope o he ins an aneous phase e olu ion. In his pape i is epo ed how en elope and ins an :<' n ~'=''..!S !::-equenc a c sui abl!! o pa ame ic me hods which a e ami 1 i i'J in spec al es ima ion p oblems. Ne e heless he main guidelines o wo k o e hese associa ed uc ions ha e been shown, u he wo k mus be de o ed o signal p ocessing ools in o de o alle ia e unce ainly in en elope/phase ep esen a ion o signals.Also side e ec s o di e e n s ways o ob ain en elope and phase will be explo ed in he u u e. REFERENCES -1- Cha les Se homie . 'Ins an aneous equency and Ene gy Dis ibu ion o a signal'. Signal P ocessing 5.(1983). -2- Pe iclis Y. K onas and Nicola Papp. 'Ins an aneous En elope and Phase ex ac ion om eal signals: Theo y, implemen a ion, and an applica ion o EEG analysis'. Signal P ocessing 2 {1980). -3-Alan V. Oppenhcim, Ronald w. Sha e . 'Digi al signal P ocessing' P en ice Hall.l975. -4-He be s. Voelcke . 'Towa d a uni ied Theo y o Modula ion. Pa !. Phase-En elo- pe Rela ionships'. P oceedings o he IEEE ol.54: n.3. Ma ch 1966. -5-S.M.Kay and S.L.Ma plc,J . 'Spec um Analysis- A Mode n Pe spec i e' P oceedings o he IEEE ol. 69: n.ll. No embe 19 81. ·' . "- - I i L I . a;