Statistical Parametric Speech Synthesis with Joint Estimation of Acoustic and Excitation Model Parameters

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1 Statstal Parametr Speeh Synthess wth Jont Estmaton of Aoust and Extaton odel Parameters Rannery aa, Hega en, J F Gales Toshba Researh Europe Ltd, Cambrdge Researh Laboratory, Cambrdge, UK {rannerymaa,hegazen,mfg}@rltoshbaouk Abstrat Ths paper desrbes a novel framework for statstal parametr speeh synthess n whh statstal modelng of the speeh waveform s performed through the ont estmaton of aoust and extaton model parameters The proposed method ombnes extraton of spetral parameters, onsdered as hdden varables, and extaton sgnal modelng n a fashon smlar to fator analyzed traetory hdden arkov model The resultng ont model an be nterpreted as a waveform level losed-loop tranng, where the dstane between natural and syntheszed speeh s mnmzed An algorthm based on the maxmum lkelhood rteron s ntrodued to tran the proposed ont model and some experments are presented to show ts effetveness Index terms: statstal parametr speeh synthess, traetory hdden arkov model, extaton modelng, fator analyss 1 Introduton In typal statstal parametr speeh synthess [1], speeh parameters are extrated from speeh waveforms and ther traetores are modeled by a statstal model, suh as a hdden arkov models (Hs) The parameters of the models are estmated so as to maxmze ther lkelhood gven the tranng data At the synthess stage, a sentene-level sequene of states of the traned statstal models s omposed aordng to an nput text, and then speeh parameters are generated so as to maxmze the output probabltes of suh states [1] Fnally, a speeh waveform s re-onstruted from the generated speeh parameters by assumng the soure-flter produton model [] In ths paper a novel approah to statstal parametr speeh synthess s proposed, n whh aoust model parameters are ontly estmated wth parameters of a stand-alone extaton model n a way to maxmze a lkelhood funton nludng both models gven the speeh waveform Ths s done by assumng that speeh an be represented by the onvoluton of a slowly varyng voal trat mpulse response flter derved from spetral parameters, and an extaton soure In the proposed approah spetral parameter extraton s ntegrated n the ont tranng of aoust and extaton models, n a smlar way to the spetral analyss based on fator analyzed traetory H of [3] However, n the present method the speeh waveform s the observed term, and ts relatonshp wth the spetral features, onsdered as a hdden varable, s obtaned va the extaton model parameters The proposed ont estmaton of aoust and extaton models based on the maxmum lkelhood (L) rteron an be vewed as a losed-loop tranng method for statstal parametr synthess, suh as the one ntrodued n [4] Indeed, one smlarty between the urrent method wth the one desrbed n [4] s the ntegraton of spetral extraton durng the tranng Nevertheless, the phlosophy of the approah desrbed n [4] s the explt mnmzaton of a dstorton of ampltude spetra of natural and syntheszed speeh, whereas for the present ase the dstane between natural and syntheszed speeh waveforms s mnmzed n the tme doman Ths paper s organzed as follows In Seton the bas dea s ntrodued; n Seton 3 the ont modelng of aoust and extaton models s defned; Seton 4 presents a tranng algorthm based on the L rteron; Seton 5 shows some experments, and onludng remarks are n Seton 6 Bas dea 1 Conventonal framework In a typal statstal parametr speeh syntheszer [1], frstly aspetralparametervetor = ˆ T 1 s extrated from the speeh waveform, where t =[ t() t(c)] s a C-th order spetral feature vetor at frame t,andt s the total number of frames Estmaton of aoust model parameters s usually done through the L rteron, e, ˆλ =argmax p( l,λ ), (1) λ where l s a transrpton of the speeh waveform and λ denotes a set of aoust model parameters At run-tme synthess, s generated for a gven text to be syntheszed l so as to maxmze ts output probablty ĉ =argmax p( l, ˆλ ) () These generated features together wth an F -generated extaton sgnal are utlzed to synthesze the speeh waveform by usng the soure-flter produton approah [] Proposed framework Sne the ntenton of any speeh syntheszer s to mm the speeh waveform as well as possble, a statstal model defned at the waveform level s proposed here The parameters of ths new model are estmated so as to maxmze ts lkelhood gven the speeh waveform tself, e, ˆλ =argmax p (s l,λ), (3) λ where s =[s() s(n 1)] s a vetor ontanng the speeh waveform, wth s(n) beng a waveform value at sample n, N the number of samples, and λ denotng the set of parameters of the ont aoust-extaton model By ntrodung two hdden varables: the state sequene q = {q,,q T 1} (dsrete); and spetral parameter =

2 ˆ T 1 (ontnuous); (3) an be rewrtten as ˆλ =argmax p (s,, q l,λ) d (4) λ q =argmax p (s, q,λ) p ( q,λ) p (q l,λ) d, λ q where q t s the state at frame t The meanngs of the terms p (s, q,λ), p ( q,λ), and p (q l,λ) n (5) are separately analyzed as follows: p (s, q,λ): thstermonernsthespeehwaveform generaton from spetral features and a gven state sequene Its maxmzaton wth respet to λ s losely related to the L spetral estmaton [5] In ths work ths probablty s related to the assumed speeh sgnal generatve model 1 p ( q,λ): ths term s gven as the produt of state-output probabltes of speeh parameter vetors f Hs or hdden sem-arkov models (HSs) [1] are used for as aoust model If traetory Hs [6] are used, ths probablty s gven as a state-sequeneoutput probablty of the spetral parameter vetor p (q l,λ): ths term represents the probablty of a state sequene q for a gven transrpton l If an H or traetory H s used for aoust modelng, ths probablty s gven as a produt of state-transton probabltes In ase HS or traetory HS s used, p (q l,λ) nludes both state-transton and stateduraton probabltes Note that t s possble to model p ( q,λ) and p (q l,λ) usng exstng aoust models, suh as H, HS, or traetory H Therefore, the problem s how to model p (s, q,λ) 3 Defnton of the proposed framework Ths seton desrbes the dstrbuton of the speeh waveform gven spetral parameters, p (s, q,λ) Intally, ths s desrbed n terms of the mpulse response of a voal trat flter Later, n Seton 3, the relatonshp between spetral parameters and voal trat flter mpulse response s then onsdered 31 Assumed speeh generatve model The speeh sgnal s assumed to be generated by the proess n Fgure 1 [7], where (5) s(n) =h (n) [h v(n) t(n)+h u(n) w(n)], (6) wth denotng lnear onvoluton, and h (n): bengthevoaltratfltermpulseresponse; t(n): beng a pulse tran; w(n): bengagaussanwhtenosesequenewthmean zero and varane one; h v(n): beng the voed flter mpulse response; h u(n): beng the unvoed flter mpulse response 1 The assumed speeh generatve model wll be dsussed n Seton 31 The voal trat, voed and unvoed flters are assumed to have the followng transfer funtons, H (z) = H v (z) = P h (p)z p, (7) p= m= h v(m)z m, (8) K H u(z) = 1 P, (9) L l=1 g(l)z l where P, and L are respetvely the orders of H (z), H v (z), and H u(z) FlterH (z) s onsdered to have mnmum-phase response sne t represents the mpulse response of the voal trat flter [] Parameters of the generatve model above omprse the voal trat, voed and unvoed flters, H (z), H v (z) and H u (z), andthepostonsandampltudesoft(n), {p,,p 1}, and{a,,a 1}, wth beng the number of pulses Although there may exst several ways to estmate oeffents h v (m) and g(l) and gan term K, of flters H v (z) and H u(z),respetvely,thspaperwllbebasedonthemethod desrbed n [7] Usng matrx notaton, wth upperase and lowerase aptal letters respetvely denotng matres and vetors, (6) an be wrtten as s = H H vt + s u, (1) where s = ˆs ` H =» h () = } n» h () v = } H v = s `N + + P 1, (11), (1) h () h (P ) }, N+ n 1 h h() h(n+ 1) h h() v h(n 1) v h v ` h v ` (13), (14) }, N 1 (15) t = ˆt() t(n 1), (16) " # s u = } s u() s u(n + L 1) } The vetor s u ontans samples of +P L (17) s u (n) =h (n) h u (n) w(n), (18) and an be nterpreted as the error of the generatve model for voed regons of the speeh sgnal wth ovarane matrx H ΦH,where g () = Φ = G G 1, (19) G = ˆ g () g (N+ 1), ()» 1 g(1) g(l) } K K K } N+ 1 (1)

3 Pulse tran t(n) a a a1 a 1 H v (z) p p 1 p p 1 Gaussan whte nose w(n) (zero mean, varane one) H u (z) v(n) Voed Extaton u(n) Unvoed Extaton e(n) H (z) Extaton Speeh s(n) Fgure 1: The assumed speeh generaton model Extaton s reated by state-dependent flterng of a pulse tran and Gaussan whte nose [7] As w(n) s Gaussan whte nose, u(n) =h u (n) w(n) s a normally dstrbuted stohast proess Usng vetor notaton, the probablty of u an be expressed as p (u G) =N (u;, Φ), () where N (x; µ, Σ) means a Gaussan dstrbuton of x wth mean vetor µ and ovarane matrx Σ And sne u(n) =H 1 (z) {s(n) h(n) h v(n) t(n)}, (3) the probablty of speeh vetor s beomes p (s H, H v, G, t) =N s; H H vt, H ΦH (4) If the last P rows of H are gnored, thus negletng the zero-mpulse response of H (z) whh produes samples s `N +,,s`n + + P 1, thenh beomes square wth dmenson N + and (4) an be re-wrtten as p (s H,λ e)=h 1 1 N `H s; H vt, Φ, (5) where λ e = {H v, G, t} are parameters of the extaton part of the omplete speeh generatve model It s nterestng to note that the term H 1 s orresponds to the resdual sequene, extrated from the speeh sgnal s(n) through nverse flterng by H (z) By assumng that H v and G n the speeh generatve model have the state-dependent parameter tyng struture proposed n [7], (5) an be re-wrtten as p (s H, q,λ e)=h 1 1 N `H s ; H v,qt, Φ q, (6) where H v,q s the voed flter mpulse response matrx for state sequene q,andφ q = `G q G q 1, wth Gq beng the nverse unvoed flter mpulse response matrx for state sequene q 3 Relatonshp between voal trat flter mpulse response and spetral parameters The prevous seton has derved the dstrbuton p (s H, q,λ) However, from (5) the omplete waveform dstrbuton requres p (s, q,λ) Therefore, a relatonshp between voal trat mpulse response represented here by the matrx H and the orrespondng spetral parameters has to be derved Dependng on the relatonshp between H and, ts often dffult to ompute H from n a losed form for some hoes of spetral parameters, suh as mel-epstral oeffents, lne spetral pars, et To address ths problem, astohastapproxmatontomodeltherelatonshpbetween H and s one possblty If the mappng between H and an be represented as a Gaussan proess wth probablty p (H, q,λ h ),whereλ h s the set of parameters of a Gaussan model that maps spetral parameters onto voal trat flter mpulse response, then p (s, q,λ e) beomes p (s, q,λ)= p (s H, q,λ e) p (H, q,λ h ) dh = H 1 N `H 1 s ; H v,q t, Φ q N (H ; f q (), Ω q ) dh, (7) where f q() s an approxmaton funton to onvert to H and Ω q s a ovarane matrx representng the nose of the onverson In the speal ase where the relatonshp between H and s determnst, f q() beomes a smple mappng funton n a losed form and p (H, q,λ h ) beomes a delta funton 31 Cepstral oeffents as spetral parameters In ths paper epstral oeffents [5] were hosen as spetral parameters n order to avod the mappng model dsussed above Therefore, as that the flter H (z) has mnmum-phase response, the relatonshp between a gven epstral oeffent vetor for frame t, t = [ t() t(c)],andtsorrespondng voal trat flter mpulse response vetor, h,t = [h,t() h,t(p )],anbeexpressedas h,t = D t EP [D t t], (8) where EP[ ] means a vetor whh s derved by takng the exponental of the elements of [ ],andd t s a () (C +1) dsrete Fourer transform (DFT) matrx, wth W WP C +1 D t = , (9) 1 WP WP PC +1 W = e π, (3) and D t s a (P + 1) (P + 1) nverse DFT matrx Dt = 1 1 W 1 W P (31) 1 W P W P Note that D t and D t are onstant for all the frames - 9 -

4 3 Relatonshp between H and h The voal trat flter mpulse response-related term that appears n the generatve model of (1) s H,noth Relatonshp between H gven as (1) and (13) wthout the last P rows (square verson of H ), and h gven as h = ˆh, h,t 1, (3) h,t = ˆh,t() h,t(p ), (33) wth h,t beng the synthess flter mpulse response of the t-th frame, an be wrtten as H = N 1 n= J nbh n (34) In (34), N s the number of samples, and» n = } 1 }, (35) n N 1 n wth B beng an N() T ()matrx to map aframe-bassvetorh nto ts sample-bass verson The N N (P + 1) matres are onstruted as follows» I J =,N() P 1, (36) N P 1, N P 1,N() P 1 J N 1 =» N 1,N() P 1 N 1,1 N 1,P, (37) 1,N() P 1 1 1,P where,y means a matrx of zero elements wth rows and Y olumns, and I s an -dmenson dentty matrx For eah sample nrement the dentty matrx I moves one row down and olumns to the rght 33 Jont aoust and extaton modelng If a traetory H [6] s used to represent the probablty p ( l,λ ),then p( q,λ )=N ( ; q, P q), (38) TY 1 p(q l,λ )=π q t= α qt q t+1, (39) where π s the ntal probablty of state, α s the transton probablty from state to state, and q and P q orrespond to the mean vetor and ovarane matrx of traetory H for q In (38), q and P q are gven as q = P qr q, (4) R q = P 1 q = W Σ 1 q W, (41) r q = W Σ 1 q µ q, (4) where W s typally a 3T (C +1) T (C +1)wndow matrx that appends dynam features (veloty and aeleraton features) to µ q and Σ 1 q n (41) and (4) orrespond to the 3T (C +1) 1 mean parameter vetor and the 3T (C +1) 3T (C +1)preson parameter matrx for the state sequene q,gvenas h µ q = µ q µ q T 1, (43) n o Σ 1 q =dag Σ 1 q,,σ 1 q T 1, (44) where µ and Σ 1 orrespond to the 3(C + 1) 1 mean-parameter vetor and the 3(C + 1) 3(C + 1) preson-parameter matrx assoated wth state, and Y = dag{ 1,,} means that matres { 1,,} are dagonal sub-matres of Y ean parameter vetors and preson parameter matres for eah state are defned as µ = ˆµ µ µ Σ 1 = dag Σ 1, Σ 1, (45), Σ 1, (46) where [ ] and [ ] mean respetvely a vetor or matrx of veloty and aeleraton features assoated wth vetor or matrx [ ] The fnal model s obtaned by ombnng the aoust and extaton models, e, p (s l,λ)= p (s, q,λ e) p ( q,λ ) q p (q l,λ ) d, (47) where p ( q,λ ) and p (q l,λ ) are gven by (38) and (39), respetvely, whereas p (s, q,λ e) s represented by (6), and fnally λ = {λ e,λ } 4 L tranng Parameters of the ont model λ = {λ e,λ } are estmated aordng to (3), where the lkelhood funton p (s l,λ) s gven by (47), wth λ e = {H v, G, t} orrespondng to parameters of the extaton model, and λ = {m, σ} onsstng of parameters of the aoust model, m = ˆµ µ S 1, (48) 1 σ = vdag dag Σ,,Σ S 1, (49) where S s the number of states m and σ are respetvely vetors formed by onatenatng all the means and dagonals of the nverse ovarane matres of all states, wth vdag{[ ]} meanng a vetor formed by the dagonal elements of [ ] 41 Lkelhood funton Unfortunately, estmaton of λ through the expetatonmaxmzaton (E) algorthm s ntratable Therefore, here an approxmate reursve approah s adopted If the summaton over all possble q n (47) s approxmated by a fxed state sequene, then p (s l,λ) beomes p (s l,λ) p (s, ˆq,λ e) p ( ˆq,λ ) p ( ˆq l,λ ) d, (5) where ˆq = {ˆq,,ˆq T 1} s a fxed state sequene Further, f the ntegraton over all possble s approxmated by sngle epstral oeffent vetor, then (5) beomes p (s l,λ) p (s ĉ, ˆq,λ e) p (ĉ ˆq,λ ) p ( ˆq l,λ ), (51) where ĉ =[ĉ ĉ T 1] s a fxed spetral parameter vetor Fnally, f the state sequene ˆq s fxed through the entre tranng proess, the term p ( ˆq l,λ ) an be gnored By takng the logarthm of the resultng expresson, the followng log lkelhood funton to be maxmzed through the update of the aoust and extaton model parameters an be obtaned L (s; ĉ, ˆq,λ e,λ )=logp (s ĉ, ˆq,λ e)+logp (ĉ ˆq,λ ) (5)

5 4 Tranng proedure The optmzaton problem s broken nto two parts: ntalzaton and reurson 41 Intalzaton 1 Extrat from eah utterane of the speeh data an ntal epstral oeffent vetor, = ˆ T 1, (53) t = ˆ t() t(c) (54) Tran traetory H parameters λ by usng, ˆλ =argmax p ( λ ) (55) λ 3 Determne the best state sequene ˆq as the Vterb path from the traned models by usng the algorthm shown n [6], ˆq =argmax p (, q λ ) (56) q 4 Estmate extaton parameters λ e assumng ˆq and, by usng the algorthm desrbed n [7], 4 Reurson ˆλ e =argmax p(s, ˆq,λ e) (57) λ e 1 Estmate the best spetral parameter vetor ĉ by usng the log lkelhood funton of (5), ĉ =argmax L(s;, ˆq, ˆλ e, ˆλ ) (58) Update the aoust model parameters by tranng traetory H usng ĉ as the observaton, ˆλ =argmax p (ĉ ˆq,λ ) (59) λ 3 Update extaton model parameters λ e through the algorthm desrbed n [7], assumng ˆq and ĉ, ˆλ e =argmax p(s ĉ, ˆq,λ e ) (6) λ e The reursve steps may be repeated several tmes 43 Estmaton of the best epstral vetor ĉ In Step 1 of the reursve proess, epstral oeffents are estmated gven both traned extaton and aoust models The log lkelhood funton of (5) an be wrtten as L(s;, ˆq,λ e,λ )= 1 s H Φ 1 ˆq H 1 s + log H + + s H Φ 1 ˆq H v,ˆqt 1 R ˆq + r ˆq + K, (61) where K s a onstant that does not depend on The best epstral oeffent vetor ĉ an be alulated by utlzng any gradent-based optmzaton algorthm [8], where the gradent of L(s;, ˆq,λ e,λ ) wth respet to s L(s;, ˆq,λ e,λ )=D DIAG (EP [D]) D B ( N 1 n= h Jn H Φ 1 ˆq (e v) e I N n ) R ˆq +r ˆq, (6) Speeh Intal epstrum vetor Traetory H tranng Rˆq, rˆq Cepstrum vetor ĉ Traetory H tranng Intal epstral analyss State sequene ˆq Estmaton of the best epstral oeffents ĉ State sequene ˆq Intalzaton Intal epstrum vetor Extaton model tranng H v (z),g(z),t(n) Cepstrum vetor ĉ Extaton model tranng Reurson Fgure : Illustraton of the proposed ont tranng of aoust and extaton models and e = H 1 s, (63) v = H v,ˆq t, (64) D = dag {D,,D T 1}, (65) D = dag {D,,D T 1}, (66) where DIAG ([ ]) means a dagonal matrx whose non-zero elements ome from vetor [ ] 5 Experment Aprelmnaryexpermentwasondutedtoverfytheeffetveness of the proposed ont estmaton proess The database onssted of 1 hand-labeled sentenes from a US Englsh female speaker Intally, epstral oeffents were alulated by performng the spetral analyss desrbed n [5], wth γ =and α =, on smooth perodograms as nput, extrated from the speeh data at every 5 ms The number of epstral oeffents alulated per frame was 4, e, C =39 Jont tranng of aoust and extaton models λ = {λ e,λ } was then onduted as desrbed n Seton 4 and llustrated n the dagram of Fgure For estmaton of the best epstral oeffent vetor ĉ a 56-th order mpulse response of H (z) (P =56)wasonsdered Thereursveproesswas repeated 4 tmes 51 Cepstral analyss gven aoust and extaton models Estmaton of the best epstral oeffents for eah teraton of the reursve proedure was performed as desrbed n the Seton 43 Fgure 3 shows the behavor of the -dependent part of the lkelhood L(s;, ˆq,λ e,λ ) for one sentene, when a smple steepest-asent algorthm was utlzed wth onvergene F nformaton was utlzed to extrat smooth perodograms - 9 -

6 Lkelhood L() x Iteraton Fgure 3: Behavor of the log lkelhood L(s;, ˆq,λ e,λ ) durng the estmaton of the best epstrum ĉ for one sentene, usng a smple steepest-asent algorthm wth update fator δ =1 These measures were taken from the frst teraton of the reursve proess llustrated n Fgure Ampltude (db) Spetrum Smoothed spetrum Spetrum from ntal baselne epstrum Spetrum from 1 best estmated epstrum n Iteraton Frequeny (Hz) Fgure 4: Short-term ampltude spetra of a gven sentene Asde from natural and smooth spetra, spetra derved respetvely from the baselne epstra alulated n the ntal analyss, and epstra alulated n the 1-best estmaton proess of the fourth reursve teraton are also shown fator δ =1 Fgure 4 shows the spetral envelope derved from one frame of ntal, and 1-best epstra alulated on the fourth teraton of the reursve proess The speeh spetrum and ts smooth verson from whh the ntal epstrum was derved are also shown It an be seen for ths ase that the epstral oeffents alulated gven extaton and aoust models result n better reproduton of the formant around 5 Hz The performane of 1-best epstral oeffents ĉ alulated n eah reursve teraton was also verfed n terms of mpat on the resdual sgnal power, estmated as P e = lm N 1 N +1 N/ n= N/ e (n) 1 N e e, (67) where N n ths ase s the number of samples of the database Fgure 5 plots P e for eah teraton Resdual sgnals were obtaned through nverse flterng wth the mel generalzed log spetrum approxmaton (GLSA) flter struture [5] It an be seen that the resdual power has a relatve sgnfant derease n the frst reursve teraton of the ont tranng 5 Qualty resultng from the ont modelng An nformal lstenng test usng natural F and phonet duratons was onduted between the proposed ont tranng framework and the method n whh both extaton and aoust models are separately traned, orrespondng to the ntalzaton proess as llustrated n Fgure Extaton sgnal at run- Resdual power P e Reursve teraton Fgure 5: Resdual power extrated usng epstral oeffents alulated n eah reursve teraton In Iteraton the baselne observed epstral oeffents were used to derve P e tme was onstruted as depted n Fgure 1, wth pulse trans beng generated from F The GLSA flter, whh an mplement the voal trat flter usng epstral oeffents dretly, was used nstead of the all-zero struture of H (z) Aordng to the results the syntheszed speeh qualty produed by both approahes s ndstngushable One possble ause for ths was the lmted sze of the tranng database 6 Conluson Aproposalofontestmatonofaoustandextatonmodels for statstal parametr speeh synthess has been presented and ts tranng proedure based on L desrbed The resultng system beomes what an be nterpreted as a statstal modelng of the speeh waveform The approxmatons made for the estmaton of the parameters of the ntrodued ont aoustextaton model onssted of fxng the state sequene along the entre tranng proess and alulaton of a 1-best spetral oeffent vetor at eah teraton Future work nludes the utlzaton of other spetral parameterzatons, estmaton of N- best spetral parameters to approxmate p (s l,λ), andexperments wth larger tranng data 7 Referenes [1] H en, K Tokuda, and A Blak, Statstal parametr speeh synthess, Speeh Communaton, vol 51, pp , Nov 9 [] J R Deller, Jr, J H L Hansen, and J G Proaks, Dsrete- Tme Proessng of Speeh Sgnals NewYork:IEEEPress Class Ressue, [3] T Toda and K Tokuda, Statstal approah to voal trat transfer funton estmaton based on fator analyzed traetory H, n Pro of ICASSP,pp ,8 [4] Y J Wu and K Tokuda, nmum generaton error tranng by usng orgnal spetrum as referene for log spetral dstorton measure, n Pro of ICASSP, pp , 9 [5] K Tokuda, T Kobayash, T asuko, and S Ima, el-generalzed epstral analyss a unfed approah to speeh spetral estmaton, n Pro of ICSLP, pp , 1994 [6] H en, K Tokuda, and T Ktamura, Reformulatng the H as a traetory model by mposng explt relatonshps between stat and dynam feature vetor sequene, Computer Speeh and Language,vol1,pp ,Jan 7 [7] R aa, T Toda, H en, Y Nankaku, and K Tokuda, An extaton model for H-based speeh synthess based on resdual modelng, n Pro of ISCA Workshop on Speesh Synthess,pp ,7 [8] J Noedal and S J Wrght, Numeral Optmzaton New York: Sprnger,