Linear prediction of the one-sided autocorrelation sequence for noisy speech recognition

80 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 1, JANUARY 1997 small. Al hough inc eased aining da a may be help ul, in he speake dependen Manda in syllable ecogni ion p oblem, a limi ed da abase will p obably s ill be a no mal si ua ion o some pe iod o ime in he u u e. VII. CONCLUSION A new app oach is p oposed in his co espondence o ob ain mo e elabo a e ini ial models co e ing cha ac e is ics o di e en ones. Imp o ed s a e ansi ion opologies a e also ound o achie e be e pe o mance compa ed wi h he simple le - o- igh model wi h wo ansi ions. A h eshold decision app oach is u he de eloped o imp o e he pe o mance o BS ecogni ion o he syllables wi h he neu al one. The es esul s on e e yday Chinese show ha a o al e o a e educ ion on he o de o 20% in he op 1 a e can be ob ained when all he concep s a e p ope ly in eg a ed. Al hough he echniques he e a e p oposed specially o ecogni ion o Manda in base syllables conside ing he e ec o ones, i is ce ainly belie ed ha simila concep s a e po en ially applicable o sol e simila p oblems in speech ecogni ion in o he languages. REFERENCES [1] R. He, Ed., Guoyu bao Tzdian (Manda in Chinese Daily Dic iona y). Taipei, R.O.C.: Guoyu bao, 1976. [2] L.-S. Lee, C.-Y. Tseng, and M. Ouh-Young, “The syn hesis ules in a Chinese ex - o speech sys em,” IEEE T ans. Acous ., Speech, Signal P ocessing, pp. 1309–1320, Sep . 1989. [3] Y. R. Chao, A G amma o Spoken Chinese. Be keley, CA: Uni e si y o Cali o nia Be keley P ess, 1968. [4] W. J. Yang, J. C. Lee, Y. C. Chang, and H. C. Wang, “Hidden Ma ko model o manda in lexical one ecogni ion,” IEEE T ans. Acous ., Speech, Signal P ocessing, pp. 988–992, July 1988. [5] F.-H. Liu, Y. Lee, and L. S. Lee, “A Di ec -conca ena ion app oach o ain hidden ma ko models o ecognize he highly con using manda in syllables wi h e y limi ed aining da a,” IEEE T ans. Speech Audio P ocessing, ol. 1, no. 1, pp. 113–119, Jan. 1993. [6] L.-S. Lee e al., “Golden manda in (I)—A eal- ime manda in speech dic a ion machine o chinese language wi h e y la ge ocabula y,” IEEE T ans. Speech Audio P ocessing, ol. 1, no. 2, pp. 158–179, Ap . 1993. [7] B.-H. Juang and L. R. Rabine , “Mix u e au o eg essi e hidden Ma ko models o speech signals,” IEEE T ans. Acous ., Speech, Signal P o- cessing, ol. ASSP-33, no. 6, pp. 1404–1413, Dec. 1985. [8] X. Huang e al., “The SPHNIX-II speech ecogni ion sys em: An o e iew,” Compu . Speech Language, pp. 137–148, Feb. 1993. Linea P edic ion o he One-Sided Au oco ela ion Sequence o Noisy Speech Recogni ion Ja ie He nando and Climen Nadeu Abs ac — The aim o his co espondence is o p esen a obus ep esen a ion o speech based on AR modeling o he causal pa o he au oco ela ion sequence. In noisy speech ecogni ion, his new ep- esen a ion achie es be e esul s han se e al o he ela ed echniques. I. INTRODUCTION Linea p edic i e coding (LPC) [1] is a spec al es ima ion ech- nique widely used in speech p ocessing and, pa icula ly, in speech ecogni ion. Howe e , he con en ional LPC echnique, which is equi alen o AR modeling o he signal x ( n ) , is known o be e y sensi i e o he p esence o backg ound noise. This ac leads o poo ecogni ion a es when his echnique is used in speech ecogni ion unde noisy condi ions, e en i only a mode a e le el o con amina ion is p esen in he speech signal. Simila esul s a e ob ained wi h he well-known mel-ceps um echnique [2]. This explains why some o he main a emp s o comba he noise p oblem consis o inding no el acous ic ep esen a ions ha a e mo e esis an o noise co up ion han adi ional pa ame e iza ion echniques. Linea p edic ion o he au oco ela ion sequence has been he common app oach o se e al obus spec al es ima ion me hods o noisy signals p esen ed in he pas . Fo speech ecogni ion, Mansou and Juang [3] p oposed he sho - ime modi ied cohe ence (SMC) as a obus ep esen a ion o speech based on ha app oach. On he o he hand, Cadzow [4] in oduced he use o an o e de e mined se o Yule–Walke equa ions o obus modeling o ime se ies. Al hough Cadzow applies linea p edic ion o he signal, his me hod can also be in e p e ed as pe o ming linea p edic ion in he au oco ela ion domain. Bo h me hods ely, ei he explici ly o implici ly, on he ac ha he au oco ela ion sequence is less a ec ed by b oadband noise han he signal i sel , especially a high lag indices. In his wo k, we conside he one-sided o causal pa o he au oco ela ion sequence and i s ma hema ical p ope ies. As his sequence sha es i s poles wi h he signal x ( n ) , i p o ides a good s a ing poin o LPC modeling. In his way, he new one-sided au oco ela ion LPC (OSALPC) me hod appea s as a s aigh o wa d esul o he app oach [5]. In addi ion, i is closely ela ed o he SMC ep esen a ion and Cadzow’s me hod. All o hem can be in e p e ed as AR modeling o ei he a spec al unc ion named “en elope” o i s squa e. This in e p e a ion, which is based on he p ope ies o he one-sided au oco ela ion, p o ides mo e insigh in o he a ious me hods. In his co espondence, hei pe o mance in noisy speech ecogni ion is compa ed. The op imum model o de and ceps al li e ing ha e also been in es iga ed in noisy condi ions. The simula ion esul s show ha OSALPC ou pe o ms he o he echniques in se e e noisy condi ions and ob ains simila sco es o mode a e o high SNR. Manusc ip ecei ed Feb ua y 14, 1995; e ised No embe 9, 1995. This wo k was suppo ed by G an nos. TIC-92-0800-C05/04 and TIC-92-1026- C02/02. The associa e edi o coo dina ing he e iew o his pape and app o ing i o publica ion was D . Kuldip K. Paliwal. The au ho s a e wi h he Depa men o Signal Theo y and Communica ions, Poly echnical Uni e si y o Ca alonia, Ba celona, Spain. Publishe I em Iden i ie S 1063-6676(97)00766-9. 1063–6676/97$10.00 1997 IEEE IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 1, JANUARY 1997 81 This co espondence is o ganized in he ollowing way. In Sec ion II, he OSALPC echnique is in oduced, and i s ela ionship wi h he con en ional LPC app oach and he o he pa ame e iza ions based on AR modeling in he au oco ela ion domain is discussed. Sec ion III epo s he applica ion o all hose pa ame e iza ion echniques o an isola ed wo d mul ispeake ecogni ion ask, using he HMM app oach, in o de o compa e hei pe o mances in he p esence o addi i e whi e noise. Finally, some conclusions a e summa ized in Sec ion IV. II. AR MODELING IN THE AUTOCORRELATION DOMAIN F om he au oco ela ion sequence R ( m ) , we de ine he one-sided (causal pa o he) au oco ela ion (OSA) sequence in he ollowing way: R + ( m )= R ( m ) m> 0 R (0) 2 m =0 0 m< 0 (1) I s Fou ie ans o m is he complex “spec um” S + ( ! )= 1 2 [ S ( ! )+ jS H ( ! )] (2) whe e S ( ! ) is he eal spec um, i.e., he Fou ie ans o m o R ( m ) , and S H ( ! ) is he Hilbe ans o m o S ( ! ) . Due o he analogy be ween S + ( ! ) in (2) and he analy ic signal used in ampli ude modula ion, a spec al “en elope” E ( ! ) [6] can be de ined as E ( ! )= j S + ( ! ) j : (3) Due o he la ge dynamic ange o speech spec a, he en elope E ( ! ) s ongly enhances he highes powe equency bands wi h espec o S ( ! ) [5]. Consequen ly, he noise componen s lying ou side he enhanced equency bands a e la gely a enua ed in E ( ! ) wi h espec o S ( ! ) , and hus, E ( ! ) is mo e obus o b oadband noise han S ( ! ) . On he o he hand, as i is well known, he OSA sequence R + ( m ) and he signal x ( n ) ha e he same poles [7]. Those wo p ope ies, i.e., obus ness o noise and pole p ese a- ion, sugges ha AR pa ame e s o he speech signal can be mo e eliably es ima ed om he OSA sequence R + ( m ) han di ec ly om he signal x ( n ) when x ( n ) is co up ed by b oadband noise. Thus, as he con en ional LPC echnique assumes an all-pole model o he speech spec um S ( ! ) , we may apply linea p edic ion o he OSA sequence, assuming an all-pole model o i s “spec um” E 2 ( ! ) . This is he basis o he one-sided au oco ela ion linea p edic i e coding (OSALPC) pa ame e iza ion echnique [5]. A s aigh o wa d algo i hm is p oposed in [5] ha calcula es he OSALPC ceps al coe icien s. I consis s o applying he (windowed) au oco ela ion me hod o linea p edic ion o an es ima ion o he OSA sequence: a) Fi s , om he speech ame o leng h N , he au oco ela ion lags un il M = N= 2 a e compu ed ( his alue o M was empi ically op imized o conside he well-known adeo be ween a iance and equency esolu ion o he spec al es ima e [8]). b) Second, he Hamming window om m =0 o M is applied on such es ima ed OSA sequence. c) Thi d, i p is he p edic ion o de , he i s p +1 au oco ela ion alues o ha OSA sequence a e compu ed om m =0 o p , using he con en ional biased es ima o , i.e., he one ha is commonly employed in speech p ocessing. d) Then, hese alues a e used as en ies o he Le inson–Du bin algo i hm o es ima e he AR pa ame e s ak , k =1 ; 111 ;p: Fig. 1. Robus ness o he OSALPC ep esen a ion o addi i e whi e noise: (a) LPC spec um and (b) OSALPC squa ed en elope o a oiced speech ame in noise ee condi ions (solid line) and SNR equal o 0 dB (do ed line). e) Finally, he ceps al coe icien s co esponding o he model a e ecu en ly compu ed om hose AR pa ame e s. The obus ness o OSALPC o addi i e whi e noise is illus a ed in Fig. 1. As can be seen in his igu e, he OSALPC squa ed en elope shows a p ominen i s o man , and i s whole cu e is mo e obus o addi i e whi e noise han ha o he LPC spec um. In his case, he con en ional biased au oco ela ion es ima o was used o compu e he OSA sequence om he signal. Fig. 1 also shows ha spu ious peaks may appea in he OSALPC squa e en elope. They a e p obably due o he ac ha he OSALPC echnique pe o ms only a pa ial decon olu ion o he speech signal [9]. In spi e o ha , OSALPC shows a be e speech ecogni ion pe o mance han con en ional LPC in se e e condi ions o addi i e whi e noise, as will be seen in he nex sec ion. The OSALPC echnique is closely ela ed o he sho - ime modi- ied cohe ence (SMC) ep esen a ion p oposed by Mansou and Juang in [3]. SMC is also based on AR modeling in he au oco ela ion domain. Howe e , whe eas in he OSALPC echnique, he en ies o he Le inson–Du bin algo i hm ( i s p alues o he au oco ela ion o he OSA sequence) a e calcula ed om he OSA sequence using he con en ional biased au oco ela ion es ima o , in he SMC ep esen- a ion, hey a e compu ed using a squa e oo spec al shape . In ac , in e ms o he abo e o mula ion, ha di e ence lies in assuming in he SMC echnique an all-pole spec al model o he en elope E ( ! ) ins ead o E 2 ( ! ) . Fu he mo e, R + (0) is se o 0 in he case o addi i e whi e noise because i is se e ely co up ed by noise. On he o he hand, he name o he SMC ep esen a ion de i es om he usage o a pa icula es ima o , which is e e ed o as cohe ence in [3], o compu e he OSA sequence om he signal. This es ima o is a mo e homogeneous measu e han he con en ional biased au oco ela ion es ima o in he sense ha e e y es ima ed alue is compu ed using he same numbe o signal samples, whe eas in he con en ional es ima o , he numbe o signal samples employed o es ima e R ( m ) dec eases along he index m . Tha p ope y does no ha e much ele ance in he es ima ion o he au oco ela ion en ies o he Le inson–Du bin algo i hm since only he i s p +1 82 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 1, JANUARY 1997 Fig. 2. Ma ix o mula ion o OSALPC and LSMYE me hods. alues a e conside ed, and usually, p  N . Howe e , i may be impo an in he es ima ion o he OSA sequence om he speech signal since he OSA leng h conside ed in bo h OSALPC and SMC echniques is M = N= 2 and no negligible wi h espec o N . The OSALPC echnique can also be easily ela ed o he o e de- e mined se o Yule–Walke equa ions p oposed by Cadzow in [4] o seek ARMA models o ime se ies. As an AR ( p ) p ocess con amina ed by addi i e whi e noise becomes an ARMA ( p; p ) p ocess, Cadzow’s me hod can be used o es ima e he pa ame e s o his noisy AR p ocess simply by se ing he same AR and MA o de s in he so-called leas squa es modi ied Yule–Walke equa ions (LSMYWE’s) [8]. The ela ionship be ween he OSALPC and LSMYWE echniques is illus a ed by he ma ix equa ion in Fig. 2, whe e M deno es he highes au oco ela ion lag used, and e ( m ) is he e o o be minimized. The minimiza ion o he no m o he ull e o ec o e ( m ) g m =1 ; 111 ;M + p wi h espec o he AR pa ame e s a k is equi alen o he applica ion o he (windowed) au oco ela ion me hod o linea p edic ion o he sequence R ( m ) , m =1 ; 111 ;M , i.e., he OSALPC echnique. On he o he hand, he LSMYWE echnique minimizes he no m o he sub ec o e ( m ) g m = p +1 ; 111 ;M , and he e o e, i amoun s o applying he (unwindowed) co a iance me hod o linea p edic ion on he same ange o au oco ela ion lags. When M =2 p , LSMYWE a e he modi ied Yule–Walke equa ions [8] o an ARMA ( p; p ) p ocess. In bo h cases, only au oco ela ion lags co esponding o he OSA sequence a e employed. In ou compa ison, we will also conside ano he e sion o his (unwindowed) co a iance-based app oach ha will be called leas squa es Yule–Walke equa ions (LSYWE’s). Whe eas in he LSMYWE echnique he i s p edic ed au oco ela ion alue is R ( p + 1) , in he LSYWE echnique, he p edic ion begins a R (1) . Bo h LSMYWE and LSYWE me hods and hei ela ionship o OSALPC a e g aphically desc ibed in Fig. 3. As i is shown, he only di e ence be ween he a ious echniques is he ange o au oco ela ion lags conside ed in he minimiza ion o he e o . I is wo h no ing ha LSYWE conside s some nega i e au oco ela ion lags ha do no belong o he OSA sequence. In pa icula , i M is equal o p , LSYWE a e he con en ional Yule–Walke equa ions. As will be seen in he nex sec ion, in spi e o he simila i y be ween all hese echniques, he OSALPC ep esen a ion ou pe o ms he LSYWE, LSMYWE, and SMC echniques in speech ecogni ion in se e e noisy condi ions. On he o he hand, as a as he compu a ional complexi y o he algo i hms is conce ned, OSALPC and SMC echniques a e much mo e e icien han LSYWE and LSMYWE echniques because hey use he Le inson–Du bin algo i hm. Finally, i is wo h no ing ha he OSALPC echnique may be included in he ield o highe o de spec al es ima ion due o he ac ha he squa ed en elope E 2 ( ! ) is he Fou ie ans o m o he au oco ela ion o he OSA sequence, which is a pa icula ou h- o de momen o he signal. Fig. 3. In e p e a ion o he (a) OSALPC, (b) LSMYWE, and (c) LSYWE app oaches as applica ion o he au oco ela ion o co a iance me hods o linea p edic ion o an au oco ela ion sequence in di e en lag anges. III. SPEECH RECOGNITION EXPERIMENTS This sec ion epo s he applica ion o all he abo e pa ame e - iza ion echniques o ecognize isola ed wo ds in a mul ispeake ask wi h a disc e e HMM-based sys em in o de o compa e hei pe o mance and o gain some insigh in o he me i o he OSALPC ep esen a ion in he p esence o addi i e whi e noise A. Speech Da abase and Recogni ion Sys em The da abase used in ou expe imen s consis s o 10 epe i ions o he Ca alan digi s u e ed by se en male and h ee emale speake s (1000 wo ds) and eco ded in a quie oom. Fi s , he sys em was ained wi h hal o he da abase and es ed wi h he o he hal . Then, he oles o bo h hal es we e changed, and he epo ed esul s we e ob ained by a e aging hose wo esul s. The analog speech signal was i s bandpass il e ed o 100–3400 Hz by an an ialiasing il e , sampled a 8 kHz and, 12 bi s quan ized. The digi ized clean speech was manually endpoin ed o de e mine he bounda ies o each wo d. The endpoin s ob ained in his way we e used in all ou expe imen s, including hose in which noise was added o he signal. Clean speech was used o aining in all he expe imen s. Noisy speech was simula ed by adding ze o mean whi e Gaussian noise o he clean signal so ha he SNR o he esul ing signal becomes 1 (clean), 20, 10, and 0 dB. No p eemphasis was pe o med. In he pa ame e iza ion s age o he ecogni ion sys em, he signal was di ided in o ames o 30 ms a a a e o 15 ms, and each ame was cha ac e ized by i s ceps al pa ame e s ob ained ei he by he con en ional LPC me hod o by any o he echniques p esen ed in he las sec ion. Be o e en e ing he ecogni ion s age, he ceps al pa ame e s we e ec o quan ized using bo h a codebook o 64 codewo ds and he Euclidean dis ance measu e be ween li e ed ceps al ec o s. Each digi was cha ac e ized by a le - o- igh disc e e hidden Ma ko model o 10 s a es wi hou skips. T aining and es ing we e pe o med using Baum–Welch and Vi e bi algo i hms, espec i ely. B. Recogni ion Resul s Fi s o all, we ca ied ou some expe imen s wi h he abo e desc ibed speech ecogni ion sys em o op imize he model o de and IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 1, JANUARY 1997 83 TABLE I RECOGNITION RATES OF THE CONVENTIONAL LPC TECHNIQUE FOR SEVERAL PREDICTION ORDER VALUES AND CEPSTRAL LIFTERS TABLE II RECOGNITION RATES OF THE CONVENTIONAL LPC, LSMYWE, AND LSYWE TECHNIQUES FOR p =12 AND THE SLOPE LIFTER TABLE III RECOGNITION RATES OF THE CONVENTIONAL LPC, SMC, AND OSALPC TECHNIQUES FOR p =12 AND THE SLOPE LIFTER he ype o ceps al li e in he con en ional LPC echnique. In Table I, he ecogni ion esul s o LPC model o de s p = 8, 12, and 16 and o he bandpass [10], in e se o s anda d de ia ion [11] (ISD), and slope [12] li e s a e p esen ed. The ecogni ion esul s show ha nei he he model o de no he ype o ceps al li e a e ele an o ou ask in noise- ee condi ions. Howe e , in he p esence o noise, he ecogni ion esul s a e e y sensi i e o bo h ac o s. I is also clea om Table I ha he nonsymme ical li e s—slope and ISD—ou pe o m he bandpass li e o e e y model o de . This may be due o he ac ha in he p esence o whi e noise, he lowe o de ceps al coe icien s a e mo e a ec ed han he highe o de ones in he unca ed ceps al ec o . The bes esul s o se e e noisy condi ions—10 and 0 dB o SNR—a e ob ained using slope li e and p edic ion o de p equal o 12. The con enience o his ela i ely high o de comes om he ac ha he sensi i i y o he au oco ela ion sequence o addi i e whi e noise ends o dec ease along he lag index. Model o de s ha a e oo high, howe e , yield poo ecogni ion esul s since he spec al es ima e shows spu ious peaks. Ac ually, ecogni ion a es we e calcula ed using he slope li e o a la ge ange o alues o he model o de , and he bes esul s we e hose ob ained o p = 12. In Table II, he ecogni ion a es o con en ional LPC, LSMYWE, and LSYWE app oaches a e p esen ed, using M = N= 2 and bo h op imum model o de and li e ob ained o he con en ional LPC echnique, i.e., p = 12 and he slope li e . Ob iously, hese a e no he op imum condi ions o each pa ame e iza ion echnique, bu he esul s can help o compa e hei pe o mance. As can be seen om Table II, he con en ional LPC echnique ou pe o ms no iceably he o he app oaches. Howe e , he excellen pe o mance o he LSYWE app oach in noise- ee condi ions is wo h no ing. Fig. 4. Compa ison o ecogni ion a es o he LPC, SMC, OSALPC-I and OSALPC-II echniques. Fig. 5. Block diag am o he calcula ion o he LPC, SMC, OSALPC-I and OSALPC-II ceps a. In Table III and Fig. 4, he ecogni ion a es co esponding o he con en ional LPC echnique, he SMC ep esen a ion, and he no el OSALPC app oach a e p esen ed, whe e we also use M = N= 2 , p =12 , and he slope li e . The wo e sions OSALPC-I and OSALPC-II o he OSALPC app oach co espond o he OSA es ima o s o which we e e ed in Sec ion II: OSALPC-I uses he con en ional biased au oco ela ion es ima o , and OSALPC-II like SMC uses he cohe ence es ima o (and se s R (0) o 0). Fig. 5 shows 84 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 1, JANUARY 1997 TABLE IV RECOGNITION RATES FOR THE OSALPC-II TECHNIQUE FOR SEVERAL PREDICTION ORDER VALUES AND CEPSTRAL LIFTERS a block diag am o he calcula ion o he LPC, SMC, OSALPC-I, and OSALPC-II ceps a ha pe mi s compa ison o hei espec i e algo i hms. The OSALPC and SMC ep esen a ions clea ly ou do he con- en ional LPC echnique in se e e noisy condi ions: OSALPC-I and OSALPC-II a es a e be e han LPC ones a 10 and 0 dB, and SMC ou pe o ms LPC a 0 dB. Mo eo e , OSALPC-I and OSALPC-II ep esen a ions ou pe o m he SMC echnique in all noisy condi ions. Fo he OSALPC ep esen a ion, he use o he con en ional biased au oco ela ion es ima o o compu ing he OSA sequence ( e sion OSALPC-I) is con enien in se e e noisy condi ions, i.e., o an SNR o 10 o 0 dB. Howe e , in noise- ee condi ions, he e is a loss o ecogni ion pe o mance in he OSALPC and SMC app oaches wi h espec o he con en ional LPC echnique due o he impe ec decon olu ion o he speech signal pe o med by hose echniques. This e ec seems o be minimized by using he cohe ence es ima o o compu e he OSA sequence, as in he case o OSALPC-II and SMC. Finally, Table IV shows he ecogni ion a es co esponding o OSALPC-II o he same model o de s and ceps al li e s as in Table I. I can be no iced ha he new echnique is less sensi i e o changes in bo h he model o de and he ype o ceps al li e han he con en ional LPC app oach, p o ided ha he model o de is no oo low. IV. CONCLUSIONS In his co espondence, se e al LPC-based echniques ha wo k in he au oco ela ion domain a e p esen ed and compa ed in noisy speech ecogni ion. The OSALPC echnique, which is based on he applica ion o he (windowed) au oco ela ion me hod o linea p edic ion o he one-sided au oco ela ion sequence, yields he bes esul s among all he compa ed LPC-based echniques in se e e noisy condi ions. REFERENCES [1] F. I aku a, “Minimum p edic ion esidual p inciple applied o speech ecogni ion,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. ASSP-23, pp. 67–72, 1975. [2] S. B. Da is and P. Me mels ein, “Compa ison o pa ame ic ep e- sen a ions o monosyllabic wo d ecogni ion in con inuously spoken sen ences,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. ASSP- 28, pp. 357–366, 1980. [3] D. Mansou and B. H. Juang, “The sho - ime modi ied cohe ence ep esen a ion and i s applica ion o noisy speech ecogni ion,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. 37, pp. 795–804, 1989. [4] J. A. Cadzow, “Spec al es ima ion: An o e de e mined a ional model equa ion app oach,” P oc. IEEE, ol. 70, pp. 907–939, 1982. [5] J. He nando and C. Nadeu, “Speech ecogni ion in noisy ca en i on- men based on OSALPC ep esen a ion and obus simila i y measu ing echniques,” in P oc. ICASSP’94, Adelaide, Ap . 1994, pp. 69–72. [6] M. A. Lagunas and M. Amengual, “Non-linea spec al es ima ion,” in P oc. ICASSP’87, Dallas, Ap . 1987, pp. 2035–2038. [7] D. P. McGinn and D. H. Johnson, “Reduc ion o all-pole pa ame e es ima ion bias by successi e au oco ela ion,” in P oc. ICASSP’83, Bos on, Ap . 1983, pp. 1088–1091. [8] S. L. Ma ple, J ., Ed., Digi al Spec al Analysis wi h Applica ions. Englewood Cli s, NJ: P en ice-Hall, 1987. [9] C. Nadeu, J. Pascual, and J. He nando, “Pi ch de e mina ion using he ceps um o he one-sided au oco ela ion sequence,” in P oc. ICASSP’91, To on o, Canada, May 1991, pp. 3677–3680. [10] B. H. Juang, L. R. Rabine , and J. G. Wilpon, “On he use o band-pass li e ing in speech ecogni ion,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. ASSP-35, pp. 947–954, 1987. [11] Y. Tohku a, “A weigh ed ceps al dis ance measu e o speech ecog- ni ion,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. ASSP-35, pp. 1414–1422, 1987. [12] B. A. Hanson and H. Waki a, “Spec al slope dis ance measu es wi h linea p edic ion analysis o wo d ecogni ion in noise,” IEEE T ans. Acous ., Speech, Signal P ocessing, ol. ASSP-35, pp. 968–973, 1987. A Fas Algo i hm o Finding he Adap i e Componen Weigh ed Ceps um o Speake Recogni ion Mihailo S. Zilo ic, Ra i P. Ramachand an, and Richa d J. Mammone Abs ac — In speake ecogni ion sys ems, he adap i e componen weigh ed (ACW) ceps um has been shown o be mo e obus han he con en ional linea p edic i e (LP) ceps um. The ACW ceps um is de i ed om a pole-ze o ans e unc ion whose denomina o is he p h-o de LP polynomial A ( z ). The nume a o is a ( p 0 1 ) h-o de polynomial ha is up o now ound as ollows. The oo s o A ( z )a e compu ed, and he co esponding esidues ob ained by a pa ial ac ion expansion o 1 =A ( z ) a e se o uni y. The e o e, he nume a o is he sum o all he ( p 0 1 ) h-o de co ac o s o A ( z ). In his co espondence, we show ha he nume a o polynomial is me ely he de i a i e o he denomina o polynomial A ( z ). This g ea ly speeds up he compu a ion o he nume a o polynomial coe icien s since i in ol es a simple scaling o he denomina o polynomial coe icien s. Roo inding is comple ely elimina ed. Since he denomina o is gua an eed o be minimum phase and he nume a o can be p o en o be minimum phase, wo sepa a e ecu sions in ol ing he polynomial coe icien s es ablishes he ACW cep- s um. This new me hod, which a oids oo inding, educes he compu e ime signi ican ly and imposes negligible o e head when compa ed wi h he app oach o inding he LP ceps um. I. INTRODUCTION Speake ecogni ion is he ask o iden i ying a speake by his o he oice [1]. A common p oblem in ealizing obus speake ecogni ion sys ems is ha a misma ch in aining and es ing condi ions se iously deg ades he pe o mance [2]. One o he pu sued app oaches o Manusc ip ecei ed Feb ua y 3, 1995; e ised July 13, 1996. The associa e edi o coo dina ing he e iew o his pape and app o ing i o publica ion was D . Joseph Campbell. M. S. Zilo ic is wi h Bell Communica ions Resea ch, Red Bank, NJ USA. R. P. Ramachand an and R. J. Mammone a e wi h he CAIP Cen e , Depa men o Elec ical Enginee ing, Ru ge s Uni e si y, Pisca away, NJ 08855 USA. Publishe I em Iden i ie S 1063-6676(97)00762-1. 1063–6676/97$10.00 1997 IEEE