The Role of Pitch Filter in Pulse-by-Pulse Reoptimization of the LP Synthesis Filter. Allam Mousa

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1 An-Najah Univ. J. Res. (N. S), Vol. 6 (2), 2002 The Role of Pih Filer in Pulse-by-Pulse Reopimizaion of he LP Synhesis Filer Allam Mousa Deparmen of Elerial Enineerin, An-Najah Naional Universiy, Nablus, Palesine. Absra Reeived: (27//2000), Aeped: (29/2/200) Two ieraive analysis alorihms were developed for he reopimizaion of he LP synhesis filer based on a pulse-by-pulse reopimizaion manner. In his sudy, he use of he pih filer in he analysis alorihms is inrodued. Similar o he no pih ase, improvemen in he ain is ahieved. On he oher hand, his ain has dropped ompared o he no pih ase. Moreover, he number of pulses needed o reopimize he LP filer found o be muh less han ha, in he no pih ase.... I. Inroduion The onvenional asade onneions of he forman and pih prediion filers were onsidered in [], where he oeffiiens of hese prediors are defined for one prediion filer hen for he oher one, sequenially. The forman filer is used o remove near-sample redundanies and he pih filer is inrodued o remove he disansample redundanies in he speeh sinal. The join reopimizaion of he forman and pih filer oeffiiens was also disussed in several mehods [2-6]. I has been shown in [2] ha he join reopimizaion

2 04 The Role of Pih Filer in Pulse-by- " soluion provides hiher prediion ain ompared o ha of he onvenional forman-pih sequenial one. For improvin he performane of he Analysis-by-Synhesis (A-b-S) speeh oders, i has been shown ha he reopimizaion of he forman filer oeffiiens, joinly, wih he exiaion parameers, provides beer performane as shown in [3-4]. A differen approah for parameer reopimizaion of he forman filer oeher wih he exiaion and onsiderin he pih filer in he analysis alorihm was also disussed in [5]. Anoher join reopimizaion of he Linear Prediion Coder (LPC) and pih filer parameers in Code Exied Linear Prediion (CELP) was presened in [6]. A simplified analysis alorihm for join reopimizaion of he forman filer and he exiaion was disussed in [7], where wo ieraive alorihms had iven a onsiderable ain in performane. These wo alorihms depend only on he pulse loaions and no on heir ampliudes as in [3-4]. Hene i is referred o as pulse-by-pulse reopimizaion analysis alorihms. However, he pih filer was no onsidered in he analysis alorihms in [7] and he overall Semenal Sinal-o-Noise Raio (SEGSNR) was improved by.7 o 2.6 db. In his paper, he pih filer is inrodued o he pulse-by-pulse reopimizaion alorihms disussed in [7], and he resuls are analyzed, wih and wihou usin he pih filer, refereed o as (COD-WP) and (COD-NP), respeively. The former oder has shown similar behavior lie he laer in ahievin an improvemen in he SEGSNR, bu he amoun of ain has dropped. Anoher major differene is in he sinifian drop of he number of ieraions required, by COD-NP, o reah sauraion. Compuer simulaions are performed usin Muli-Pulse (MP) exiaion o show he effe of inludin he pih filer in he reopimizaion proess, and he omplexiy of he new alorihm is disussed. An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

3 Allam Mousa 05 II. Pulse-Based LP Analysis Inludin he Pih Filer The pulse-based LP analysis, inludin he pih filer, was inrodued in [5]. Assumin a blo of daa wih N+p samples of speeh is available, he predied speeh samples veor an be wrien as; or ŝ S D HG () Where ŝ S D H (2) (STP) oeffiiens, G 2... p is he veor of he Shor Term Prediion... np is he veor of he Lon Term Prediion (LTP) oeffiiens, G 2... M is he veor of he pulse ampliudes, s = [s(0), s(), s(2),..., s(n -)] is he veor of he speeh samples, S is he Nxp daa marix and H is he NxM posiion marix, iven by S ij =s(i-j), H ij = i ), respeively, and D is he ( j inermediae daa marix onsrued from he forman residual whih is obained by inverse filerin. The equaion, whih ives he minimum norm soluion of Eq. (2), is S S D S H S S D D H D D S D H H H H S s D s H s I is a very diffiul as o obain he minimum norm soluion as iven in Eq. (3). In fa, dire aemp o solve Eq. (3) requires he inversion of a (p+m+np) x (p+m+np) marix whih is impraial for ypial values of p and M. A formal desripion of he alorihm whih solves, he STP, LTP and pulse ain, joinly, is disussed in [5]. Hereafer, his alorihm will be referred o as ALG. (3) An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

4 06 The Role of Pih Filer in Pulse-by- " III. The Effe of Pih Filer in he Pulse-by-Pulse LP Reopimizaion In he pulse-by-pulse linear prediion analysis alorihm desried in [7], wo ompuaionally effiien ieraive alorihms were developed wih no marix inversion exep he one a heir iniial seps. However, he pih filer was no aen ino onsideraion in he reopimizaion proess, and so he pulse exied linear prediion problem was saed as he deerminaion of he filer parameers,, pulse ampliudes,, ( ) and -h pulse loaion,, assumin ha he previous pulse loaions,, 2,...,, are nown. However, in his seion, he pih filer is ineraed wih he pulse-by-pulse reopimizaion analysis alorihm. This, in fa, joinly reopimizes he LTP oeffiiens,, assumin ( ) ha he LTP la, T, is nown. A he -h ieraion, he prediion speeh sample a ime n an be wrien as sˆ ( n) p s( n i) np d( n ( ) ( ) i ( ) i ( ) i i i i i This an be wrien in marix form as T i ) ( n ) (4) or sˆ S D H C s ˆ ( ) h (6) (5) Where C is he aumened An-Najah Univ. J. Res. (N. S), Vol. 6(2), oeffiien-ain veor holdin he STP and LTP oeffiiens and he pulses ain, ( ) S D H ( ), H ( ) is he pulse posiion marix orrespondin o, 2,..., h is he posiion veor orrespondin o he unnown, and S and D are as defined before. Noe ha ( 0) [ S D ].

5 Allam Mousa 07 The equaion ha ives he minimum norm soluion of Eq. (6) is C ( ) ( ) ( ) h ( ) s h ( ) h h h s (7) Definin, R S S, r ) S s h (, w h S ( ) h and h s s ( ), Eq. (7) an be rewrien as, and noin ha C R w ( ) w r s( ) (8) However, i is more informaive o wrie where w ( ) s( -) s( - 2)... s( - p) d( - T) d( - T - np) (9) is a x(-) null marix. Solvin and rearranin Eq. (8) ive he wo-oupled equaions; e ( ) (0) ( ) ( ) ( ) C( ) ( ) R ( ) w ( ) C () where w R ( ) w, and e ( ) ( ) is he prediion error of he orrespondin STP-LTP asaded filer a ime, whih is iven by; p e ( ) s( ) s( j) d( T j ) (2) ( ) ( ) j ( ) j j j Eq. () requires updain he inverse of R whih an be wrien as; np R ( ) R ZZ Z Z (3) An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

6 08 The Role of Pih Filer in Pulse-by- " where Z R ( ) w, and, iniially, he inverse of R ( 0 ) is required whih an be relaed o he inverse of he usual ovariane marix R onsrued from S and D. Assumin he pulse loaions,,2,... M are available, he alorihm based on he reursive Eqs. (0) and () an be saed as follows: Sep 0: Given; M, p, np, s, d, and M. i i Sep : Do he onvenional ovariane analysis o find he STP parameers, say, and R expliily. Sep 2: Deermine he LTP parameers as in he onvenional mehod. Sep 3: Consru R ( 0). Sep 4: Obain he parameers and from he equaion R r ( 0) ( 0 ) Sep 5: Obain he residual e(n) and esimae he pulse loaions, (, 2,..., M ). Sep 6: Se C ( 0), and =. Sep 7: Compue he residual sample e ( ) ( ) usin Eq. (2). Sep 8: Deermine ( ) and C usin Eqs. (0) and (), respeively. Sep 9: Updae R usin Eq. (3). Sep 0: Se =+. If <M o o sep 7. Sep : Se C C and. ( M ) ( M ) Noe ha he soluion a he M-h ieraion orresponds o Eq. (3). Hereafer, his alorihm will be referred o as ALG2. An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

7 Allam Mousa 09 Similar o he no pih ase disussed in [7], a ompuaionally more effiien, bu subopimal, alorihm an be derived if he problem is saed as he deerminaion of he -h pulse loaion, ampliude and he predior oeffiiens C, iven all pulse loaions and ampliudes up o (-). The resulin equaions are idenial o hose iven in Eqs. (0) and () exep ha R is replaed by R, ha is: where, C C( ) R w (4) ( w R ( ) (5) w This drops sep 9 from ALG2. Sine, a eah ieraion, he previous pulse ampliudes are fixed, he alorihm is unable o ive he resul in Eq. (3) a he final ieraion. Tha is why i is no opimal. A formal desripion of he alorihm (say ALG3) is as follows; Sep 0: Given; M, p, np, s, d, and M. i i Sep : Do he onvenional ovariane analysis o find he STP parameers, say, and R expliily. Sep 2: Deermine he LTP parameers as in he onvenional mehod. Sep 3: Consru R ( 0). Sep 4: Obain he parameers and from he equaion R r ( 0) ( 0 ) Sep 5: Obain he residual e(n) and esimae he pulse loaions, (, 2,..., M ). Sep 6: Se C ( 0), and =. ) (0) An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

8 0 The Role of Pih Filer in Pulse-by- " Sep 7: Compue he residual sample e ( ) ( ) usin Eq. (2). Sep 8: Deermine ( ) and C usin Eqs. (0) and (4), respeively. Sep 9: Se =+. If <M o o sep 7. Sep 0: Se C C and. ( M ) ( M ) IV. Experimenal Resuls To show he effe of inludin he pih filer in he pulse-by-pulse reopimizaion, experimens were ondued for he followin MP exied oders:. onvenional oder (COD) 2. oder ha uses only he pulse loaions of he A-b-S exiaion wih ALG2 (COD2) 3. oder ha uses only he pulse loaions of he A-b-S exiaion wih ALG3 (COD3) The speeh daabase was aen from boh male and female speaers. Speeh was band limied hen sampled a 8 Hz and quanified by 6 bis. The number of he LP oeffiiens, p, was fixed o en and ha of he LTP oeffiiens, np, was fixed o one, he LTP la searh was performed in he rane 20 o 47 samples. No windowin or preemphasis was applied. The analysis frame lenh, N, was se o 200 samples and so was he parameer-updain rae. The ovariane analysis mehod was used wih COD and he sabilized ovariane mehod was used wih COD2 and COD3, he exiaion frame was se o 50 samples and he LTP updae frame was also 50 samples, he weihin faor,, was se o 0.8. The Semenal Sinal-o-Noise raio (SEGSNR) was used as an objeive es of he performane of he oders. Semenal SNR was alulaed as: N f Semenal SNR (db) = SNR ( db) N f j An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002 j

9 Allam Mousa Where, 2 s ( n) n SNR j (db)= 0 lo( ) K 2 (ˆ( s n) s( n)) n K is he Sinal-o-Noise raio of he j-h semen, s (n) and s ˆ( n) are he aual and reonsrued speeh samples respeively, N is he number of frames and K is he number of samples per semenal frame whih is aen o be 00 samples. Fi. shows he SEGSNR versus he number of pulses per frame for COD, COD2 and COD3 wih pih (WP) filer inluded in he analysis alorihms. The improvemen due o he reopimizaion used in COD2 is lear. I an also be seen in Fi.2 ha he loss in COD3 due o he subopimal alorihm, ALG3, is small and even neliible. Hene, he simple analysis alorihm, ALG3, urned ou o be effiien and i provides almos he same performane as ALG2 bu wih muh less omplexiy. 5 4 f Semenal SNR, db COD2 COD COD Number of pulses/frame Fiure : SEGSNR versus pulse rae for COD, COD2 and COD3, wih pih filer inluded Fi. 2 ompares he performane of COD and COD3 for he wo ases wih and wihou a pih filer. I is obvious in Fi.2 ha he wo An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

10 2 The Role of Pih Filer in Pulse-by- " ases have similar behavior, bu wih a drop in he ain obained by he reopimizaion when he pih filer is inluded, COD3-WP, ompared o he ain in he reopimizaion of he STP parameers in he absene of he LTP filer, COD3-NP. For example, a he pulse rae of 20 pulses per frame, he ain obained in he reopimizaion proess usin COD3-NP is.8 db, whereas i is only.2 db in COD3-WP. Semenal SNR, db COD3-WP COD-WP COD3-NP COD-NP Number of pulses/ frame Fiure 2: SEGSNR versus pulse rae for COD and COD3 wih and wihou a pih filer. Anoher imporan differene in behavior of he reopimizaion alorihm in COD3-WP ompared o COD3-NP is shown in Fi. 3 whih refles he SEGSNR versus he number of pulses used in he reopimizaion analysis alorihm, ALG3. The number of exiaion pulses was hosen o be 20 pulses per frame, and so he pulse-by-pulse reopimizaion analysis alorihm uses he pulses sarin from no pulse (onvenional mehod) up o 20 pulses, and he resulin SEGSNR was reorded for eah number of pulses used. As i an be easily seen, COD3- NP performane is radually inreasin wih he inreasin number of pulses used in he analysis alorihm. However, when he pih filer is used, he performane, rapidly, reahes sauraion afer usin he firs hree pulses ou of he 20 available pulses. This propery an be used o sinifianly redue he omplexiy of he analysis alorihm in he presene of he pih filer. An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

11 Allam Mousa 3.5 Semenal SNR,dB COD3-WP COD3-NP Number of ieraions Fiure 3: SEGSNR versus he number of analysis pulse in ALG3 wih and wihou pih filer. Fi. 4 shows a shor speeh semen of a female speaer and he orrespondin emporal variaions of he SEGSNR obained usin COD and COD3 boh wih pih filer inluded. The number of pulses used is 20 pulses per frame and he over all ain here is.8 db. Similar o [7], he reopimizaion in COD3 does no always ouperform ha in COD; beer resuls were obained in 6% and 57% of he female and male speaers, respeively, ompared o 67% and 6% when no pih was inrodued. This observaion also shows anoher differene in performane of he reopimizaion alorihm when he pih filer is onsidered, and i ives a hin on he drop in ain observed in Fi. 2. Fousin on he suessful and unsuessful frames, i was found ha he SEGSNR for female speaer is improved by 2.6 db over he suessful frames and dropped by 0.7 db over he unsuessful frames. The orrespondin fiures wih no pih used were 3.2 db and 0.58 db respeively. Similar resuls were obained for he male speaer bu wih less ain for COD3-WP ompared o ha of COD3-NP. An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002

12 4 The Role of Pih Filer in Pulse-by- " Semenal SNR,dB Ampliude 0.5 x (a) ime index frame index Fiure 4: (a) Female speeh semen, (b) SEGSNR performane, + COD and o COD3. To see he effe of reopimizaion on he LP oeffiiens, Fi. 5 illusraes a ypial LP envelope obained usin he onvenional and he reopimizaion shemes. I is lear ha he reopimizaion here smoohes he unnaural sharp peas of he firs and hird formans obained usin he onvenional analysis sheme. (b) 25 Ampliude, db ALG ALG Fiure 5: LP speral envelopes usin ALG and ALG3 An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002 Frequeny, Hz.

13 Allam Mousa 5 V. Conlusion The pih filer has been inrodued in he reopimizaion of he LP synhesis filer in pulse exied A-b-S oders. This new approah has shown he same performane as ha of he no pih ase. However, less ain has been obained in reopimizin he LP oeffiiens wih pih. Moreover, in pulse-by-pulse reopimizaion mehod wih no pih filer, he SNR is inreased radually as a funion of he number of used pulses, whereas i has almos reahed sauraion afer he hird pulse when pih filer is used. VI. Referenes ] R. Ramahandran and P. Kabal, "Pih Prediion Filers in Speeh Codin", IEEE Trans. On ASSP, 37(4), (989), ] P. Kabal and R. P. Ramahandran, "Join Opimizaion of Linear Prediors in Speeh Codin", IEEE Trans. on ASSP, 37(5), (989), ] S. Sinhal and B. Aal, "Opimizin LPC Filer Parameers for Muli-Pulse Exiaion", in pro. ICASSP, (983), ] M. Frai, G. Mian and G. Riardi, "An Approah o Parameer Reopimizaion in Mulipulse-Based Coders", IEEE Trans. On Speeh and Audio Pro., (4), (993), ] A. Hasib, K, Haiolu, "Soure Combined Linear Prediive Analysis in Pulse- Based Speeh Coders", IEE Pro., Vision, Imae and Sinal Proessin, 43(3), June (996) ] M. Serizawa, A. Gersho, Join Opimizaion of LPC and Closed-Loop Pih Parameers in CELP Coders, IEEE, Sinal Proessin Leers, 6(3), (999), ] K. Haiolu, A. Hasib, "Pulse-by-Pulse Reopimizaion of he Synhesis Filer in Pulse Based Coders", IEEE Trans. On Speeh and Audio Proessin, 6(2), Marh (998), An-Najah Univ. J. Res. (N. S), Vol. 6(2), 2002