for periodic signals with period Np, it is obvious that

Transcription

1 Dgtal Seech Processg Lecture 3 Lear Predctve Codg (LPC)- Itroducto LPC Methods LPC ethods are the ost wdely used seech codg, seech sythess, seech recogto, seaer recogto ad verfcato ad for seech storage LPC ethods rovde extreely accurate estates of seech araeters, ad does t extreely effcetly basc dea of Lear Predcto: curret seech sale ca be closely aroxated as a lear cobato of ast sales,.e., ( ) = s s ( ) for soe value of, 's LPC Methods for erodc sgals wth erod N, t s obvous that s ( ) s ( N ) but that s ot what LP s dog; t s estatg s ( ) fro the ( << N ) ost recet values of s ( ) by learly l redctg ts value for LP, the redctor coeffcets (the 's) are detered (couted) by zg the su of squared dffereces (over a fte terval) betwee the actual seech sales ad the learly redcted oes LPC Methods LP s based o seech roducto ad sythess odels - seech ca be odeled as the outut of a lear, te-varyg syste, excted by ether quas-erodc ulses or ose; - assue that the odel araeters rea costat over seech aalyss terval LP rovdes a robust, relable ad accurate ethod for estatg the araeters of the lear syste (the cobed vocal tract, glottal ulse, ad radato characterstc for voced seech) 3 4 LPC Methods LP ethods have bee used cotrol ad forato theory called ethods of syste estato ad syste detfcato used extesvely seech uder grou of aes cludg. covarace ethod. autocorrelato ethod 3. lattce ethod 4. verse flter forulato 5. sectral estato forulato 6. axu lelhood ethod 7. er roduct ethod 5 Basc Prcles of LP Sz ( ) Hz ( ) = = GU( z) az ( ) = s as ( ) + Gu ( ) the te-varyg dgtal flter reresets the effects of the glottal ulse shae, the vocal tract IR, ad radato at the ls the syste s excted by a ulse tra for voced seech, or a rado ose sequece for uvoced seech ths all-ole odel s a atural reresetato for o-asal voced seech but t also wors reasoably well for asals ad uvoced souds 6

2 LP Basc Equatos LP Estato Issues th a order lear redctor s a syste of the for ( ) ( ) ( ) ( ) Sz s = s Pz = z = Sz ( ) = = the redcto error, e ( ), s of the for e ( ) = s ( ) s ( ) = s ( ) s ( ) the redcto error s the outut of a syste wth trasfer fucto Ez ( ) ( ) = = ( ) = ( ) Az Pz z Sz f the seech sgal obeys the roducto odel exactly, ad f = a, e ( ) = Gu ( )ad Az ( ) s a verse flter for Hz ( ),.e., Hz ( ) = Az ( ) 7 eed to detere { } drectly fro seech such that they gve good estates of the te-varyg sectru eed to estate { } fro short segets of seech eed to ze ea-squared redcto error over short segets of seech resultg { } assued to be the actual { a} the seech roducto odel => ted to show that all of ths ca be doe effcetly, relably, ad accurately for seech 8 Soluto for { } short-te average redcto error s defed as E e ( ) s ( ) s ( ) ( ) = = = sˆ( ) ˆ( ) s = select seget of seech sˆ ( ) ( ˆ = s + ) the vcty of sale ˆ the ey ssue to resolve s the rage of for suato (to be dscussed later) 9 Soluto for { } ca fd values of that ze E by settg: E =, =,,..., gvg the set of equatos - sˆ ˆ ˆ ( )[ s( ) s ( )] =, = s( ) e( ) =, where ˆ are the values of that ze E (fro ow o ust use rather tha ˆ for the otu values) redcto error ( e( )) s orthogoal to sgal ( s( )) for delays ( ) of to Soluto for { } defg φ (, ) = s ( s ) ( ) we get φ φ (, ) = (, ), =,,..., leadg to a set of equatos uows that ca be solved a effcet aer for the { } Soluto for { } u ea-squared redcto error has the for = E = s ( ) s ( ) s ( ) whch ca be wrtte the for E = φ (, ) φ (, ) Process:. coute φ (, ) for,. solve atrx equato for eed to secfy rage of to coute φ (, ) eed to secfy s ( )

3 Autocorrelato Method assue s ( ) exsts for Lad s exactly zero everywhere else (.e., wdow of legth L sales) ( Assuto #) sˆ ( ) = ( + ˆ s ) w( ), L where w ( ) s a fte legth wdow of legth L sales ^ +L- ^ Autocorrelato Method f s ( ) s o-zero oly for Lthe e( ) = s( ) s( ) s o-zero oly over the terval L +, gvg L+ E = e( ) = e( ) = = at values of ear (.e., =,..., ) we are redctg sgal fro zero-valued sales outsde the wdow rage => e ( ) wll be (relatvely) large at values ear = L (.e., = L, L+,..., L+ ) we are redctg zero-valued sales (outsde wdow rage) fro o-zero sales => e ( ) wll be (relatvely) large for these reasos, orally use wdows that taer the seget to zero (e.g., Hag wdow) = =L = =L- 3 =- =L+- 4 Autocorrelato Method The Autocorrelato Method s = s + w ^ ^ˆ + L [ ˆ ] [ ] [ ] L- L R [ ] = s [ ] s [ + ],,, = 5 6 The Autocorrelato Method Autocorrelato Method Large errors at eds of wdow + L s [ ] [ ˆ ˆ ] [ ] = s + w ˆ + L e[ ] = s[ ] s[ ] = s ˆ [ ] s [ ] w [ ] + = L L ( L + ) 7 for calculato of φ(, ) sce s( ) = outsde the rage L, the L+ φ(, ) = s( s ) ( ),, = whch s equvalet to the for L+ ( ) φ (, ) = s( s ) ( + ),, = there are L o-zero ters the coutato of φ (, ) for each value of ad ; ca easly show that φ(, ) = f ( ) = R( ),, where R( ) s the short-te autocorrelato of s( ) evaluated at where L R( ) = s( ) s( + ) = 8 3

4 Autocorrelato Method sce R ( ) s eve, the φ(, ) = R( ),, thus the basc equato becoes φ ( ) = φ(, ), R( ) = R( ), wth the u ea-squared redcto error of the for E = φ(, ) φ(, ) = R( ) R( ) 9 Autocorrelato Method as exressed atrx for R( ) R().. R( ) R() ˆ() ˆ( ).. ˆ( ) R R R R( ) = ˆ( ) ˆ( ).. ˆ( ) ˆ( ) R R R R R =r wth soluto = R r R s a Toeltz Matrx => syetrc wth all dagoal eleets equal => there exst ore effcet algorths to solve for { } tha sle atrx verso Covarace Method Covarace Method there s a secod basc aroach to defg the seech seget s ( ) ad the lts o the sus, aely fx the terval over whch the ea-squared error s couted, gvg g ( Assuto #) : L L E = e( ) = s( ) s( ) = = = L φ (, ) = s( s ) ( ),, = chagg the suato dex gves L φ(, ) = s( ) s( + ),, = L φ(, ) = s( ) s( + ),, = ey dfferece fro Autocorrelato Method s that lts of suato clude ters before = => wdow exteds sales bacwards fro s ( ˆ - ) to s ( ˆ + L- ) sce we are extedg wdow bacwards, do't eed to taer t usg a HW- sce there s o trasto at wdow edges =- =L- Covarace Method Covarace Method caot use autocorrelato forulato => ths s a true cross correlato eed to solve set of equatos of the for φ (, ) = φ(, ), =,,...,, E = φ (, ) φ (, ) φ (, ) φˆ(, ).. ˆ(, ) φ φ(, ) ˆ(,) ˆ(, ).. ˆ(, ) φ φ φ φ(,) = φˆ(, ) φˆ(, ).. φˆ(, ) φˆ(, ) φ = ψ or = φ ψ 3 4 4

5 Covarace Method we have φ(, ) = φ(, ) => syetrc but ot Toeltz atrx whose dagoal eleets are related as φ( +, + ) = φ(, ) + s( ) s( ) s( L) s( L) φ (, ) = φ (, ) + s ( ) s ( ) s ( L ) s ( L ) all ters φ (, ) have a fxed uber of ters cotrbutg to the couted values ( L ters) φ (, ) s a covarace atrx => secalzed soluto for { } called the Covarace Method 5 Suary of LP th use order lear redctor to redct s( ) fro revous sales ze ea-squared error, E, over aalyss wdow of durato L-sales soluto for otu redctor coeffcets, { }, s based o solvg a atrx equato => two solutos have evolved - autocorrelato ethod => sgal s wdowed by a taerg wdow order to ze dscotutes at begg (redctg seech fro zero-valued sales) ad ed (redctg zero-valued sales fro seech sales) of the terval; the atrx φ ˆ (, ) s show to be a autocorrelato fucto; the resultg autocorrelato atrx s Toeltz ad ca be readly solved usg stadard atrx solutos - covarace ethod => the sgal s exteded by sales outsde the oral rage of L- to clude sales occurrg ror to = ; ths elates large errors coutg the sgal fro values ror to = (they are avalable) ad elates the eed for a taerg wdow; resultg atrx of correlatos s syetrc but ot Toeltz => dfferet ethod of soluto wth soewhat dfferet set of otal redcto coeffcets, { } 6 LPC Suary LPC Suary. Seech Producto Model: s ( ) = as ( ) + Gu ( ) Sz ( ) Hz ( ) = = GU( z). Lear Predcto Model: s ( ˆ) = ( ˆ s) Sz ( ) Pz ( ) = = z Sz ( ) az 3. LPC Mzato: E = e ( ) = s ( ) s ( ) = = s ( ) s ( ) E =, =,,..., φ = (, ) = s( s ) ( ) s ( ) s ( ) = s ( ) s ( ) e ( ˆ) = s ( ˆ) s ( ˆ) = s ( ˆ) s ( ˆ ) Ez ( ) Az ( ) = = Sz ( ) z 7 φ(, ) = φ (, ), =,,..., E = φ (, ) φ (, ) 8 LPC Suary LPC Suary 4. Autocorrelato Method: s ( ) = s( + ) w( ), L e( ) = s( ) s( ), L + s( ) defed for L; e( ) defed for L + large errors for ad for L L + L+ E = e ( ) = L ( ) (, ) ( ) ( ) ( ) = R( ) = R( ), ( ) φ = R = s s + = R 4. Autocorrelato Method: resultg atrx equato: R = r or =R r Rˆ ( ) ˆ().. ˆ ( ) R R R() ˆ() ˆ( ).. ˆ( ) R R R Rˆ( ) = ˆ( ) ˆ( ).. ˆ( ) R R R Rˆ( ) E = R ( ) R ( ) 9 atrx equato solved usg Levso or Durb ethod 3 5

6 5. Covarace Method: fx terval for error sgal = LPC Suary L L E = e( ) = s( ) s( ) = = = eed sgal for fro s ( ˆ ) to s ( ˆ + L) L+ sales φ(, ) = φ(, ), =,,..., E = φ (,) φ (, ) exressed as a atrx equato: φ = ψ = φ ψ φ or, syetrc atrx φ(,) φ(, ).. φ(, ) φ (, ) φ ˆ(,) φ ˆ(,).. φ ˆ(, ) φ ˆ(,) = φ (, ) φ(, ).. φˆ(, ) φˆ(, ) 3 Coutato of Model Ga t s reasoable to exect the odel ga, G, to be detered by atchg the sgal eergy wth the eergy of the learly redcted sales fro the basc odel equatos we have ( ) = ( ) Gu s a s( ) odel whereas for the redcto error we have ( ) = ( ) e s s ( ) best ft to odel whe = a (.e., erfect atch to odel), the e ( ) = Gu ( ) sce t s vrtually ossble to guaratee that = a, caot use ths sle atchg roerty for deterg the ga; stead use eergy atchg crtero (eergy error sgal=eergy exctato) L+ L+ G u ( ) = e ( ) = E = = 3 Ga Assutos assutos about exctato to solve for G voced seech-- u ( ) = δ ( ) Lorder of a sgle tch erod; redctor order,, large eough to odel glottal ulse shae, vocal tract IR, ad radato uvoced seech-- u ( )-zero ea, uty varace, statoary whte ose rocess 33 Soluto for Ga (Voced) for voced seech the exctato s Gδ ( ) wth outut h ( ) (sce t s the IR of the syste), ( ) = ( ) + δ( ); G G h h G Hz ( ) = = Az ( ) z wth autocorrelato R ( ) (of the u lse resose) satsfyg the relato show below R ( ) = hh ( ) ( + ) = R [ ], < = R ( ) = R( ), < R ( ) = R( ) + G, = 34 Soluto for Ga (Voced) Sce R ( ) ad R( ) have the detcal for, t follows that R ( ) = c R ( ), where c s a costat to be detered. Sce the total eerges the sgal ( R( )) ad the ulse resose ( R( )) ust be equal, the costat c ust be, ad we obta the relato G = R( ) R( ) = E sce R ( ) = R ( ),, ad the eergy of the ulse resose=eergy of the sgal => frst + coeffcets of the autocorrelato of the ulse resose of the odel are detcal to the frst + coeffcets of the autocorrelato fucto of the seech sgal. Ths codto called the autocorrelato atchg roerty of the autocorrelato ethod. 35 Soluto for Ga (Uvoced) for uvoced seech the ut s whte ose wth zero ea ad uty varace,.e., E u( ) u( ) = δ ( ) f we excte the syste wth ut Gu( ) ad call the outut g ( ) the g ( ) = g ( ) + Gu ( ) Sce the autocorrelato fucto for the outut s the covoluto of the autocorrelato fucto of the ulse resose wth the autocorrelato fucto of the whte ose ut, the Egg [ [ ] [ ]] = R [ ] δ [ ] = R [ ] lettg R ( ) deote the autocorrelato of g ( ) gves R ( ) = E gg ( ) ( ) = [ ( ) Eg g ( )] + E Gu ( ) g ( ) = R ( ), sce E Gu( ) g ( ) = for > because u( ) s ucorrelated wth ay sgal ror to u ( ) 36 6

7 Soluto for Ga (Uvoced) for = we get R ( ) = R ( ) + GE u( ) g ( ) = R ( ) + G sce E u( ) g ( ) = E u( )( Gu( ) + ters ror to = G sce the eergy the sgal ust equal the eergy the resose to Gu( ) we get R ( ) = R( ) ( ) ( ) G = R R = E 37 Frequecy Doa Iterretatos of Lear Predctve Aalyss 38 The Resultg LPC Model LPC Sectru The fal LPC odel cossts of the LPC araeters, { },,,...,, ad the ga, G, whch together defe the syste fucto G Hz ( ) = z = wth frequecy resose H G G ( e ) = = Ae ( ) e wth the ga detered by atchg the eergy of the odel to the short-te eergy of the seech sgal,.e., G = E = e ( ) = R ( ) R ( ) ( ) = 39 G He ( ) = e x = s.* hag(3); X = fft( x, ) [ A, G, r ] = autolc( x, ) H = G./ fft(a,); LP Aalyss s see to be a ethod of short-te sectru estato wth reoval of exctato fe structure (a for of wdebad sectru aalyss) 4 LP Short-Te Sectru Aalyss LP Short-Te Sectru Aalyss Defed seech seget as: s [ ] [ ˆ ˆ ] [ ] = s + w The dscrete-te Fourer trasfor of ths wdowed seget s: ( ) S e = s[ + ] w[ ] e = Short-te FT ad the LP sectru are led va short-te autocorrelato (a) Voced seech seget obtaed usg a Hag wdow (b) Corresodg shortte autocorrelato fucto used LP aalyss (heavy le shows values used LP aalyss) (c) Corresodg shortte log agtude Fourer trasfor ad short-te log agtude LPC sectru (F S =6 Hz) 4 4 7

8 LP Short-Te Sectru Aalyss (a) Uvoced seech seget obtaed usg a Hag wdow (b) Corresodg shortte autocorrelato fucto used LP aalyss (heavy le shows values used LP aalyss) (c) Corresodg shortte log agtude Fourer trasfor ad short-te log agtude LPC sectru (F S =6 Hz) 43 Frequecy Doa Iterretato of Mea-Squared Predcto Error The LP sectru rovdes a bass for exag the roertes of the redcto error (or equvaletly the exctato of the VT) The ea-squared redcto error at sale s: E = L+ = e [ ] whch, by Parseval's Theore, ca be exressed as: π π = ( ) ω ( ) ( ) ω π E = π = π π E e d S e A e d G where S ( e ) s the FT of s [ ] ad A( e ) s the corresodg redcto error frequecy resose Ae ( ) = e 44 Frequecy Doa Iterretato of Mea-Squared Predcto Error The LP sectru s of the for: G He ω ( ) = Ae ( ) Thus we ca exress the ea-squared error as: G S ( e ) E = dω = G π π π He ( ) We see that zg total squared redcto error s equvalet to fdg ga ad redctor coeffcets such that the tegral of the rato of the eergy sectru of the seech seget to the agtude squared of the frequecy resose of the odel lear syste s uty. Thus S ( e ) ca be terreted as a frequecy-doa weghtg fucto LP weghts frequeces where S ( e ) s large ore heavly tha whe S ( e ) s sall. 45 LP Iterretato Exale Much better sectral atches to STFT sectral eas tha to STFT sectral valleys as redcted by sectral terretato of error zato. 46 LP Iterretato Exale Effects of Model Order Note sall dffereces sectral shae betwee STFT, autocorrelato sectru ad covarace sectru whe usg short wdow durato (L=5 sales). The AC fucto, R ˆ[ ] of the seech seget, s ˆ[ ], ad the AC fucto, R [ ], of the ulse resose, h [ ], corresodg to the syste fucto, H ( z), are equal for the frst ( + ) values. Thus, as, the AC fuctos are equal lfor all values ad dthus: l He ω ω ( ) = S( e ) Thus f s large eough, the FR of the all-ole odel, He ω ( ), ca aroxate the sgal sectru wth arbtrarly sall error

9 Effects of Model Order Effects of Model Order 49 5 Effects of Model Order Effects of Model Order lots show Fourer trasfor of seget ad LP sectra for varous orders - as creases, ore detals of the sectru are reserved - eed to choose a value of that reresets the sectral effects of the glottal ulse, vocal tract ad radato--othg else 5 5 Lear Predcto Sectrogra Lear Predcto Sectrogra Seech sectrogra revously defed as: L ( π / N) log Sr[ ] = log s[ rr+ ] w[ ] e = for set of tes, t = rrt, ad set of frequeces, F = F / N,,,..., N / r S where R s the te shft ( sales) betwee adacet STFTs, T s the salg erod, F = / T s the salg frequecy, S ad N s the sze of the dscrete Fourer trasfor used to coute each STFT estate. Slarly we ca defe the LP sectrogra as a age lot of: log H [ ] = log at aalyss te rr. r Ar ( e ) π / r ( N) where G ad A e are the ga ad redcto error olyoal r ( π / N) r( ) G L=8, R=3, N=, 4 db dyac rage

10 Coarso to Other Sectru Aalyss Methods Sectra of sythetc vowel /IY/ (a) Narrowbad sectru usg 4 sec wdow (b) Wdebad sectru usg a sec wdow (c) Cestrally soothed sectru (d) LPC sectru fro a 4 sec secto usg a = order LPC aalyss 55 Coarso to Other Sectru Aalyss Methods Natural seech sectral estates usg cestral soothg (sold le) ad lear redcto aalyss (dashed le). Note the fewer (surous) eas the LP aalyss sectru sce LP used = whch restrcted the sectral atch to a axu of 6 resoace eas. Note the arrow badwdths of the LP resoaces versus the cestrally soothed resoaces. 56 Selectve Lear Predcto t s ossble to aly LP ethods to selected arts of sectru - -4 Hz for voced souds use a redctor of order Hz for uvoced souds use a redctor of order the ey dea s to a the frequecy rego { fa, fb} learly to { 5,. } or, equvaletly, the rego { π f A, πf B } as learly to {, π } va the trasforato ω πf ω A = πfb πfb πfa we ust odfy the calculato for the autocorrelato to gve: π R ( ) = ˆ ( ) ω π S e e d π 57 Selectve Lear Predcto - Hz rego odeled usg =8 o dscotuty odel sectra at 5 Hz -5 Hz rego odeled usg =4 5- Hz rego odeled usg =5 dscotuty odel sectra at 5 Hz 58 LPC Solutos-Covarace Method for the covarace ethod we eed to solve the atrx equato Solutos of LPC Equatos Covarace Method (Cholesy Decoosto Method) φ (, ) = φ(, ), =,,..., φ = ψ ( atrx otato) φ s a ostve defte, syetrc atrx wth (, ) eleet φ ˆ (, ), ad ad ψ are colu vectors wth eleets ad φ (, ) the soluto of the atrx equato s called the Cholesy decoosto, or square root ethod t φ=vdv ; V = lower tragular atrx wth 's o the a dagoal D=dagoal atrx 59 6

11 LPC Solutos-Covarace Method ca readly detere eleets of V ad D by solvg for (, ) eleets of the atrx equato, as follows φ (, ) = VdV, = gvg Vd = φ (, ) VdV, = ad for the dagoal eleets φ (, ) = VdV gvg d = φ (, ) V d, wth d = φ (, ) 6 Cholesy Decoosto Exale cosder exale wth = 4, ad atrx eleets φ (, ) = φ φ φ φ3 φ4 φ φ φ3 φ4 = φ 3 φ3 φ33 φ 43 φ4 φ4 φ34 φ44 d V V3 V4 V d V3 V4 V3 V3 d3 V 43 V4 V4 V43 d4 6 Cholesy Decoosto Exale LPC Solutos-Covarace Method solve atrx for d, V, V, V, d, V, V, d, V, d ste V d V d V d V d V d V d ste d = φ = φ; 3 = φ3; 4 = φ4 = φ / ; 3 = φ 3 / ; 4 = φ 4 / d = φ V d 3 = φ3 3 3 = ( φ3 3 ) / d V d V dv V V dv V d = φ V dv V = ( φ V dv )/ d terate rocedure to solve for d, V, d ste 3 ste 4 63 ow eed to solve for usg a -ste rocedure t VDV = ψ wrtg ths as VY= ψ wth t DV = Y or t V = D Y fro V (whch s ow ow) solve for colu vector Y usg a sle recurso of the for = ψ Y VY, = wth tal codto Y = ψ 64 LPC Solutos-Covarace Method ow ca solve for usg the recurso =+ = Y / d V, wth tal codto = Y / d calculato roceeds bacwards fro = dow to = 65 Cholesy Decoosto Exale cotug the exale we solve for Y Y ψ V Y ψ = V3 V3 Y 3 ψ 3 V4 V4 V43 Y4 ψ 4 frst solvg for Y Y4 we get Y = ψ Y = ψ VY Y3 = ψ 3 V3Y V3Y = ψ V Y V Y V Y Y

12 Cholesy Decoosto Exale ext solve for fro equato V V3 V4 / d Y Y/ d V3 V4 / d Y Y / d = = V 43 3 / d3 Y 3 Y3/ d 3 4 / d 4 Y 4 Y 4 / d 4 gvg the results 4 = Y4 / d4 3 = Y3/ d3 V434 = Y / d V33 V44 = Y/ d V V33 V44 coletg the soluto 67 Covarace Method Mu Error the u ea squared error ca be wrtte the for ˆ = φˆ(, ) E φ(, ) t = φ (,) ψ t t sce = YD V ca wrte ths as t E = φ(, ) Y D Y = φ (,) Y / d ths coutato for E ˆ ca be used for all values of LP order fro to ca uderstad how LP order reduces ea-squared error 68 Solutos of LPC Equatos Autocorrelato Method va Levso-Durb Algorth 69 7 Levso-Durb Algorth Autocorrelato equatos (at each frae ) : R[ ] = R[ ] =rrrs a ostve defte e syetrc Toeltz atrx The set of otu redctor coeffcets satsfy: R [] R[ ] =, wth u ea-squared redcto error of: ( ) R[] R[ ] = E 7 Levso-Durb Algorth By cobg the last two equatos we get a larger atrx equato of the for: ( ) R[] R[] R[]... R[ ] E ( ) R[] R[] R[]... R[ ] ( ) R [] R [] R []... R [ ] = ( ) R [ ] R [ ] R [ ]... R[] exaded atrx s stll Toeltz ad ca be solved teratvely by cororatg ew correlato value at each terato ad solvg for ext hgher order redctor ters of ew correlato value ad revous redctor 7

13 Levso-Durb Algorth 3 Levso-Durb Algorth 4 Show how order soluto;.e., gve we derve soluto to R th order soluto ca be derved fro ( ) ( ) ( ) ( ) ( ) () () (), the soluto to R = e st The ( ) soluto ca be exressed as: ( ) R[] R[] R[]... R[ ] E ( ) R[] R[] R[]... R[ ] ( ) R[] R[] R[]... R[ 3] = ( ) R [ ] R [ ] R [ 3]... R[] st = e ( ) ( ) Aedg a to vector ad ultlyg by the atrx R gves: ( ) R[] R[] R[]... R[ ] E ( ) R[] R[] R[]... R[ ] ( ) R[] R[] R[]... R[ ] = ( ) R [ ] R [ ] R [ 3]... R[] ( ) R [ ] R [ ] R [ ]... R[] γ ( ) ( ) = where γ = R [] R [ ] ad R [] are troduced Levso-Durb Algorth 5 Key ste s that sce Toeltz atrx has secal syetry we ca reverse the order of the equatos (frst equato last, last equato frst), gvg: ( ) R[] R[] R[]... R[ ] γ R[] R[] R[]... R[ ] ( ) ( ) R[] R[] R[]... R[ ] = ( ) R [ ] R [ ] R [ 3]... R[] ( ) R [ ] R [ ] R [ ]... R[] E 75 Levso-Durb Algorth 6 To get the equato to the desred for (a sgle () cooet the vector e ) we cobe the two sets of atrces (wth a ultlcatve factor ) gvg: ( ) ( ) E γ ( ) ( ) ( ) ( ) () R =.... ( ) ( ) ( ) ( ) γ E ( ) Choose γ so that vector o rght has oly a sgle o-zero etry,.e., ( ) R [] [ ] ( ) R γ = = = ( ) ( ) E E 76 Levso-Durb Algorth 7 The frst eleet of the rght had sde vector s ow: E = E ( γ = E ( ) ( ) ( ) ) ( ) The araeters are called PARCOR coeffcets. ( Wth ths choce of γ, the vector of order redctor ) th coeffcets s: () ( ( ) ) () ( ) ( ) =... () ( ) ( ) () yeldg the udatg rocedure () ( = (, =,,..., ) ) () = 77 Levso-Durb Algorth 7 The fal soluto for order s: = ( ) wth redcto error ( ) = = = = E E[] ( ) R[] ( ) If we use oralzed autocorrelato coeffcets: r [ ] = R [ ]/ R[] we get oralzed errors of the for: () () E () ν = = r [ ] = ( ) R[] = () where < ν or < < 78 3

14 Levso-Durb Algorth ( ) = ( ) ( ) () ( ) A z A z ( ) z A z 79 Autocorrelato Exale cosder a sle = soluto of the for R( ) R( ) R( ) = () ( ) R R R( ) wth soluto ( ) E = R( ) = R()/ R( ) () = R()/ R( ) () R ( ) R ( ) E = R( ) 8 Autocorrelato Exale R R R = R ( ) R ( ) ( ) R R R = ( ) ( ) ( ) ( ) ( ) ( ) R ( ) R ( ) ( ) R() R( ) R() R( ) = R ( ) R ( ) wth fal coeffcets ( ) = ( ) = () E = redcto error for redctor of order 8 Predcto Error as a Fucto of V E R[ ] R [ ] R [ ] = = Model order s usually detered by the followg rule of thub: F s / oles for vocal tract -4 oles for radato oles for glottal ulse 8 Autocorrelato Method Proertes ea-squared redcto error always o-zero decreases ootocally wth creasg odel order autocorrelato atchg roerty odel ad data atch u to order sectru atchg roerty favors eas of short-te FT u-hase roerty zeros of A(z) are sde the ut crcle Levso-Durb recurso effcet algorth for fdg redcto coeffcets PARCOR coeffcets ad MSE are by-roducts 83 4