both covariance and autocorrelation methods use two step solutions 1. computation of a matrix of correlation values

Transcription

1 Dgtal Speech Processng Lecture 4 Lnear Predctve Codng (LPC)-Lattce Lattce Methods, Applcatons

2 Lattce Formulatons of LP both covarance and autocorrelaton methods use two step solutons. computaton of a matrx of correlaton values 2. effcent soluton of a set of lnear equatons another class of LP methods, called lattce methods, has evolved n whch the two steps are combned nto a recursve algorthm for determnng LP parameters begn wth Durbn algorthm--at at the stage the set of () coeffcents { α j, j = 2,,..., } are coeffcents of the order optmum LP th th 2

3 Lattce Formulatons of LP defne the system functon of the order nverse flter (predcton error flter) as () ( ) α () = k A z z nˆ k = k f the nput to ths flter s the nput segment ( ) ( ˆ () () s m = s n+ m) w( m), wth output e ( m) = e ( nˆ + m) () () α k k = e ( m) = sm ( ) sm ( k) where we have dropped subscrpt nˆ - the absolute locaton of the sgnal the z-transform gves () () () E ( z) = A ( z) S( z) = k αk z S ( z ) k = th nˆ 3

4 Lattce Formulatons of LP usng the steps of the Durbn recurson () (-) (-) () j j k - j ( α = α α, and α = k ) we can obtan a recurrence formula for A ( z ) of the form () ( ) ( ) ( ) = ( ) ) ( z ) A z A z k z A gvng for the error transform the expresson () () ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) E ( z) A ( zsz ) ( ) kz A ( z ) Sz ( ) e m e m k b m 4

5 Lattce Formulatons of LP where we can nterpret the frst term as the z-transform of fthe forward predcton error for an ( ) order predctor, and the second term can be smlarly nterpreted based on defnng a backward predcton error () () ( ) ( ) = = ( ) ( ) = z B ( z) ke ( z) B ( z) z A ( z ) S( z) z A ( z ) S( z) k A ( z) S( z) wth nverse transform () () αk ( ) k = b ( m) = s( m ) s( m+ k ) = b ( m ) ke ( m) () st ( ) wth the nterpretaton that we are attemptng to predct sm ( ) from the samples of the nput that follow s( m ) => we are dong a backward predcton and b ( m) s called the backward predcton error sequence 5

6 Lattce Formulatons of LP same set of samples s used to forward predct s(m) and backward predct s(m-) 6

7 Lattce Formulatons of LP () () the predcton error transform and sequence E ( z), e ( m) can now be expressed n terms of forward and backward errors, namely () ( ) ( ) E ( z) = E ( z) k z B ( z) () ( ) ( ( ) = ( ) e m e m k b ) ( m ) smlarly we can derve an expresson for the backward error transform and sequence at sample m of the form () ( ) ( ) = B ( z) z B ( z) k E ( z) () ( ) ( ) b ( m ) = b ( m ) k e ( m ) 2 these two equatons defne the forward and backward predcton error for th an order predctor n terms of the correspondng predcton errors of an th ( ) order predctor, wth the remnder that a zeroth order predctor does no predcton, so ( 0) ( 0) e ( m ) = b ( m ) = s ( m ) 0 m L 3 ( p) Ez ( ) = E ( z) = Az ( )/ Sz ( ) 7

8 Lattce Formulatons of LP Assume we know k (from external computaton): () (). we compute e [ m] and b [ m] from sm [ ], 0 m Lusng Eqs. and 2 (2) (2) 2. we next compute e [ m] and b [ m] for 0 m L+ usng Eqs. and 2 3. extend soluton (lattce) to p sectons gvng ( p) ( p) e [ m] and b [ m] for 0 m L+ p ( p) 4. soluton s en [ ] = e [ n] at the output of the th p lattce secton 8

9 Lattce Formulatons of LP lattce network wth p sectons the output of whch s the forward predcton error dgtal network mplementaton of the predcton error flter wth transfer functon A(z) no drect correlatons no alphas k s computed from forward and backward error sgnals 9

10 Lattce Flter for A(z) ( 0 ) ( 0 ) e [ n] b [ n] s[ n] n L = = 0 e [ n] = e [ n] k b [ n ] = 2,,..., p, 0 n L + ( ) ( ) ( ) b [ n] = ke [ n] + b [ n ] = 2,,..., p, 0 n L + ( ) ( ) ( ) ( p) en [ ] = e [ n ], 0 n L + p Ez ( ) = AzSz ( ) ( ) 0

11 ( p e ) [ n] All-Pole Lattce Flter for H(z) ( p e ) [ n] e () [ n ] k k p p k e (0) [ n ] s [ n] k k p p k ( p b ) [ n] ( p) e n = e n z z ( p b ) [ n] b () [ n ] b (0) [ n] ( ) [ ] [ ] ( ) ( ) ( ) e [ n] = e [ n] + k b [ n ] = p, p,...,, 0 n L + ( ) ( ) ( ) [ ] b n = ke [ n] + b [ n ] = p, p,...,, 0 n L + ( 0) ( 0) s[ n] = e [ n] = b [ n], 0 n L Sz Az ( ) ( ) = Ez ( )

12 All-Pole Lattce Flter for H(z) () ( ). snce b [ ] = 0,, we can frst solve for e [0] for = p p e = e ( ) ( ),,...,, usng the relatonshp: [0] [0] (0) (0) ( ) 2. snce b [0] = e [0] we can then solve for b [0] for = 2,2,..., p usng the equ aton: b [0] = k e [0] () ( ) ( ) 3. we can now begn to solve for as: e [] = e [] + k b [0], = p, p,..., ( ) ( ) ( ) (0) (0) ( ) 4. we set b [] = e [] and we can then solve for b [] for =,2,..., p usng the equaton: b [] = k e [] + b [0], =, 2,..., p () ( ) ( ) 5. we terate for n= 2,3,..., N and end up wth s n e n b n (0) (0) [ ] = [ ] = b [ ] e [] 2

13 Lattce Formulatons of LP the lattce structure comes drectly out of the Durbn algorthm the k parameters are obtaned from the Durbn equatons the predctor coeffcents, α, do not appear explctly n the lattce structure L + k can relate the k parameters to the forward and backward errors va m= 0 ( ) ( ) e ( m) b ( m ) k = 4 2 / L + L + ( ) 2 ( ) 2 [ e ( m)] [ b ( m )] m = 0 m = 0 where k s a normalzed cross correlaton between the forward and backward predcton error, and s therefore called a partal correlaton or PARCOR coeffcent can compute predctor coeffcents re cursvely from the PARCOR coeffcents 3

14 Drect Computaton of k Parameters assume sn [ ] non-zero for 0 n L assume k chosen to mnmze total energy of the forward k (or backward) predcton errors we can then mnmze forward predcton error as : L + () () 2 L + forward = ( m= 0 m= 0 ( ) ( ) E e m) = e ( m) k ( ) b m E ( ) ( ) ( ) k () L + forward ( ) ( ) ( ) = 0 = 2 e m k b m b m k m= 0 L + ( ) ( ) e m b m forward m= 0 = L + 2 m= 0 ( ) ( ) ( b ) ( m ) 2 4

15 Drect Computaton of k Parameters we can also choose to mnmze the backward predcton error L + 2 L + () () ( ) ( ) backward m= 0 m= 0 E = b ( m) = ke ( m) + b ( m ) E () L + backward ( ) ( ) = 0= 2 ke m + b m k m=0 L + ( ) ( ) e ( m) b ( m ) backward m = 0 k = L + 2 ( e ) ( m) m= 0 ( ( ) ( ) e ) ( m) 2 5

16 Drect Computaton of k Parameters f we wndow and sum over all tme, then L + 2 L + 2 ( ) ( ) e ( m) = b ( m ) m= 0 m= 0 therefore k = k k = k = k PARCOR forward backward forward backward = L + m= 0 ( ) ( ) e m b m ( ) ( ) L + 2 L + ( ) ( ) 2 e ( m) b ( m ) m= 0 m= 0 2 / 6

17 Drect Computaton of k Parameters mnmze sum of forward and backward predcton errors over fxed nterval (covarance method) { 2 2 ( ) ( ) } () = L Burg () + () m= 0 E e m b m E k Burg 2 ( ) ( ) ( ) ( L ) e ( m) k ( ) b m ke ( m) b m= 0 m= = + + () L Burg ( ) ( ) ( ) k m= 0 = = 0= 2 e ( m ) k b ( m ) b ( m ) L ( ) ( ) ( ) 2 ke ( ) + ( ) m b m e ( m) m= 0 ( ) ( ) 2 L e ( m) b ( m ) m= 0 L 2 L 2 ( ) ( ) m= 0 Burg k e ( m ) + b ( m ) always m= 0 ( m ) 2 7

18 Comparson of Autocorrelaton and Burg Spectra sgnfcantly less smearng of formant peaks usng Burg method 8

19 Summary of Lattce Procedure steps nvolved n determnng the predctor coeffcents and the k parameters for the lattce method are as follows ( 0) ( 0). ntal condton, e ( m) = s( m) = b ( m) from Eq. *3 () 2. compute k = α from Eq. *4 () () 3. determne forward and backward predcton errors e ( m ), b ( m ) from Eqs. * and *2 4. set = 2 () = α () j = () () 5. determne k from Eq. *4 6. determne α j for j = 2,,..., from Durbn teraton 7. determne e ( m) and b ( m) from Eqs. * and *2 8. set = + 9. f s less than or equal to p, go to step 5 0. procedure s termnated predctor coeffcents obtaned drectly from speech samples => wthout calculaton of autocorrelaton functon method s guaranteed to yeld stable flters ( k <) wthout usng wndow 9

20 Summary of Lattce Procedure 20

21 Summary of Lattce Procedure (0) [n] () e [ n ] [ n] s ( (2) e ] e [ n ] ( ) e p [ n] e[n] [ ] (0) z k 2 k2 k k z () z k k b [n] n b [nn ] b [nn ] [ n ] ( 0) ( 0) (2) e ( m) = b ( m) = s( m) 3 ( ) b p p p k L + + m= 0 ( ) ( ) e ( m) b ( m ) = 4 2 / L + L + ( ) 2 ( ) 2 [ e ( m )] [ b ( m )] m= 0 m= 0 ( ) ( ) ( ) e ( m) = e ( m) k b ( m ) () ( ) ( ) b ( m) = b ( m ) k e ( m) 2 2

22 Predcton Error Sgnal. Speech Producton Model p k = sn ( ) = asn ( k) + Gun ( ) Sz ( ) Hz ( ) = = Uz ( ) 2. LPC Model: k p G k = az k k en ( ) = sn ( ) sn ( ) = sn ( ) α sn ( k) Ez ( ) Az ( ) = = Sz ( ) 3. LPC Error Model: p k = α z k k Sz ( ) = = Az ( ) Ez ( ) α z p k = p k = k k p k = sn ( ) = en ( ) + α sn ( k) k k Gu(n) s(n) e(n) H(z) A(z) /A(z) s(n) e(n) s(n) Perfect reconstructon even f a k not equal to α k 22

23 LPC Comparsons fle:ah, ss:3000, L:300, Lp:2 23

24 LPC Comparsons fle:test_6k, ss:3000, L:300, p:2 24

25 LPC Comparsons fle:test_6k, ss:000, L:400, p:2 25

26 Comparsons Between LP Methods the varous LP soluton technques can be compared n a number of ways, ncludng the followng: computatonal ssues numercal ssues stablty of soluton number of poles (order of predctor) wndow/secton length for analyss 26

27 LP Soluton Computatons assume L L 2 >>p; choose values of L =300, L 2 =300, L 3 =300, p=0 computaton for covarance method L p+p *,+ autocorrelaton t method L 2 p+p *,+ lattce method 5L 3 p 5000 *,+ 27

28 LP Soluton Comparsons stablty guaranteed for autocorrelaton method cannot be guaranteed for covarance method; as wndow sze gets larger, ths almost always makes the system stable guaranteed for lattce method choce of LP analyss parameters need 2 poles for each vocal tract resonance below F s /2 need 3-4 poles to represent source shape and radaton load use values of p

29 The Predcton Error Sgnal 29

30 LP Speech Analyss fle:s5, ss:000, frame sze (L):320, lpc order (p):4, cov method Top panel: speech sgnal Second panel: error sgnal Thrd panel: log magntude spectra of sgnal and LP model Fourth panel: log magntude spectrum of error sgnal 30

31 LP Speech Analyss fle:s5, ss:000, frame sze (L):320, lpc order (p):4, ac method Top panel: speech sgnal Second panel: error sgnal Thrd panel: log magntude spectra of sgnal and LP model Fourth panel: log magntude spectrum of error sgnal 3

32 LP Speech Analyss fle:s3, ss:4000, frame sze (L):60, lpc order (p):6, cov method Top panel: speech sgnal Second panel: error sgnal Thrd panel: log magntude spectra of sgnal and LP model Fourth panel: log magntude spectrum of error sgnal 32

33 LP Speech Analyss fle:s3, ss:4000, frame sze (L):60, lpc order (p):6, ac method Top panel: speech sgnal Second panel: error sgnal Thrd panel: log magntude spectra of sgnal and LP model Fourth panel: log magntude spectrum of error sgnal 33

34 Varaton of Predcton Error wth LP Order predcton error flattens at around p=3-4 4 normalzed predcton error for unvoced speech larger than for voced speech => model sn t qute as accurate 34

35 LP Soluton Comparsons choce of LP analyss parameters use small values of L for reduced computaton use l order of several ptch perods for relable predcton-especally when usng tapered wndow use l from samples at 0 khz for autocorrelaton for lattce and covarance methods, L as small as 2 p has been used (wthn ptch perods); however er f ptch pulse occurs wthn wndow, bad predcton results => use much larger values of L 35

36 Predcton Error Sgnal Behavor - the predcton error sgnal s computed as p k k = en ( ) = sn ( ) α sn ( k) = Gun ( ) - en ( ) should be large at the begnnng of each ptch perod (voced speech) => good sgnal for ptch detecton - can perform autocorrelaton on en ( ) and detect largest peak - error spectrum s approxmately flat-so effects of formants on ptch detecton are mnmzed 36

37 Normalzed Mean-Squared Error - for autocorrelaton t method V nˆ = L+ p m= 0 L m= 0 L m= 0 L e 2 nˆ 2 n ( m) sˆ ( m) - for covarance method V nˆ = m= 0 e 2 nˆ 2 n ( m) sˆ ( m) - the predcton error sequence (defnng α = ) s e ( m) = α s ( m k) nˆ nˆ p k = 0 k nˆ - gvng many forms for the normalzed error V V V nˆ nˆ = = p p = 0 j= 0 p = 0 p = φnˆ (, j) α α j φ (,) 00 nˆ φnˆ (, 0) α φ (,) 00 nˆ 2 = ( k ) 0 37

38 Expermental Evaluatons of LPC Parameters 38

39 Normalzed Mean-Squared Error V n versus p for synthetc vowel, ptch perod of 83 samples, L=60, ptch synchronous analyss covarance method-error error goes to zero at p=, the order of the synthess flter autocorrelaton method, V n 0. for p 7 snce error domnated by predcton at begnnng of nterval 39

40 Normalzed Mean-Squared Error normalzed error versus p for ptch asynchronous analyss, L=20, normalzed error falls to 0. near p= for both covarance and autocorrelaton methods 40

41 Normalzed Mean-Squared Error normalzed error versus L, p=2 for L < ptch perod (83 samples), covarance method gves smaller normalzed error than autocorrelaton method for values of L at or near multples of ptch perod, normalzed error jumps due to large predcton error n vcnty of ptch pulse when L > 2 * ptch perod, normalzed error same for both autocorrelaton and covarance methods 4

42 Normalzed Mean-Squared Error results for natural speech 42

43 Normalzed Mean-Squared Error varablty of normalzed error wth postonng of frame sample-by-sample LP analyss of 40 msec of vowel // sgnal energy n part a; normalzed error n part b for p=4 pole analyss usng 20 msec frame sze for covarance method; normalzed error n part c for autocorrelaton method usng HW; normalzed error n part d for autocorrelaton method usng RW average ptch perod of 84 samples => 2.5 ptch perods n 20 msec wndow substantal varatons n normalzed error for covarance method-especally peaked when ptch pulses enter wndow => longer wndows gve larger normalzed errors substantal hgh frequency varatons n normalzed error for autocorrelaton method wth some ptch modulatons 43

44 Propertes of the LPC Polynomal 44

45 Mnmum-Phase Property of A(z) A( ( z ) has all ts zeros nsde d the unt crcle 2 P roof : Assume that z ( z > ) s a zero (root) of A( z) o o ( ) ( ) ( ) A z = z o z A z The mnmum mean-squared error s π 2 ω 2 ω 2 ˆ = ˆ[ ] = j j n n ( ) nˆ( ) m= 2π π E e m A e S e dω π jω jω jω = ( ) 0 2 ˆ( ) ω > π ze o A e Sn e d π jω jω o = o ( / o) ze z z e Thus, A( z) could not be the optmum flter because we could replace z by ( / z ) and decrease the error. o o 45

46 PARCORs and Stablty () prove that k z for some j ( ) ( ) ( ) ( ) = = j j = A ( z) A ( z) k z A ( z ) ( z z ) It s easly shown that k s the coeffcent () () of z n A ( z),.e., α = k. Therefore, k p = z j = () j If, then ether all the roots must be on the k unt crcle or at least one of the unt crcle. j them must be outsde 46

47 PARCORs and Stablty f, then ether all the roots must be on the k unt crcle or at least one of them must be outsde () the unt crcle. Snce ths s true for all A ( z), = 2,,..., p, a necessary condton for the roots of ( p A ) ( z) to be nsde the unt crcle s: for the th k <, = 2,,...,, p -order optmum lnear predctor, () 2 ( ) 2 ( 0 ) ( ) ( j ) j = E = k E = k E > 0 ( p) so k < and t herefore A ( z ) has all ts roots nsde the unt crcle. 47

48 Root Locatons for Optmum LP Model H ( z) = G Az ( ) = p G = G = = α z p p = = Gz ( z z ) ( z z ) p 48

49 Pole-Zero Plot for Model Whch poles correspond to formant frequences? 49

50 Pole Locatons 50

51 Pole Locatons (F S =0,000 Hz) F = ( θ /80) ( F S / 2) 5

52 Estmatng Formant Frequences compute A(z) and factor t. fnd roots that are close to the unt crcle. compute equvalent analog frequences from the angles of the roots. plot formant frequences es as a functon of tme. 52

53 Spectrogram wth LPC Roots 53

54 Spectrogram wth LPC Roots 54

55 Comparson to ABS Methods error measure for ABS methods s log rato of power spectra,.e., π 2 ω 2 log j S ( ) n e E = ω ( ω ) 2 d j π He thus for ABS mnmzaton of E s equvalent to mnmzng mean squared error between two log spectra comparng ELPC and EABS we see the followng: both error measures related to rato of sgnal to model spectra both tend to perform unformly over whole frequency range both are sutable to selectve error mnmzaton over specfed frequency ranges error crteron for LP modelng places hgher weght on frequency regons jω 2 jω 2 where ee S n(e e ) < H ( e ), whereas easthe ABS Serror crteron places equal weght on both regons for unsmoothed sgnal spectra (as obtaned by STFT methods), LP error crteron yelds better spectral matches than ABS error crteron for smoothed sgnal spectra (as obtaned at output of flter banks), both error crtera wll perform well 55

56 Relaton of LP Analyss to Lossless Tube Models 56

57 Dscrete-Tme Model -I Make all sectons the same length wth delay τ =Δ x / c where = NΔx 57

58 Dscrete-Tme Model - II N-secton lossless tube model corresponds to dscrete-tme system when: cn FS = 2 where c s the velocty of sound, N s the number of tube sectons, F s the samplng frequency, and s S the total length of the vocal tract. The reflecton coeffcents { r, k N } are related to the areas of the lossless tubes by: r k = A A k+ + k+ Ak A k Can fnd transfer functon of dgtal system subject to constrants of the form r G =, r N = r = L ρc/ AN Z ρc/ A + Z N L L k 58

59 Dscrete-Tme Model - III 59

60 Dscrete-Tme Model - IV Gven the system functon: Vz ( ) 05.( + rg) ( + rk) z U L( z) k= = = U ( z) D( z) G N N / 2 A k+ Ak rk = reflecton coeffcent Ak+ + Ak ρ c / AN Z ρc / A + Z N L f rg = (.e., RG = ) and rn = rl = N L then Dz ( ) satstes the recurson D ( z) = 0 ( ) ( ) k k = k + k k ( ) = 2 D ( z) D ( z) r z D ( z ) k,,, N Dz ( ) = D ( z) N 60

61 e[n] All-Pole Lattce Flter for H(z) s[n] k p k p z k p k p z k k z A ( 0) ( z) = A ( z) = A ( z) k z A ( z ) = 2,,, p () ( ) ( ) ( p) Az ( ) = A ( z) S ( z ) = E ( z ) Az ( ) If r = k and N = p, then D( z) = A( z). 6

62 Tube Areas from PARCORS Relaton to areas: A A ( A / A ) = = = + + k r A+ + A ( A+ / A) + Solve for A + n terms of A A + k A = A > 0 k + = > + k A + k Log area ratos (good for quantzaton) g A + k = log = log A + k senstvty under 0 Mnmzes spectral unform quantzaton 62

63 Estmatng Tube Areas from Speech (Wakta). Sample speech [ sn [ ] = s( nt)] at samplng rate F = / T = ρ c /( 2 ). s 2. Remove effects of glottal source and radaton by pre-emphass emphass xn [ ] = sn [ ] sn [ ]. 3. Compute the PARCOR coeffcents on a short- tme bass: k, = 2,,..., p. 4. Assumng A = (arbtrary), compute k A+ = A, = 2,,..., p + + k H. Wakta, IEEE Trans. Audo and Electroacoustcs,October, 973. a 63

64 Estmaton of Tube Areas Analyss of syllable /IY/ /B/ /AA/ /IY/ sound n part (a) /B/ sound n parts (b) and (c) /AA/ sound n part (d) Usng Wakta method to estmate tube areas from the speech waveform as shown n lower plot. 64

65 Estmatons of Tube Areas Use of Wakta method to estmate tube areas for the fve vowels, /IY/, /EH/, /AA/, /AO/, /UW/ 65

66 Alternatve Representatons of the LP Parameters 66

67 LP Parameter Sets 67

68 PARCORs to Predcton Coeffcents assume that k, = 2,,, p are gven. Then we can skp the computaton of k n the Levnson recurson. for = 2,,, p () α = k f >, then for j = 2,,, α = α k α () ( ) ( ) j j j end end ( p ) α j = α j j = 2,,, p 68

69 Predcton Coeffcents to PARCORs assume that α j, j = 2,,, p are gven. Then we can work backwards through the Levnson Recurson. α k ( p) j p = α for j = 2,,, p = α j ( p) p for = p, p,, 2 for j = 2,,, end α end () () α α ( ) j + k j j = 2 k ( ) k = α 69

70 LP Parameter Transformatons roots of the predctor polynomal p p α k k k k = k = ( ) Az ( ) = z = zz where each root can be expressed as a z-plane or s-plane root,.e., z = z + jz ; s = σ + j Ω gvng z k kr k k k k k = st k e Ω = z = + mportant for formant estmaton ( z z ) k 2 2 k tan ; σ k log kr k T z kr 2T 70

71 LP Parameter Transformatons cepstrum of IR of overall LP system from predctor coeffcents n k = ˆ k hn ( ) = α ˆ n + hk ( ) αn k n n predctor coeffcents from cepstrum of IR where α n n ˆ k = hn ( ) ˆ hk ( ) αn k n n k = n G hnz p n= 0 k Hz ( ) = ( ) = αk z k = 7

72 LP Parameter Transformatons IR of all pole system p k = hn ( ) = α hn ( k) + Gδ( n) 0 n autocorrelaton of IR k R () = hnhn ( ) ( ) = R ( ) n= 0 p R () α R ( k ) = k k = p αk k = R ( 0) = R ( k) + G 2 72

73 LP Parameter Transformatons autocorrelaton of the predctor polynomal p Az ( ) = α z k = wth IR of the nverse flter k k p k k = an ( ) = δ ( n) αδ( n k) wth autocorrelaton R ( ) = akak ( ) ( + ) 0 p a p k = 0 73

74 LP Parameter Transformatons log area rato coeffcents from PARCOR coeffcents A+ k g = log = log A + k p wth nverse relaton g e k = p g + + e 74

75 Quantzaton of LP Parameters consder the magntude-squared of the model frequency response jω 2 He ( ) = = P ( ω, g ) j ω 2 A ( e ) where g s a parameter that affects P. spectral senstvty can be defned d as π S P( ω, g ) = lm log d ω g Δg 0 Δ g 2π P ( ω, g +Δg π ) whch measures senstvty to errors n the g parameters 75

76 Quantzaton of LP Parameters spectral senstvty for k parameters; low senstvty around 0; hgh senstvty around spectral senstvty for log area rato parameters, g low senstvty for vrtually entre range s seen 76

77 Lne Spectral Par Parameters Az ( ) = + α z + α z α z 2 p 2 p = all-zero predcton flter wth all zeros, zk, nsde the unt crcle Az ( ) = z Az ( ) = α z α z + α z + z ( p+ ) p+ p ( p+ ) p 2 = recprocal polynomal wth nverse zeros, / z consder the followng: Az ( ) ω ( ) = j Lz = allpass system Le ( ) =, all ω Az ( ) form the symmetrc polynomal Pz ( ) as: ( + ) ( ) = ( ) + p Pz Az Az ( ) = Az ( ) + z Az ( ) Pz ( ) has zeros for Lz ( ) = ; ( Az ( ) = Az ( )) { } j jω ω k arg L( e ) = ( k + / 2 ) 2π, k = 0,,..., p form the ant-symmetrc polynomal Qz ( ) as: Qz = Az Az = Az z Az Qz Lz = + Az = Az ( p+ ) ( ) ( ) ( ) ( ) ( ) ( ) has zeros for ( ) ;( ( ) ( )) j { ω k } 2π 0 arg Le ( ) = k, k=,,..., p z k 77

78 Lne Spectral Par Parameters zeros of Pz ( ) and Qz ( ) fall on unt crcle and are nterleaved wth { ω } each other set of called Lne Spectral Frequences (LSF) 2 k LSFs are n ascendng order stablty of Hz ( ) guaranteed by quantzng LSF parameters j ω j ω jω P ( e ) + Q ( e ) Ae ( ) = 2 jω Ae ( ) = 2 2 jω jω Pe ( ) + Qe ( ) 4 78

79 Lne Spectrum Par (LSP) Parameters propertes of LSP parameters. Pz ( ) corresponds to a lossless tube, open at the lps and open ( k p+ = ) at the glotts 2. Q ( z ) corresponds to a lossless tube, open at the lps and closed ( k = p+ ) Q p+ at the glotts 3. all the roots of Pz ( ) and Qz ( ) are on the unt crcle 4. f p s an even nteger, then P ( z ) has a root at z =+ and Q ( z ) has a root at z = 5. a necessary and suffcent condton for k <, = 2,,..., p s that the roots of Pz ( ) and Qz ( )alternate on the unt crcle 6. the LSP frequences get close together when roots of Az ( ) are close to the unt crcle 7. the roots of Pz ( ) are approxmately equal to the formant frequences 79

80 LSP Example p = 2 Pz ( ) roots o Qz ( ) roots x A ( z ) roots 80

81 Lne Spectrum Par (LSP) Parameters 8

82 LPC Synthess the basc synthess equaton s p sn ( ) = α sn ( k) + Gun ( ) k = k need to update parameters every 0 msec or so ptch synchronous mode works best 82

83 LPC Analyss-Synthess. Extract α k parameters properly 2. Quantze α k parameters properly so that there s lttle quantzaton error Small number of bts go nto codng the α k coeffcents 3. Represent e(n) va: Ptch pulses and nose LPC Codng Multple pulses per 0 msec nterval MPLPC Codng Codebook vectors CELP Almost all of the codng bts go nto codng of e(n) 83

84 LPC Vocoder header bt rates of 72 F S where F S =00, 67, and d33f for bt rates of 7200 bps, 4800 bps and 2400 bps 3600 bps 2400 bps VEV LPC 84

85 LPC Bascs A(z) H(z) s (m) e (m) s (m) n n n p k Ez ( ) Az ( ) = αkz = Pz ( ) = ; predcton error flter Sz ( ) k = p nˆ nˆ k nˆ k = e ( m) = s ( m ) α s ( m k );predcton error Hz ( ) = = = ; all pole model Az ( ) ( ) α p k Pz z k = k p [ ] 2 Enˆ = enˆ( m) = snˆ( m) αksnˆ( m k) m= m= k= 2 85

86 LPC Bascs-Speech Model e(n)=gu(n) + s(n) P(z) Hz ( ) jω He ( ) = G G S( z) = = = p ( ) ( ) α k P z U z z k = p k G αk e k = jωk 86

87 Summary the LP model has many nterestng and useful propertes that t follow from the structure t of the Levnson-Durbn algorthms the dfferent equvalent representatons have dfferent propertes under quantzaton polynomal coeffcents (bad) polynomal roots (okay) PARCOR coeffcents (okay) lossless tube areas (good) LSP root angles (good) almost all LPC representatons can be used wth a range of compresson schemes and are all good canddates for the technque of Vector Quantzaton 87