Digital Speech Processing— Lecture 14 Linear Predictive Coding (LPC)-Lattice Methods, Applications 1 Prediction Error Signal 1. Speech Production Model p sn()=−+ asn ( k ) Gun () ∑ k Gu(n) s(n) k =1 H(z) Sz() G Hz()==p Uz() −k 1− ∑azk k =1 2. LPC Model: p en()=−=− sn () sn () sn ()α sn ( − k ) % ∑ k s(n) e(n) k =1 A(z) p Ez() −k Az()==−1 ∑αk z Sz() k =1 3. LPC Error Model: 11Sz() ==p Az() Ez () −k e(n) s(n) 1− ∑αk z k =1 1/A(z) p sn()=+ en ()α sn ( − k ) ∑ k Perfect reconstruction even if k =1 2 ak not equal to αk Lattice Formulations of LP • both covariance and autocorrelation methods use two step solutions 1. computation of a matrix of correlation values 2. efficient solution of a set of linear equations • another class of LP methods, called lattice methods, has evolved in which the two steps are combined into a recursive algorithm for determining LP parameters • begin with Durbin algorithm--at the i th stage the set of ()ith coefficients {α j ,ji= 12 , ,..., } are coefficients of the i order optimum LP 3 Lattice Formulations of LP • define the system function of the i th order inverse filter (prediction error filter) as i Az()iik( ) =−1 α () z− ∑ k k =1 • if the input to this filter is the input segment ˆ ()ii () ˆ smnˆ ()=+ snmwm ( )(), with output emnˆ ( )=+ enm ( ) i e()ii() m=− sm ()α () sm ( − k ) ∑ k k =1 • where we have dropped subscript nˆ - the absolute location of the signal • the z-transform gives ⎛⎞i Ez()ii( )==− AzSz () ( ) ( ) ⎜⎟1 α ()ikzSz− () ⎜⎟∑ k ⎝⎠k =1 4 Lattice Formulations of LP • using the steps of the Durbin recursion ()ii (-)11 (-) i () i (ααjj =−=kkii α ij- , and α i ) ()i we can obtain a recurrence formula for Az( ) of the form ()ii (−−−11 ) ii ( ) −1 Az( )=− A ( zkzA ) i ()z • giving for the error transform the expression ()ii (−−−−111 ) ii ( ) E( z )=− A ( zSz ) ( ) kzi A ( z ) Sz ( ) ()ii ()−−11 () i em()=− e () mkbmi ( −1 ) 5 Lattice Formulations of LP • where we can interpret the first term as the z-transform of the forward prediction error for an (i −1 )st order predictor, and the second term can be similarly interpreted based on defining a backward prediction error ()iii−− ()1111 −−− ii ( ) ( i − ) Bz( )== zAzSzzA ( ) ( ) ( zSzkAzSz ) ( ) −i ( ) ( ) −−11()ii () − 1 =−zB() z kEi () z • with inverse transform i bm()iii( )=−− smi ( )α () smki ( +−= ) b (−1 ) ( m −1)()− ke()i −1 m ∑ k i k =1 •− with the interpretation that we are attempting to predict sm( i ) from the ismi samples of the input that follow (− ) => we are doing a backward prediction and bm()i ( ) is called the backward prediction error sequence 6 Lattice Formulations of LP same set of samples is used to forward predict s(m) and backward predict s(m-i) 7 Lattice Formulations of LP • the prediction error transform and sequence Ezem()ii( ), () ( ) can now be expressed in terms of forward and backward errors, namely ()ii (−−−111 ) ( i ) Ez()=− E () zkzBi () z ()ii (−−11 ) ( i) em( )=− e ( mkb ) i ()1m −∗1 • similarly we can derive an expression for the backward error transform and sequence at sample m of the form ()iii−−11 ( ) ( − 1 ) Bz()=− zB () zkEzi () ()ii (−−11 ) ( i ) bm( )=−−∗ b ( m1 ) kemi ( ) 2 • these two equations define the forward and backward prediction error for an i th order predictor in terms of the corresponding prediction errors of an (i −1 )th order predictor, with the reminder that a zeroth order predictor does no prediction, so embmsm()00()==≤≤− () () ()01 mL ∗ 3 Ez()== E()p () z Az ()/() Sz 8 Lattice Formulations of LP Assume we know k1 (from external computation): 1. we compute em(1)[]and bm (1) []from sm[],0≤≤ m L using Eqs. ∗1 and ∗2 2. we next compute em(2)[]and bm (2) []for 01≤≤+mL using Eqs. ∗1 and ∗2 3. extend solution (lattice) to p sections giving em()pp[]and bm () []for 0≤ mLp ≤+− 1 4. solution is en[]= e()p [] n at the output of the pth lattice section 9 Lattice Formulations of LP • lattice network with p sections—the output of • no direct correlations which is the forward prediction error • no alphas • digital network implementation of the prediction • k’s computed from forward error filter with transfer function A(z) and backward error signals10 Lattice Filter for A(z) en()00[]== bn () [] sn []01 ≤≤− nL ()ii ()−−11 () i en[ ]=− e [ n ] kbni [ −=1120 ] i , ,..., p , ≤≤−+ nL 1 i ()iii ()−−11 () bn[ ]=− keni [ ] + b [ n −1120 ] i = , ,..., p , ≤ nL ≤ − 1 + i en[]=≤≤−+ e()p [], n01 n L p Ez()= AzSz ()() 11 All-Pole Lattice Filter for H(z) en()p [] en(1)p − [] en(1) [] en(0)[] s []n k k p −1 p k1 − k − k − k p p −1 1 −1 −1 z z bn()p [] bn(1)p − [] bn(1) [] bn(0)[] en()p []=≤≤−+ en [],01 nL p ()ii−−11 () () i enenkbn[]=+ []i [ −1 ] ipp=−,110 ,..., , ≤≤−+− nL 1 i 1 ()iii ()−−11 () bn[]=− keni [] + b [ n −1 ] ipp=−,110 ,..., , ≤≤−+ nL 1 i 1 ()00 () Sz()= 12 Ez () snen[]== [] bn [],01 ≤≤− nL Az() All-Pole Lattice Filter for H(z) 1. since bi()ii[1]0,−= ∀ , we can first solve for e (− 1) [0] for ipp=−,1,...,1,using the relationship: e(1)ii− [0][0] = e () This image cannot currently be displayed. 2. since be(0)[0]= (0) [0] we can then solve for b (i ) [0] for ()ii (− 1) ip=1,2,..., using the equation: bke[0]=− i [0] 3. we can now begin to solve for e(1)i− [1] as: (1)ii−− () (1) i eekbipp[1]=+ [1]i [0], =− , 1, ...,1 4. we set be(0)[1]= (0) [1] and we can then solve for b (i ) [1] for ip=1,2,..., using the equation: ()iii (−− 1) ( 1) bkebi[1]=−i [1] + [0], = 1,2,..., p 5. we iterate for nN=−2,3,..., 1 and end up with sn[]== e(0) [] n b (0) [] n 13 Lattice Formulations of LP • the lattice structure comes directly out of the Durbin algorithm • the ki parameters are obtained from the Durbin equations • the predictor coefficients, αk , do not appear explicitly in the lattice structure • can relate the ki parameters to the forward and backward errors via Li−+1 ∑ embm()ii−−11() () (−1 ) k =∗m=0 4 i 12/ ⎧⎫⎡⎤⎡⎤Li−+11 Li −+ ⎪⎪()ii−−12 () 1 2 ⎨⎬⎢⎥⎢⎥∑∑[em ( )] [ bm (−1 )] ⎩⎭⎪⎪⎣⎦⎣⎦⎢⎥⎢⎥mm==00 • where ki is a normalized cross correlation between the forward and backward prediction error, and is therefore called a partial correlation or PARCOR coefficient • can compute predictor coefficients recursively from the PARCOR coefficients 14 Direct Computation of k Parameters assume sn[ ] non-zero for 01≤ n≤− L assume ki chosen to minimize total energy of the forward (or backward) prediction errors we can then minimize forward prediction error as : 2 Li−+112 Li −+ Eem()ii⎡ ()( )()()⎤⎡emkbm()ii−−11 () ⎤ forward = ∑∑⎣ ⎦⎣=−−i 1 ⎦ mm==00 ∂E ()i Li−+1 forward ⎡⎤emkbm()ii−−11() () ( ) bm () i − 1 ( ) ==−02∑ ⎣⎦ −i − 1 − 1 ∂ki m=0 Li−+1 ()ii−−11 () ∑ ⎣⎦⎡⎤embm()⋅− (1 ) forward m=0 ki = Li−+1 2 ()i −1 ∑ ⎣⎦⎡⎤bm()−1 m=0 15 Direct Computation of k Parameters • we can also choose to minimize the backward prediction error 2 Li−+112 Li −+ E()ii⎡⎤ bm ()( ) ⎡ kembm ( ii−−11 ) ( ) ( ) (1 ) ⎤ backward==−+−∑∑⎣⎦ ⎣ i ⎦ mm==00 ∂E ()i Li−+1 backward ⎡⎤ke()ii−−11() m b () ( m )em()i −1 () ==−02∑ ⎣⎦ −i + − 1 ∂ki m=0 Li−+1 ()ii−−11 () ∑ ⎣⎦⎡⎤embm() (−1 ) backward m=0 ki = Li−+1 ()i −1 2 ∑ ⎣⎦⎡⎤em() m=0 16 Direct Computation of k Parameters • if we window and sum over all time, then Li−+1122 Li −+ ()ii−−11 () ∑∑⎣⎦⎣⎡⎤⎡em( )=− bm (1 ) ⎦⎤ mm==00 therefore PARCOR forward backward forward backward kkkkkiiiii=== Li−+1 ∑ embm()ii−−11() () (−1 ) m=0 = / Li−+11 Li −+ 12 ⎧⎫(i −1)()22i −1 ⎨⎬∑∑⎣⎦⎣⎦⎡⎤⎡⎤e ()mbm (−1 ) mm==00 ⎩⎭17 Direct Computation of k Parameters • minimize sum of forward and backward prediction errors over fixed interval (covariance method) L−1 22 Eembm()iii=+⎡⎤⎡⎤ ()( ) () ( ) Burg ∑{}⎣⎦⎣⎦ m=0 L−1 2 ∞ 2 ⎡⎤emkbm()ii−−11() () (1 )⎡⎤ kemb () ii −− 11 () ()()m 1 =−−+−+∑ ⎣⎦ii∑ ⎣⎦− m=0 m=−∞ ∂E ()i L−1 Burg 02⎡⎤emkbm()ii−−11() () () 1 bm () i − 1 () 1 ==−∑ ⎣⎦ −i − − ∂ki m=0 L−1 21⎡⎤ke()ii−−11() m b () ( m ) e () i − 1 () m −−∑ ⎣⎦i + − m=0 L−1 ()ii−−11 () 21∑ ⎣⎦⎡⎤embm()⋅− ( ) Burg m=0 ki = L−−1122L ()ii−−11 () ∑∑⎣⎦⎣⎡⎤⎡em()+− bm (1 ) ⎦⎤ m=0 m=0 Burg 18 •−≤11ki ≤ always Comparison of Autocorrelation and Burg Spectra • significantly less smearing of formant peaks using Burg method 19 Summary of Lattice Procedure • steps involved in determining the predictor coefficients and the k parameters for the lattice method are as follows 1. initial condition, emsmbm()00( )== ( ) () ( ) from Eq. *3 ()1 2. compute k1 = α1 from Eq. *4 3. determine forward and backward prediciton errors embm()11( ), () ( ) from Eqs. *1 and *2 4. set i = 2 ()i 5. determine ki = αi from Eq. *4 ()i 6. determine α j for j =−12, ,...,i 1 from Durbin iteration 7. determine em()ii( ) and bm ()( ) from Eqs. *1 and *2 8. set ii=+1 9. if ip is less than or equal to , go to step 5 10. procedure is terminated • predictor coefficients obtained directly from speech samples => without calculation of autocorrelation function • method is guaranteed to yield stable filters (ki <1) without using window 20 Summary of Lattice Procedure 21 Summary of Lattice Procedure (0) (1) (2) s[n] e [n] e [n] e [n] e( p)[n] e[n] − k1 − k2 − k p −1 − k1 −1 − k − k z z 2 z−1 p ( p) b(0)[n] b(1)[n] b(2)[n] b [n] embmsm()00( )= () ( )=∗ ( ) 3 Li−+1 ∑ embm()ii−−11() () (−1 ) k =∗m=0 4 i 12/ ⎧⎫⎡⎤⎡⎤Li−+11 Li −+ ⎪⎪()ii−−12 () 1 2 ⎨⎬⎢⎥⎢⎥∑∑[()][()]em bm−1 ⎩⎭⎪⎪⎣⎦⎣⎦⎢⎥⎢⎥mm==00 ()ii ()−−11 () i em( )= e ( mkbm )−−∗i (1 ) 1 bm()ii( )= b (−−11 ) ( m−−1 ) kem ( i ) ( ) ∗ 2 i 22 Prediction Error Signal 1.