Linear predictive coding

Transcription

1 Liner predictive coding Thi ethod cobine liner proceing with clr quntiztion. The in ide of the ethod i to predict the vlue of the current ple by liner cobintion of previou lredy recontructed ple nd then to quntize the difference between the ctul vlue nd the predicted vlue. Liner prediction coefficient re weighting coefficient ued in liner cobintion. A iple predictive quntizer or differentil pule-coded odultor i hown in Fig. 5.. If the predictor i iply the lt ple nd the quntizer h only one bit, the yte becoe delt-odultor. It i hown in Fig. 5..

2 Differentil pule-coded odultor x e QUANTIZ q xˆ d x DQUANTIZ LINA PDICTO

3 Delt-odultor x x xˆ COMPAATO x xˆ Stirce function forer

4 Liner predictive coding The in feture of the quntizer hown in Fig 5., 5. i tht they exploit not ll dvntge of predictive coding. Prediction coefficient ued in thee chee re not optil Prediction i bed on pt recontructed ple nd not true ple Uully coefficient of prediction re choen by uing oe epiricl rule nd re not trnitted For exple quntizer in Fig.5. inted of ctul vlue of error e ue recontructed vlue d nd inted of true ple vlue x their etite x obtined vi d.

5 Liner predictive coding The ot dvnced quntizer of liner predictive type repreent bi of the o-clled Code xcited Liner Predictive CLP coder. It ue the optil et of coefficient or in other word liner prediction coefficient of thi quntizer re deterined by iniizing the MS between the current ple nd it predicted vlue. It i bed on the originl pt ple Uing the true ple for prediction require the looking-hed procedure in the coder. The predictor coefficient re trnitted

6 Liner predictive coding Aue tht quntizer coefficient re optiized for ech ple nd tht the originl pt ple re ued for prediction. Let x T, xt,... be equence of ple t the quntizer input. Then ech ple x i predicted by the previou ple ccording to the forul ˆ xˆ k k x kt where x i the predicted vlue, k re prediction coefficient, denote the order of prediction. The prediction error i e x xˆ., 5.

7 Liner predictive coding Prediction coefficient re deterined by iniizing the u of qured error over given intervl n e 5. nn Inerting 5. into 5. we obtin n x x T... x T nn n n x nn j nn j x x jt j k j k n nn x jt x kt. 5.3

8 Liner predictive coding Differentiting 5.3 over k, k,,..., yield / k n nn x x kt Thu we obtin yte of unknown quntitie,...,, j j n nn x kt x liner eqution with jt j j c jk c ok, k,,..., 5.4 where c jk c kj n nn x jt x kt. 5.5 The yte 5.4 i clled the Yule-Wlker eqution.

9 If,..., Liner predictive coding, re olution of 5.4 then we cn find the inil chievble prediction error. Inert 5.5 into 5.3. We obtin tht c c c. 5.6 Uing 5.3 we reduce 5.6 to k k k k k j j jk c k k c k

10 Interprettion of the Yule-Wlker eqution like digitl filter q. 5. decribe the th order predictor with trnfer function equl to ˆ z X z X z P k k k z. z-trnfor for the prediction error i k k k z z X z X z. The prediction error i n output ignl of the dicrete-tie filter with trnfer function k k k z z X z z A. The proble of finding the optil et of prediction coefficient = proble of contructing th order FI filter.

11 Interprettion of the Yule-Wlker eqution like digitl filter Another ne of the liner prediction 5. i the utoregreive odel of ignl x. It i ued tht the ignl x cn be obtined the output of the utoregreive filter with trnfer function H z, z k k k tht i cn be obtined the output of the filter which i invere with repect to the prediction filter. Thi filter i dicrete-tie II filter.

12 Method of finding coefficient c, i,,...,, j,,..., ij In order to olve the Yule-Wlker eq. 5.4 it i necery firt to evlute vlue c, i,,...,, j,,..., There re two pproche to etiting thee vlue: ij The utocorreltion ethod nd The coplexity of olving 5.4 i proportionl to the covrince ethod. The coplexity of olving 5.4 i proportionl to 3

13 c The vlue ij c We et ij c c ij ji Autocorreltion ethod re coputed i ni x it x jt. 5.7 i,i nd x if n, n N, where N i clled the intervl of nlyi. In thi ce we cn iplify 5.7 N n x it x jt N i j n x x i Norlized by N they coincide with etite of entrie of covrince trix for x ˆ i j c ij / N / N N i j n x x i jt jt.. 5.8

14 Autocorreltion ethod

15 Autocorreltion ethod The Yule-Wlker eqution for utocorreltion ethod hve the for ˆ i j ˆ j, j,,...,. 5.9 i i q.5.9 cn be given by trix eqution where,,...,, b, b ˆ, ˆ,..., ˆ, ˆ ˆ... ˆ ˆ ˆ... ˆ.... ˆ ˆ... ˆ

16 Autocorreltion ethod It i id tht 5.9 relte the preter of the utoregreive odel of th order with the utocorreltion equence. Mtrix propertie. of the utocorreltion ethod h two iportnt It i yetric, tht i ˆ i, j ˆ j, i It h Toeplitz property, tht i ˆ i, j ˆ i j. The Toeplitz property of ke it poible to reduce the coputtionl coplexity of olving 5.4. The ft Levinon-Durbin recurive lgorith require only opertion.

17 We chooe Covrince ethod i i N nd nd ignl i not contrined in tie. In thi ce we hve Set c ij c k ij N i k i N n n i x kt x it x jt x. 5. then 5. cn be rewritten x kt i j T, i,...,, j, reeble 5.8 but it h other rnge of definition for k. It ue ignl vlue out of rnge k N, The ethod led to the cro-correltion function between two iilr but not exctly the e finite egent of x kt.

18 ˆ i, j c Covrince ethod ij / N / N N n x n i T x n j T The Yule-Wlker eqution for the covrition ethod re iˆ i, j ˆ, j, j,,...,. 5. i q. 5. cn be given by the trix eqution P where,,...,, c. c, ˆ, ˆ,... ˆ, ˆ, ˆ,... ˆ, P.... ˆ, ˆ,... ˆ,. ˆ,, ˆ,,..., ˆ,,

19 Covrince ethod Unlike the trix of utocorreltion ethod the trix P i yetric ˆ i, j ˆ j, i but it i not Toeplitz. Since coputtionl coplexity of olving n rbitrry yte of liner eqution of order i equl to 3 then in thi ce to olve 5. it i necery opertion. 3

20 Algorith for the olution of the Yule-Wlker eqution The coputtionl coplexity of olving the Yule- Wlker eqution depend on the ethod of evluting vlue c ij. Aue tht cij re found by the utocorreltion ethod. In thi ce the Yule-Wlker eqution h the for 5.9 nd the trix i yetric nd the Toeplitz trix. Thee propertie ke it poible to find the olution of 5.9 by ft ethod requiring opertion. There re few ethod of thi type: the Levinon- Durbin lgorith, the ucliden lgorith nd the Berlekp-Mey lgorith.

21 The Levinon-Durbin lgorith It w uggeted by Levinon in 948 nd then w iproved by Durbin in 96. Notice tht thi lgorith work efficiently if trix of coefficient i iultneouly yetric nd Toeplitz. The Berlekp- Mey nd the ucliden lgorith do not require the trix of coefficient to be yetric. We equentilly olve eqution 5.9 of order l,...,. l l l l Let,,..., l denote the olution for the l yte of the l th order. Given we find the olution for the l th order. At ech tep of the lgorith we evlute the prediction error of the l th order yte. l

22 The Levinon-Durbin lgorith Initiliztion: l, ˆ,. ecurrent procedure: For l,..., l l copute ˆ l l i l i ˆ l i/ l, l l l l, j l, j j l l j l l l l. When l we obtin the olution,,...,,.

23 , /, xple k /, /, /.,

24 xple. /. /