Lecture 14_winter_2012_6tp

Lecture 14_winter_2012_6tp - Prediction Error Signal 1....

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1 1 Digital Speech Processing— Lecture 14 Linear Predictive Coding (LPC)-Lattice Methods, Applications 2 Prediction Error Signal H(z) Gu(n) s(n) 1 1 1 1 1 1 11 1 1. Speech Production Model () ( ) 2. LPC Model: () () () () ( ) 3. LPC Error Model: α = = = = =− + == = −=− % p k k p k k k p k k p k k k k sn asn k Gun Sz G Hz Uz az en sn sn sn k Ez Az z 1 1 () () ( ) = = =+ p k k p k k z A(z) s(n) e(n) 1/A(z) e(n) s(n) Perfect reconstruction even if a k not equal to α k 3 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 stage the set of th i 12 coefficients { , , ,. .., } are coefficients of the order optimum LP = it h j ji i 4 1 1 ˆ define the system function of the order inverse filter (prediction error filter) as ( ) if the input to this filter is the input segment ˆ () ( )( = th i ii k k k n i z sm snmwm 1 1 ˆ ˆ ), with output ( ) ( ) ( ) ˆ where we have dropped subscript - the absolute location of the signal the z-transform gives ( ) ( ) ( ) = n i k k k em enm em s m s mk n Ez Az S z 1 = ⎛⎞ ⎜⎟ ⎝⎠ i ik k zS z Lattice Formulations of LP 5 Lattice Formulations of LP (-) - ( ) ( using the steps of the Durbin recursion ( = , and ) we can obtain a recurrence formula for ( ) of the form ( ) ( ) αα −− −= ± i i jj i j i i i i i kk Az A zk zA 1 1 1 ) ( ) ( ) giving for the error transform the expression ( ) ( ) ( ) ( ) ( ) ( ) i i i i i z E zAz S z k z S z em e mk b m 6 Lattice Formulations of LP 1 where we can interpret the first term as the z-transform of the forward prediction error for an ( ) order predictor, and the second term can be similarly interpreted based on defining a ba st i 1 1 1 1 1 ( ) ( ) ( ) ckward prediction error ( )() ()() with inverse transform ( ) ( ) ( ) ( = + = i i i i i i i i k k Bz zAzS z zA zS zk A z S z zB z kE z bm s mi s mki b m 1 1 ) with the interpretation that we are attempting to predict ( ) from the samples of the input that follow ( ) => we are doing a prediction and ( ) is called the •− i i i ke m sm i is m i bm backward backward prediction error sequence
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2 7 Lattice Formulations of LP same set of samples is used to forward predict s(m) and backward predict s(m-i) 8 Lattice Formulations of LP 11 1 () ( ) ( ) ( the prediction error transform and sequence ( ), ( ) can now be expressed in terms of forward and backward errors, namely ( ) ( ) −− =− ii i i i i Ez em EzE zk zB z em e mk b 1 1 1 ) ( ) ( ) ()1 similarly we can derive an expression for the backward error transform and sequence at sample of the form ( ) ( ) ( ) 2 these −∗ iii i i i m m Bz zB zk E z bm b m k e m 1 two equations define the forward and backward prediction error for an order predictor in terms of the corresponding prediction errors of an (
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This note was uploaded on 12/29/2011 for the course ECE 259 taught by Professor Rabiner,l during the Fall '08 term at UCSB.

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Lecture 14_winter_2012_6tp - Prediction Error Signal 1....

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