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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 1 1 1 1. Speech Production Model ( ) ( ) ( ) ( ) ( ) ( ) 2. LPC Model: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3. LPC Error Model: ( ) ( ) ( ) α α α = = = = = + = = = = = = = = % p k k p k k k p k k p k k k k s n a s n k Gu n S z G H z U z a z e n s n s n s n s n k E z A z z S z S z A z E z 1 1 ( ) ( ) ( ) α = = = + p k k p k k z s n e n s n k 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 1 2 ( ) coefficients { , , ,..., } are coefficients of the order optimum LP α = i th j j i 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 ( ) ( ) ( α = = = + th i i i k k k n i A z z s m s n m w m 1 1 ( ) ( ) ˆ ( ) ( ) ( ) ( ) ˆ ), with output ( ) ( ) ( ) ( ) ( ) ˆ where we have dropped subscript - the absolute location of the signal the z-transform gives ( ) ( ) ( ) α α = = + = = = i i n i i i k k i i k e m e n m e m s m s m k n E z A z S z 1 ( ) ( ) = i i k k z S z Lattice Formulations of LP 5 Lattice Formulations of LP 1 1 1 1 ( ) ( - ) ( - ) ( ) - ( ) ( ) ( ) ( using the steps of the Durbin recursion ( = , and ) we can obtain a recurrence formula for ( ) of the form ( ) ( ) α α α α = = ± i i i i i i j j i j i i i i i i i k k A z A z A z k z A 1 1 1 1 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) giving for the error transform the expression ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = i i i i i i i i i z E z A z S z k z A z S z e m e m k 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 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ckward prediction error ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) with inverse transform ( ) ( ) ( ) ( α = = = = = + = i i i i i i i i i i i i i i k k B z z A z S z z A z S z k A z S z z B z k E z b m s m i s m k i 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 k e m s m i i s m i b m backward backward prediction error sequence
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