# Lecture13 - Lecture 13 EE214A Abeer Alwan Speech...

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1 Lecture 13 EE214A Abeer Alwan Speech Application In speech, we assume Transfer function is Goal: Estimate a k ’s Assume prediction signal is Prediction error is ) ( ) ( ) ( 1 n u G k n s a n s p k k ) ( 1 ) ( ) ( 1 z A G z a G z U z S p k k k p k k k n s a n s 1 ) ( ) ( ~ ) ( ~ ) ( ) ( n s n s n e

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2 Two Key Equations n k n k k n s n s a n s E ) ( ) ( ) ( 2 min n k n n k i n s k n s a i n s n s ) ( ) ( ) ( ) ( Solve the Normal Equations (obtained by minimizing E ): After obtaining the coefficients, compute the minimum error: 1. The A.C. Method (assumes signal is zero outside the observation interval) 2 min 1 (0) ( ) p k k E R a R k G k N n N k n k n s n s k n s n s k R 1 0 1 ) ( ) ( ) ( ) ( ) ( p k k k i R a i R 1 |) (| ) ( where and p k i , 1 The normal equations can be rewritten as: and The method requires N samples (0 to N-1)
3 Matrix notation ) ( ) 2 ( ) 1 ( ) 0 ( ) 1 ( ) 1 ( ) 1 ( ) 0 ( ) 1 ( ) 1 ( ) 1 ( ) 0 ( 2 1 p R R R a a a R p R R R R R p R R R p 1 r R a r Ra diagonal) and (symmetric Toeplitz is R 2. Covariance method 1 0 ( , ) ( ) ( ) ( , ) N n i k s n i s n k k i 1 ( ,0) (0, ) ( , ) p k k i i a i k min 1 (0,0) (0, ) p k k E a k Then, the normal equations could be rewritten as: Also, The method requires samples from -p to N-1 Define

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4 Matrix notation 1 2 (1,1) (1,2) (1, ) (1,0) (2,1) (2,2) (2,0) ( ,1) ( , ) ( ,0) p a p a a p p p p 1 a a   is symmetric but not Toeplitz (not diagonal) Implementation Methods Autocorrelation (A.C.) Levinson-Durbin Recursion Covariance method Cholesky’s decomposition p k k k i R a i R 1 |) (| ) ( p k k k i a i i 1 ) , ( ) , 0 ( ) 0 , (
5 1.A.C.

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• Winter '11
• ALWAN
• Autocorrelation, Cross-correlation, Toeplitz matrix, Levinson recursion

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