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Lpc - LPC.PPT 5.1 LPC.PPT 5.2 Deller 266 Lecture 5 Linear...

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Page 5.1 Linear Prediction E.4.14 – Speech Processing LPC.PPT(4/15/2002) 5.1 Lecture 5 Linear Prediction (LPC) Aims of Linear Prediction Derivation of Linear Prediction Equations Autocorrelation method of LPC Interpretation of LPC filter as a spectral whitener LPC.PPT(4/15/2002) 5.2 Deller: 266++ Notes: We will neglect the pure delay term z –½ p in the numerator of V(z) . 50% of the world puts a + sign in the denominator of V(z) (this is almost essential when using MATLAB). V(z) R(z) u (n) u l (n) s(n) 1 ½ 1 ½ 1 ) ( ) ( 1 ) ( ) ( ) ( ) ( = = = = = = = z z R z A Gz z a Gz z V n s n u n u p p j j j p l microphone the at pressure lips the at flow volume two the of mixture or noise waveform, glottal a time-varying all-pole filter Linear Prediction: Analysis & Coding (LPC) The aim of Linear Predication Analysis (LPC) is to estimate V(z) from the speech signal s(n) .
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Page 5.2 Linear Prediction E.4.14 – Speech Processing LPC.PPT(4/15/2002) 5.3 Deller: 266++ If the vocal tract resonances have high gain, the second term will dominate: V(z) R(z) u(n) u l (n) s(n) = + = p j j j n s a n u G n s 1 ) ( ) ( ) ( s n a s n j j j p ( ) ( ) = 1 Prediction Error We can reverse the order of V ( z ) and R ( z ) since both are linear and V ( z ) doesn’t change substantially during the impulse response of R ( z ) or vice-versa: The right hand side of this expression is a prediction of s(n) as a linear sum of past speech samples. Define the prediction error at sample n as e n s n a s n j s n a s n a s n a s n p E z S z A z j j p p ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = 1 1 2 1 2 or in terms of z transforms: R(z) V(z)/G= 1 /A(z) × G u(n) u'(n) s(n) LPC.PPT(4/15/2002) 5.4 Given a frame of speech { F } , we would like to find the values a i that minimize: e n s n i i p s n s n i a s n j s n i i p a s n j s n i s n s n i a where s n i s n j n F j j p n F j n F j p n F ij j i ij n F ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) ( ) { } { } { } { } { } = = = = = = = = = 0 1 0 1 1 1 0 for for φ φ φ j p = 1 = } { 2 ) ( F n E n e Q ( ) = = = } { } { } { 2 ) ( ) ( 2 ) ( ) ( 2 ) ( F n F n i F n i i E i n s n e a n e n e a n e a Q The optimum values of a i must satisfy p equations: To do so, we differentiate w.r.t each a i : or in matrix form: Φ Φ Φ a c a c = = 1 1 providing exists the matrix Φ is symmetric and positive semi-definite.
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Page 5.3 Linear Prediction E.4.14 – Speech Processing LPC.PPT(4/15/2002) 5.5 Matrices with Special Properties – Symmetric: Positive Definite: Positive Semi-Definite: as above but with . Toeplitz: Constant diagonals: Inverting Matrices Any special properties possessed by a matrix can be used when inverting it to: reduce the computation time improve the accuracy φ φ ji ij T = = Φ Φ x x i ij j i j T φ , > > 0 0 0 x x x Φ for any ( ) φ φ i j ij f i j + + = = 1 1 , Matrix ( p×p ) Computation General p 3 Symmetric, +ve definite ½ p 3 Toeplitz, Symmetric, +ve definite p 2 LPC.PPT(4/15/2002) 5.6 Autocorrelation LPC We start with a frame of windowed speech (typ 20-30 ms): We take { F
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