ELG5377 Illustration of Performance of LMS for AR(2) Process Eric - PowerPoint PPT Presentation

ELG5377 Illustration of Performance of LMS for AR(2) Process Eric Dubois School of Electrical Engineering and Computer Science University of Ottawa November 2012 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 1 / 12

Second order predictor for an AR(2) process u ( n ) = − a 1 u ( n − 1) − a 2 u ( n − 2) + v ( n ) We assume that σ 2 u = 1 = r (0) From the first Yule-Walker equation, r (0)( − a 1 ) + r (1)( − a 2 ) = r (1), a 1 we find r (1) = − 1+ a 2 = ρ . Thus the correlation matrix is � r (0) � � 1 � r (1) ρ R = = r (1) r (0) 1 ρ The optimal predictor is given by � − a 1 � w 0 = − a 2 The variance of the white noise is given by v = (1 − a 2 )((1 + a 2 ) 2 − a 2 1 ) J min = σ 2 1 + a 2 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 2 / 12

Eigenvalues and eigenvectors We find the eigenvalues by solving det( R − λ I ) = 1 − 2 λ + λ 2 − ρ 2 = 0. λ 1 = 1 + ρ = λ max and λ 2 = 1 − ρ = λ min The orthonormal eigenvectors are found to be 1 � 1 � q 1 = √ 1 2 � 1 1 � q 2 = √ − 1 2 Thus � 1 � � 1 + ρ � 1 1 0 Q T RQ = Q = √ 1 − 1 0 1 − ρ 2 λ min = 1+ ρ Eigenvalue spread is χ = λ max 1 − ρ Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 3 / 12

Parameters for this Example For this example, we use one of the cases studied for steepest descent. a 1 = − 1 . 5955 and a 2 = 0 . 95 ρ = 0 . 818, λ max = 1 . 818, λ min = 0 . 182. The eigenvalue spread is χ = 10 MMSE J min = 0 . 0322. Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 4 / 12

Sample realization of AR(2) Process Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 5 / 12

Prediction error with Wiener filter and LMS filter µ = 0 . 1 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 6 / 12

Tracking of AR(2) parameter estimates µ = 0 . 1 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 7 / 12

Learning curve for AR(2) LMS predictor, µ = 0 . 1 J ex = 0 . 0054 misadjustment = 0 . 1669 misadjustment estimate = 0 . 1226 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 8 / 12

Learning curve for AR(2) LMS predictor, µ = 0 . 05 J ex = 0 . 0019 misadjustment = 0 . 0604 misadjustment estimate = 0 . 0551 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 9 / 12

Learning curve for AR(2) NLMS predictor, ˜ µ = 0 . 1 J ex = 0 . 0045 misadjustment = 0 . 1405 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 10 / 12

Learning curve for AR(2) NLMS predictor, ˜ µ = 0 . 3 J ex = 0 . 0123 misadjustment = 0 . 3803 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 11 / 12

Learning curve for AR(2) NLMS predictor, ˜ µ = 1 . 0 J ex = 0 . 0562 misadjustment = 1 . 7446 Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 12 / 12