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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

SLIDE 1

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

SLIDE 2

Second order predictor for an AR(2) process

u(n) = −a1u(n − 1) − a2u(n − 2) + v(n) We assume that σ2

u = 1 = r(0)

From the first Yule-Walker equation, r(0)(−a1) + r(1)(−a2) = r(1), we find r(1) = −

a1 1+a2 = ρ.

Thus the correlation matrix is R = r(0) r(1) r(1) r(0)

1 ρ ρ 1

The optimal predictor is given by

w0 = −a1 −a2

The variance of the white noise is given by

Jmin = σ2

v = (1 − a2)((1 + a2)2 − a2 1)

1 + a2

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

SLIDE 3

Eigenvalues and eigenvectors

We find the eigenvalues by solving det(R − λI) = 1 − 2λ + λ2 − ρ2 = 0. λ1 = 1 + ρ = λmax and λ2 = 1 − ρ = λmin The orthonormal eigenvectors are found to be q1 = 1 √ 2 1 1

q2 =

1 √ 2 1 −1

Thus

Q = 1 √ 2 1 1 1 −1

QTRQ =

1 + ρ 1 − ρ

Eigenvalue spread is χ = λmax

λmin = 1+ρ 1−ρ

Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 3 / 12

SLIDE 4

Parameters for this Example

For this example, we use one of the cases studied for steepest descent. a1 = −1.5955 and a2 = 0.95 ρ = 0.818, λmax = 1.818, λmin = 0.182. The eigenvalue spread is χ = 10 MMSE Jmin = 0.0322.

Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 4 / 12

SLIDE 5

Sample realization of AR(2) Process

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

Learning curve for AR(2) LMS predictor, µ = 0.1

Jex = 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

SLIDE 9

Learning curve for AR(2) LMS predictor, µ = 0.05

Jex = 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

SLIDE 10

Learning curve for AR(2) NLMS predictor, ˜ µ = 0.1

Jex = 0.0045 misadjustment = 0.1405

Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 10 / 12

SLIDE 11

Learning curve for AR(2) NLMS predictor, ˜ µ = 0.3

Jex = 0.0123 misadjustment = 0.3803

Eric Dubois (EECS) ELG5377 Illustration of Performance of LMS for AR(2) Process November 2012 11 / 12

SLIDE 12

Learning curve for AR(2) NLMS predictor, ˜ µ = 1.0

Jex = 0.0562 misadjustment = 1.7446

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