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ELG5377 Adaptive Signal Processing

ELG5377 Adaptive Signal Processing. Lecture 6: LMS Algorithm Continued. Coefficient Error Vector Covariance Matrix. c ( k ) = w ( n )- w o . cov[ c ( k )] = E[ c ( k ) c H ( k )] = K ( k ). Recall that c ( k +1) = [ I - m x ( k ) x H ( k )] c ( k ) + m x ( k ) e o * ( k ).

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ELG5377 Adaptive Signal Processing

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  1. ELG5377 Adaptive Signal Processing Lecture 6: LMS Algorithm Continued

  2. Coefficient Error Vector Covariance Matrix • c(k) = w(n)-wo. • cov[c(k)] = E[c(k)cH(k)] = K(k). • Recall that • c(k+1) = [I - mx(k)xH(k)]c(k) +mx(k)eo*(k). • K(k+1) = E{[I - mx(k)xH(k)]c(k)cH(k) [I - mx(k)xH(k)]H} + E{[I - mx(k)xH(k)]mxH(k)eo(k)} + E{mx(k)eo*(k) {[I - mx(k)xH(k)]} + m2E[|eo(k)|2x(k)xH(k)]. • K(k+1)= [I - mR]K(k)[I - mR]H + m2JminR. • K(k+1)= [I - mR]K(k)[I - mR] + m2JminR.

  3. Coefficient Error Vector Covariance Matrix 2 • At steady state (or for large k), K(k+1)≈K(k). • Therefore • K(k)= [I - mR]K(k)[I - mR] + m2JminR. • 0 = -mK(k)R-mRK(k)+m2RK(k)R+m2JminR. • K(k)R+RK(k) = mJminR.

  4. Mean Square Error • e(k) = d(k)-y(k) = d(k)-wH(k)x(k). • e(k) = d(k)-y(k) = d(k)-(w(k)-wo)Hx(k)-woHx(k). • e(k) = eo(k)-cH(k)x(k). • E[|e(k)|2]=E[|eo(k)|2] + E[cH(k)x(k)xH(k)c(k)]. • E[|eo(k)|2]= Jmin. • E[cH(k)x(k)xH(k)c(k)] = E[tr{cH(k)x(k)xH(k)c(k)}] = E[tr{c(k)cH(k)x(k)xH(k)}] = tr{E[c(k)cH(k)x(k)xH(k)]} ≈ tr{K(k)R]. • tr{K(k)R} = tr{RK(k)}. • K(k)R+RK(k) = mJminR. • tr{K(k)R+RK(k)}=mJmintr{R}. • Therefore tr{K(k)R} = mJmintr{R}/2

  5. Mean Square Error 2 • Therefore the MSE at the output of the LMS filter is • J = Jmin + mJmintr{R}/2. • J = Jmin[1+(m/2)Sli] • Suppose R has a dominant eigenvalue (lmax >> li) • J ≈ Jmin(1+ (mlmax/2)).

  6. Excess Mean Square Error • Jex = J – Jmin. • Jex = mJmintr{R}/2 = Jmin(m/2)Sli. • If R has a dominant eigenvalue, then • Jex ≈Jmin(mlmax/2).

  7. Misadjustment • M = Jex/Jmin. • For LMS Filters, • M = (m/2)tr{R} = (m/2)Mr(0) = (m/2)Sli. • M ≈ (mlmax/2) • In our example in the previous lecture, Jmin = 0.0985. • For the LMS filter with m = 0.1, the misadjustment should be • 0.05* 3.57 = 0.1785 • Simulated misadjustment = (0.1255-0.0985)/0.0985 = 0.274. • For LMS filter with m = 0.3, • Theoretical = 0.536 • Simulated = 2.57

  8. Conclusion • Performance of LMS algorithm as a function of m. • Increasing m improves convergence time at a cost of increasing the misadjustment. • Misadjustment and convergence time are inversely proportional.

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