1. 10 1 Widrow-Hoff Learning (LMS Algorithm)

2. 10 2 ADALINE Network  w i w i1  w i2  w iR  =

3. 10 3 Two-Input ADALINE

4. 10 4 Mean Square Error Training Set: Input:Target: Notation: Mean Square Error:

5. 10 5 Error Analysis The mean square error for the ADALINE Network is a quadratic function:

6. 10 6 Stationary Point Hessian Matrix: The correlation matrix R must be at least positive semidefinite. If there are any zero eigenvalues, the performance index will either have a weak minumum or else no stationary point, otherwise there will be a unique global minimum x*. If R is positive definite:

7. 10 7 Approximate Steepest Descent Approximate mean square error (one sample): Approximate (stochastic) gradient:

8. 10 8 Approximate Gradient Calculation

9. 10 9 LMS Algorithm

10. 10 Multiple-Neuron Case Matrix Form:

11. 10 11 Analysis of Convergence For stability, the eigenvalues of this matrix must fall inside the unit circle.

12. 10 12 Conditions for Stability Therefore the stability condition simplifies to 12  i –1–  Since,. (where i is an eigenvalue of R)

13. 10 13 Steady State Response If the system is stable, then a steady state condition will be reached. The solution to this equation is This is also the strong minimum of the performance index.

14. 10 14 Example BananaApple

15. 10 15 Iteration One Banana

16. 10 16 Iteration Two Apple

17. 10 17 Iteration Three

18. 10 18 Adaptive Filtering Tapped Delay LineAdaptive Filter

19. 10 19 Example: Noise Cancellation

20. 10 20 Noise Cancellation Adaptive Filter

21. 10 21 Correlation Matrix

22. 10 22 Signals 1.2  2 0.5 2  3 ------   cos0.36–==mk  1.2 2  k 3 --------- 3  4 ------–   sin=

23. 10 23 Stationary Point 0 0 h Esk  mk  +  vk  Esk  mk  +  vk1–  =

24. 10 24 Performance Index

25. 10 25 LMS Response

26. 10 26 Echo Cancellation