1. Hossein Sameti Department of Computer Engineering Sharif University of Technology

2.  Definition of generalized linear-phase (GLP):  Let’s focus on Type I FIR filter: 2 It can be shown that (L+1) unknown parameters a(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

3. 3 Given determine coefficients of G(ω) (i.e. a(n)) such that L is minimized (minimum length of the filter). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

4.  G(ω) is a continuous function of ω and is as many times differentiable as we want.  How many local extrema (min/max) does G(ω) have in the range ?  In order to answer the above question, we have to write cos(ωn) as a sum of powers of cos(ω). 4 : sum of powers of cos(ω) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

5. 5 Find extrema Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6. 6 Polynomial of degree L-1 Maximum of L-1 real zeros Max. total number of real zeros: L+1 Conclusion: The maximum number of real zeros for (derivative of the frequency response of type I FIR filter) is L+1, where (N is the number of taps). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

7. 7 Given determine coefficients of G(ω) (i.e. a(n)) such that L is minimized (minimum length of the filter). Problem AProblem BProblem C Problem A Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

8. 8 Given determine coefficients of G(ω) (i.e. a(n)) such that is minimized. ComputeGuess L Algorithm B Increase L by 1 Decrease L by 1 Yes Stop!

9. 9 Define F as a union of closed intervals in Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10. 10 where W is a positive weighting function Desired frequency response Find a(n) to minimize (same assumption as Problem B)

11.  We start by showing that 11 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

12. 12 By definition: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

13. 13 By definition: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

14. 14 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

15. 15 in Problem C in Problem B

16.  Conclusion: 16 Find a(n) such that is minimized. Problem B: Find a(n) such that is minimized. Problem C: Problem B= Problem C Problem A= Problem C  We now try to solve Problem C. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17. Assumptions:  F: union of closed intervals  G(x) to be a polynomial of order L:  D = Desired function that is continuous in F.  W= positive function 17

18.  The necessary and sufficient conditions for G(x) to be unique Lth order polynomial that minimizes is that E(x) exhibits at least L+2 alternations, i.e., there are at least L+2 values of x such that 18 for a polynomial of degree 4 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

19. 19 Recall G(ω) can have at most L+1 local extrema. According to the alternation theorem, G(ω) should have at least L+2 alternations(local extrema) in F. Contradiction!?

20. 20 can also be alternation frequencies, although they are not local extrema. G(ω) can have at most L+3 local extrema in F. Ex: Polynomial of degree 7 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21. 21  According to the alternation theorem, we have at least L+2 alternations.  According to our current argument, we have at most L+3 local extrema.  Conclusion: we have either L+2 or L+3 alternations in F for the optimal case. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

22. 22 Extra-ripple case Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

23. 23 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

24.  For Type I low-pass filters, alternations always occur at  If not, we potentially lose two alternations. 24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

25. 25 Equi-ripple except possibly at Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

26. 26  For optimal type I low-pass filters, alternations always occur at If not, two alternations are lost and the filter is no longer optimal.  Filter will be equi-ripple except possibly at Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

27. 27 Given determine coefficients of G(ω) (i.e. a(n)) such that is minimized. At alternation frequencies, we have: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

28. 28 Equating Eq.1 and Eq.2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29. 29

30. 30 L+2 linear equations and L+2 unknowns Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

31. 31 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

32. 32 It can be shown that if 's are known, then can be derived using the following formulae: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. 33 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. 34 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35. 35

36. 36 Original alternation frequency Next alternation frequency Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

37.  App. estimate of L:  App. Length of Kaiser filter: 37 Example: Optimal filter: Kaiser filter: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

38. 38 Does it meet the specs? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

39. 39 Increase the length of the filter by 1. Does it meet the specs? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology