1. Lecture 11: FIR Filter Designs XILIANG LUO 2014/11 1

2. Windowing 2 Desired frequency response: Fourier series for a periodic function with period 2pi Convergence of the Fourier series

3. Windowing 3

4. 4

5. 5 Rectangular window:

6. Common Windows 6

7. 7

8. 8 Rectangular Window M=50

9. Common Windows 9 Hamming Window M=50

10. Common Windows 10 Blackman Window M=50

11. Comparisons 11

12. Kaiser Window 12

13. Kaiser Window 13

14. Kaiser Window 14

15. Kaiser Window 15

16. Kaiser Window 16

17. Optimal FIR Filter 17 Design Type-1 FIR filter:

18. Optimal FIR Filter 18

19. Optimal FIR Filter 19 Parks-McClellan algorithm is based on the reformulating the filter design problem as a problem in polynomial approximation.

20. Optimal FIR Filter 20 Approx. Error: only defined in interested subintervals of [0, pi]

21. Optimal FIR Filter 21 Parks-McClellan, MinMax criterion:

22. Optimal FIR Filter 22

23. Parks-McClellan 23 Alternation theorem gives necessary and sufficient conditions on the error for optimality in the Chebyshev or minimax sense! Optimal FIR should satisfy:

24. Parks-McClellan 24 2(L+2) unknowns

25. Parks-McClellan 25 Given set of the extremal frequencies, we can have:

26. Parks-McClellan 26 Given set of the extremal frequencies, we can have: Evaluate on other frequencies

27. Parks-McClellan 27

28. 28 Flow Chart of Parks-McClellen

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