1. Chapter 8 Design of infinite impulse response digital filter

2. 2/47  IIR filter –Recursive equation of IIR filter –Transfer function of IIR filter 1. Basic property of IIR filters

3. 3/47 –Transfer function of IIR filter Factored form

4. 4/47 2. Design of IIR filter using analog filter  Impulse invariant method –Identical impulse response of discrete filter to that of analog filter Analog Filter Transfer Function Digital Filter Transfer Function Impulse Response Impulse Response Series Fig. 8-1.

5. 5/47 –Design a LPF using Impulse invariant method Fig. 8-2.

6. 6/47 Inverse Laplace transform Sampling with T of inverse Laplace transform z-transform

7. 7/47 Transfer function with single pole Inverse Laplace transform Sampling with T of inverse Laplace transform

8. 8/47 z-transform Using commutative law

9. 9/47 Using an infinite series

10. 10/47 –Repeated poles in designing filter Repeated pole of l order z-transform

11. 11/47 –Complex number in designing filter (1) z-transform –Complex number in designing filter (2) z-transform

12. 12/47 –Example 8-1 second order Butterworth filter Partial fraction Impulse response function (T=1)

13. 13/47 Magnitude of impulse response function –Summary of impulse invariance method (1) (2) Multiply H(z) by T Fig. 8-3.

14. 14/47  Bilinear z transform –Replacing s in the transfer function depending on the filter required Arranging to z variable

15. 15/47 Replacing Considering frequency scaling –Replacing,

16. 16/47 –Relationship between analog frequency and digital frequency

17. 17/47 Frequency warping Fig. 8-4.

18. 18/47 –Example 8-2 Using bilinear z transform in transfer function

19. 19/47 1)Impulse response of analog filter : 2)Frequency response of analog filter: 3)Impulse response of digital filter : 4)Frequency response of digital filter: Magnitude = phase = Magnitude = phase =

20. 20/47 5) Relationship between and

21. 21/47 –Example 8-3 Specification of the desired filter –Filter response : -3dB at 1000Hz : -10dB at 3000Hz –Sampling frequency : 10kHz –Monotonic decrease in transition region(1000~3000Hz) Digital parameter from specification –

22. 22/47 Considering Frequency warp –Prewarp Determining order of Butterworth filter

23. 23/47 Using bilinear z transform Fig. 8-5.

24. 24/47  Two transformation method –Impulse invariant method –Bilinear z transform 3. Comparing two transformation method

25. 25/47 –Example 8-4 Transfer function of analog filter Frequency response Using impulse invariant method (1)

26. 26/47 Frequency response with Using bilinear z transform (2)

27. 27/47 Frequency response

28. 28/47 Analog Constant Impulse Response Bilinear z Transform Fig. 8-6.

29. 29/47 –Example 8-5 Partial fraction of impulse response Inverse Laplace transform

30. 30/47 Frequency response Analog filter with -3db at

31. 31/47 Impulse invariant method –Partial fraction using sampling period ( )

32. 32/47 Bilinear z transform

33. 33/47  Design of various filters using frequency transformation 4. Frequency transformation Analog low pass filter (normalization filter) Analog frequency transform Low pass Low pass High pass Band pass Band reject Desired digital filter Bilinear z transform or Impulse invariant method Fig. 8-7.

34. 34/47 Analog low pass filter (normalization) Digital low pass filter (normalization) Analog Low pass High pass Band pass Band rejec t Digital Low pass High pass Band pass Band reject Bilinear transform Analog frequency transform Digital frequency transform Fig. 8-8.

35. 35/47 –Low pass filter -3dB at –High pass filter By replacing to -3dB at

36. 36/47 –Band pass filter -3dB at –Band reject filter -3dB at

37. Table. 8-1 Analog Frequency Transform Low pass filter (cutoff frequency ) Low pass filter (cutoff frequency : ) High pass filter (cutoff frequency : ) Band pass filter (Upper cufoff frequency :,Lower cufoff frequency:, Band pass frequency : ) Band reject filter (Upper cufoff frequency :,Lower cufoff frequency :, Band reject frequency : )

38. 38/47 –Example 8-6 Specification of filter design –-3dB at 10Hz –Sampling frequency ( ) –Bilinear z transform –Transfer function : Considering prewarp

39. 39/47 Transfer function of analog LPF Bilinear z transform

40. 40/47 For computational efficiency For accurate frequency

41. 41/47 Fig. 8-9.

42. 42/47 –Example 8-7 Specification of filter design –cutoff frequency ( ) –Sampling frequency ( ) –Bilinear z transform –Transfer function : Cutoff frequency using prewarp

43. 43/47 Analog HPF using table 8-1 Bilinear z transform

44. 44/47 Fig. 8-10.

45. 45/47 –Example 8-8 Specification of filter design –Band pass frequency ( ) –Sampling frequency ( ) –Order of filter : 2 –Bilinear z transform Using table 8-1

46. 46/47 Analog bandpass filter

47. 47/47 Bilinear z transform