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

2. FIRIIR Achieving a linear phase is always possible Difficult to control the linear-phase property. Almost no particular technique is available. Always stableCan be unstable Filter order: higherFilter order: less 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

3.  We focus on IIR filters with a rational transfer function: P and Q are polynomials in z. Filter Design: To determine the values of a(n) and b(n) such that specs given to us are met. 3 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

4. 4 IIR Filter Design Bilinear Transformation Optimization techniques Impulse Invariance Pole-zero placement Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

5. 1. A set of specs for the digital (discrete-time) filter is given. 2. We transform the specs from the D.T. to C.T. (z  s) 3. Design a C.T. IIR filter : 4. s  z 5 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6.  The art of CT IIR filter design is highly advanced.  Many CT IIR methods have relatively closed-form design formulas. 6 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

7. 1) Causal/stable analog filter should be transformed to a causal stable DT filter. Causal and stable 7 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

8. 3) Rational transfer function in the s-domain should be transformed into a rational transfer function in the z- domain. 2) jΏ axis in the s-plane (CT) needs to be transformed to the unit circle in the z-domain. * Needed to translate the specs from discrete to analog domain 8 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

9. Proposal: Does this transformation satisfy the desirable properties that we just discussed? T: arbitrary parameter Does a rational analog filter lead to a rational digital filter? A rational analog filter translated into a rational digital filter. 9 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10. Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011 Is jΏ axis in the s-plane (CT) transformed to the unit circle in the z-domain? Let: 10

11. We need to show that LHP in the s-domain is mapped into inside the unit circle in the z-domain. Does a causal and stable analog filter lead to a causal and stable digital filter? 11

12. Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. 12 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

13. 1. A set of specs for the digital (discrete-time) filter is given. 2. We transform the specs from the D.T. to C.T. (z  s) 3. Design a C.T. IIR filter : 4. s  z 13 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

14. 14

15. 15 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

16.  Butterworth filter: 16 Monotonic in stop-band and pass-band Cut-off frequency Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17. 17 Poles of : Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

18. 18 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

19. Elliptic: ripple in both pass-band or stop-band Chebyshev: ripple in either pass-band or stop-band See Appendix B in the textbook for related formulae. 19 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

20. - Which one would you choose? 20 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21. Butterworth filter of order N: 21 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

22. 22

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

24. 24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

25. 25 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

26. Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. 26 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

27. 27 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

28. 28 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29. Butterworth filter of order N: 29 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

30. 30 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

32. Frequency Transformation of Low-pass IIR Filters 32 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. 1. Start with spec. in D.T. For HPF. 2. Translate the filter specs from D.T. to C.T.  specs of a HPF in C.T. (using bilinear transform.) 3. Translate specs of C.T. HPF to C.T. LPF 4. Design the LPF (Butterworth) 5. Transform C.T. LPF  C.T. HPF 6. Transform C.T. HPF  D.T. HPF (using bilinear transform.) C.T. LPF H a (s 1 ) C.T. HPF H a’ (s 2 ) C/D D.T. HPF H d (z) C/C 33 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. Proposal: k: positive constant 34 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35.  Is jΏ 1 axis in the s 1 -plane (CT) transformed to the jΏ 2 axis in the s 2 -plane (CT)? 35 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

36. Does a causal and stable analog LPF filter lead to a causal and stable analog HPF? Does LHP in the s 1 -domain (CT) map into the LHP in the s 2 -domain (CT)? It is easy to prove the above statement. It is also easy to show that a rational transfer function is mapped into another rational transfer function. 36 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

37. Frequency response is symmetric. 37 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

38. 38 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

39. *Assumption: prototype lowpass filter has band edge frequency Type of TransformationTransformationBand edge frequencies of the new filter Lowpass Highpass Bandpass Bandstop 39 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

40. 1. Start with spec. in D.T. For HPF. 2. Translate the filter specs from D.T. to C.T.  specs of a HPF in C.T. (bilinear transformation) 3. Translate specs of C.T. HPF to C.T. LPF 4. Design the LPF (Butterworth) 5. Transform C.T. LPF  C.T. HPF 6. Transform C.T. HPF  D.T. HPF (bilinear transformation) C.T. LPF H a (s 1 ) C.T. HPF H a’ (s 2 ) C/D D.T. HPF H d (z) C/C 40 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41. 1. Start with spec. in D.T. For HPF. 2. Translate the filter specs from D.T. HPF to D.T. LPF (using the transformation discussed shortly) 3. Design the LPF : (a) translate the DT LPF specs  CT LPF specs; (b) Design CT LPF;(c) Transform CT LPF to DT LPF. 4. Transform D.T. LPF  D.T. HPF C.T. LPF H a (s) D.T.LPF H(z 1 ) D/D D.T. HPF H d (z 2 ) C/D 41 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42. Proposal: It can be shown that this transformation has the 3 properties that we usually investigate for transformations: 1- A rational transfer function is transformed to a rational transfer function. 2- Unit circle in one domain is mapped into the unit circle in the other domain. 3- Inside of the unit circle in one domain is mapped to the inside of the unit circle in the other domain. 42 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43. Proposal: Proof of the second property: Unit circle in one domain is mapped into the unit circle in the other domain. 43 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

44. Proposal: 44 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

45. 1. Start with spec. in D.T. For HPF. 2. Translate the filter specs from D.T. HPF to D.T. LPF (using the transformation discussed earlier) 3. Design the LPF : (a) translate the DT LPF specs  CT LPF specs; (b) Design CT LPF;(c) Transform CT LPF to DT LPF. 4. Transform D.T. LPF  D.T. HPF C.T. LPF H a (s) D.T.LPF H(z 1 ) D/D D.T. HPF H d (z 2 ) C/D 45 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

46. 46 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47. Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 201147

48.  Suppose we have designed a filter that has met the following specs:  We have designed a Chebyshev lowpass filter with the following system function: 48 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49. 49 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

50.  To transfer this filter to a highpass filter with passband edge frequency of :  This results in the following high-pass filter: 50 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

51. 51 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

52.  IIR filters generally have lower order compared to FIR filters, however, linear-phase cannot be guaranteed.  The most popular technique is the transformation technique, although other methods such as pole-zero placement also exist.  Using transformation techniques, a low-pass prototype filter can be transformed into HP, BP and BS filters. 52 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology