1. Lecture 8: Transform Analysis of LTI System XILIANG LUO 2014/10 1

2. Phase, Magnitude Response Magnitude: Phase: Freq Response: 2

3. Ideal Filter Not realizable! 3

4. Distortions Ideal Delay: Delay distortion is regarded inconsequential! 4

5. Group Delay Measure the phase response to a narrow band signal! Approximation around frequency w0: Group Delay: 5

6. Group Delay 6

7. 7

8. 8

9. Linear Const. Coef. Diff. Eq. Typically, we implement our filters in the following way: System Function: 9

10. Linear Const. Coef. Diff. Eq. zeros poles 10

11. Linear Const. Coef. Diff. Eq. Difference equation does not uniquely specify the impulse response of an LTI system. We need to add additional constraints: 1.causal  ROC is outwards 2.stable  ROC includes the unit circle 11

12. Linear Const. Coef. Diff. Eq. Inverse Filter: 12

13. Linear Const. Coef. Diff. Eq. Inverse Filter: ROC must overlap! 13

14. Linear Const. Coef. Diff. Eq. Original system’s zeros become inverse system’s poles! What is needed if we want both original and inverse are stable and causal? 14

15. Linear Const. Coef. Diff. Eq. FIR: IIR: 15

16. Freq. Resp. for Rational Sys A BIBO stable rational system’s frequency response: What is the relationship among Magnitude, Phase, and Group Delay? 16

17. Magnitude Response zero factor pole factor 17

18. Magnitude Response zero factor pole factor Gain in dB: 18

19. Phase Response zero factor pole factor 19

20. Phase Response Principle value: 20

21. Phase Response 21

22. Phase Response 22

23. Phase Response 23

24. Single Zero System 24

25. Single Zero System 25

26. Relation betw. Mag. and Phase In general, magnitude provides no information about the phase. But, for rational systems, there is some constraint between magnitude and phase. In particular, if we know: 1.the magnitude response 2.number of poles and zeros then, there are only a finite number of choices for the phase response! 26

27. Relation betw. Mag. and Phase 27

28. Relation betw. Mag. and Phase If we know the magnitude square as a function of, by replacing it to z, we can construct C(z)! zeros/poles appear in conjugate reciprocal pairs 28

29. Relation betw. Mag. and Phase C(z) have same magnitude response! 29

30. Relation betw. Mag. and Phase C(z) a k,b k are real valued Assume we know H(z) has 3 zeros and 3 poles 30

31. Relation betw. Mag. and Phase C(z) Assume we do not know how may zeros and poles for H(z), H(z) could have undermined number of all-pass factors 31

32. All Pass System C(z) 32

33. All Pass System All-pass system with real-valued IR 1.conjugate pairs of poles 2.zero-pole are conjugate reciprocal pairs 33

34. All Pass System Group delay of a causal all-pass system is always positive! 34

35. Minimum Phase System Both zeros and poles are inside the unit circle  Minimum-Phase x o x x o o o 35

36. Minimum Phase System If we are given a magnitude square function as follows, and we know the system is of minimum phase, then, H(z) will be uniquely determined! 36

37. Minimum Phase System 37 Any rational system function can be expressed as: x o x x o o o x

38. Minimum Phase System 38 x o x x o o o x

39. Minimum Phase System 39 Decomposition:

40. Minimum Phase System Magnitude Distortion Compensation: 40

41. Minimum Phase System 41 Relative to other systems having the same magnitude response, minimum phase systems enjoying the following properties: 1. Minimum Phase-Lag Property all pass system has negative phase

42. Minimum Phase System 42 Relative to other systems having the same magnitude response, minimum phase systems enjoying the following properties: 2. Minimum Group-Delay Property all pass system has positive group delay

43. Minimum Phase System 43 Relative to other systems having the same magnitude response, minimum phase systems enjoying the following properties: 3. Minimum Energy-Delay Property For all impulse responses having the same magnitude response:

44. Minimum Phase System 44 Relative to other systems having the same magnitude response, minimum phase systems enjoying the following properties: 3. Minimum Energy-Delay Property

45. Generalized Linear Phase 45 A system is referred to as a generalized linear-phase system if its frequency response can be expressed as: a real function of w

46. Generalized Linear Phase 46 Type 1 FIR Linear Phase System M is even

47. Generalized Linear Phase 47 Type 2 FIR Linear Phase System M is odd

48. Generalized Linear Phase 48 Type 3 FIR Linear Phase System M is even

49. Generalized Linear Phase 49 Type 4 FIR Linear Phase System M is odd

50. Generalized Linear Phase 50 Type 5 FIR Linear Phase System M is odd