1. Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin Digital Signal Processing 1

2. Lecture 16 IIR Filters: Feedback and H(z) and H(z) Digital Signal Processing 2

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4. READING ASSIGNMENTS This Lecture: –Chapter 8, Sects. 8-1, 8-2 & 8-3 Other Reading: –Recitation: Ch. 8, Sects 8-1 thru 8-4 POLES & ZEROS –Next Lecture: Chapter 8, Sects. 8-4 8-5 & 8-6 4

5. LECTURE OBJECTIVES –Show how to compute the output y[n] FIRST-ORDER CASE (N=1) Z-transform: Impulse Response h[n]  H(z)  INFINITE IMPULSE RESPONSE FILTERS IIR  Define IIR DIGITAL Filters FEEDBACK  Have FEEDBACK: use PREVIOUS OUTPUTS 5

6. THREE DOMAINS Z-TRANSFORM-DOMAIN POLYNOMIALS : H(z) FREQ-DOMAIN TIME-DOMAIN 6

7. Quick Review: Delay by n d IMPULSE RESPONSE SYSTEM FUNCTION FREQUENCY RESPONSE 7

8. Quick Review: L-pt Averager IMPULSE RESPONSE SYSTEM FUNCTION 8

9. LOGICAL THREAD FIND the IMPULSE RESPONSE, h[n] –INFINITELY LONG –IIR –IIR Filters EXPLOIT THREE DOMAINS: –Show Relationship for IIR: 9

10. ONE FEEDBACK TERM CAUSALITY –NOT USING FUTURE OUTPUTS or INPUTS FIR PART of the FILTER FEED-FORWARD PREVIOUS FEEDBACK  ADD PREVIOUS OUTPUTS 10

11. FILTER COEFFICIENTS ADD PREVIOUS OUTPUTS MATLAB –yy = filter([3,-2],[1,-0.8],xx) SIGN CHANGE FEEDBACK COEFFICIENT 11

12. COMPUTE OUTPUT 12

13. COMPUTE y[n] FEEDBACK DIFFERENCE EQUATION:  NEED y[-1] to get started 13

14. AT REST CONDITION y[n] = 0, for n<0 BECAUSE x[n] = 0, for n<0 14

15. COMPUTE y[0] THIS STARTS THE RECURSION: SAME with MORE FEEDBACK TERMS 15

16. COMPUTE MORE y[n] CONTINUE THE RECURSION: 16

17. PLOT y[n] 17

18. IMPULSE RESPONSE 18

19. IMPULSE RESPONSE DIFFERENCE EQUATION: Find h[n] CONVOLUTION in TIME-DOMAIN IMPULSE RESPONSE LTI SYSTEM 19

20. PLOT IMPULSE RESPONSE 20

21. Infinite-Length Signal: h[n] POLYNOMIAL Representation SIMPLIFY the SUMMATION APPLIES to Any SIGNAL 21

22. Derivation of H(z) Recall Sum of Geometric Sequence: Yields a COMPACT FORM 22

23. H(z) = z-Transform{ h[n] } FIRST-ORDER IIR FILTER: 23

24. H(z) = z-Transform{ h[n] } ANOTHER FIRST-ORDER IIR FILTER: 24

25. CONVOLUTION PROPERTY MULTIPLICATION of z-TRANSFORMS CONVOLUTION in TIME-DOMAIN IMPULSE RESPONSE 25

26. STEP RESPONSE: x[n]=u[n] 26

27. DERIVE STEP RESPONSE 27

28. PLOT STEP RESPONSE 28

29. MATH FORMULA for h[n] Use SHIFTED IMPULSES to write h[n] 0.2 n 0 4 29

30. LTI: Convolution Output = Convolution of x[n] & h[n] –NOTATION: y[n] = x[n]*h[n] –Here is the FIR case: FINITE LIMITS Same as b k 30

31. L-pt RUNNING AVG H(z) ZEROS on UNIT CIRCLE (z-1) in denominator cancels k=0 term 31

32. FILTER DESIGN: CHANGE L Passband Narrower for L bigger 32

33. H(z) = z-Transform{ h[n] } FIRST-ORDER CASE: 33

34. POP QUIZ Given: Plot the Magnitude and Phase Find the output, y[n] –When 34

35. POP QUIZ: MAG & PHASE Given: Derive Magnitude and Phase 35

36. Ans: FREQ RESPONSE 36

37. POP QUIZ : Answer #2 Find y[n] when 37

38. 11-pt RUNNING SUM H(z) NO zero at z=1 38

39. 3 DOMAINS MOVIE: FIR ZEROS MISSING h[n] H(  ) H(z) 39

40. L-pt RUNNING AVG: Step Response STEP RESPONSE 40