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

2. Lecture 18 3-Domains for IIR Digital Signal Processing 2

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4. READING ASSIGNMENTS This Lecture: –Chapter 8, all Other Reading: –Recitation: Ch. 8, all POLES & ZEROS –Next Lecture: Chapter 9

5. LECTURE OBJECTIVES SECOND-ORDER IIR FILTERS –TWO FEEDBACK TERMS H(z) can have COMPLEX POLES & ZEROS THREE-DOMAIN APPROACH –BPFs have POLES NEAR THE UNIT CIRCLE

6. THREE DOMAINS Z-TRANSFORM-DOMAIN: poles & zeros POLYNOMIALS: H(z) Use H(z) to get Freq. Response FREQ-DOMAIN TIME-DOMAIN

7. Z-TRANSFORM TABLES

8. SECOND-ORDER FILTERS Two FEEDBACK TERMS

9. MORE POLES Denominator is QUADRATIC –2 Poles: REAL –or COMPLEX CONJUGATES

10. TWO COMPLEX POLES Find Impulse Response ? –Can OSCILLATE vs. n –“RESONANCE” FREQUENCY RESPONSEFind FREQUENCY RESPONSE –Depends on Pole Location –Close to the Unit Circle? BANDPASS FILTERMake BANDPASS FILTER

11. 2nd ORDER EXAMPLE

12. h[n]: Decays & Oscillates “PERIOD”=6

13. 2nd ORDER Z-transform PAIR GENERAL ENTRY for z-Transform TABLE

14. 2nd ORDER EX: n-Domain aa = [ 1, -0.9, 0.81 ]; bb = [ 1, -0.45 ]; nn = -2:19; hh = filter( bb, aa, (nn==0) ); HH = freqz( bb, aa, [-pi,pi/100:pi] );

15. Complex POLE-ZERO PLOT

16. UNIT CIRCLE MAPPING BETWEEN

17. FREQUENCY RESPONSE from POLE-ZERO PLOT

18. h[n]: Decays & Oscillates “PERIOD”=6

19. Complex POLE-ZERO PLOT

20. h[n]: Decays & Oscillates “PERIOD”=12

21. Complex POLE-ZERO PLOT

22. 3 DOMAINS MOVIE: IIR POLE MOVES h[n] H(  ) H(z)

23. THREE INPUTS Given: Find the output, y[n] –When

24. SINUSOID ANSWER Given: The input: Then y[n]

25. SINUSOID Starting at n=0 Given: The input: Then y[n]

26. SINUSOID Starting at n=0

27. TransientSteady-State

28. Step Response Partial Fraction Expansion

29. Step Response

30. Stability Nec. & suff. condition: Pole must be Inside unit circle

31. SINUSOID starting at n=0 We’ll look at an example in MATLAB –cos(0.2  n) –Pole at –0.8, so a n is (–0.8) n There are two components: –TRANSIENT Start-up region just after n=0; (–0.8) n –STEADY-STATE Eventually, y[n] looks sinusoidal. Magnitude & Phase from Frequency Response

32. Cosine input

33. STABILITY When Does the TRANSIENT DIE OUT ?

34. STABILITY CONDITION ALL POLES INSIDE the UNIT CIRCLE UNSTABLE EXAMPLE: POLE @ z=1.1

35. BONUS QUESTION Given: The input is Then find y[n]

36. Transient & Steady State

37. CALCULATE the RESPONSE Use the Z-Transform Method And PARTIAL FRACTIONS

38. GENERAL INVERSE Z (pole) n

39. SPLIT Y(z) to INVERT Need SUM of Terms:

40. INVERT Y(z) to y[n] Use the Z-Transform Table

41. TWO PARTS of y[n] TRANSIENTTRANSIENT –Acts Like (pole) n –Dies out ? IF |a 1 |<1 STEADY-STATESTEADY-STATE –Depends on the input –e.g., Sinusoidal

42. STEADY STATE HAPPENS When Transient dies out Limit as “n” approaches infinity Use Frequency Response to get Magnitude & Phase for sinusoid

43. NUMERICAL EXAMPLE

44. REALISTIC FIR BANDPASS FIR L = 24 M=23 23 zeros

45. FIR BPF: 23 ZEROS

46. Complex POLE-ZERO PLOT

47. POLES & ZEROS of IIR 3 POLES

48. IIR Elliptic LPF (N=3) 3 POLES

49. 3-D VIEW UNIT CIRCLE EVALUATE H(z) EVERYWHERE

50. FLYING THRU Z-PLANE POLES CAUSE PEAKS in H(z) H(  ) H(z)