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Digital Signal Processing

Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. Digital Signal Processing. Lecture 18 3-Domains for IIR. READING ASSIGNMENTS. This Lecture: Chapter 8, all Other Reading: Recitation: Ch. 8, all POLES & ZEROS

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Digital Signal Processing

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

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

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

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

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

  6. Z-TRANSFORM TABLES

  7. SECOND-ORDER FILTERS • Two FEEDBACK TERMS

  8. MORE POLES • Denominator is QUADRATIC • 2 Poles: REAL • or COMPLEX CONJUGATES

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

  10. 2nd ORDER EXAMPLE

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

  12. 2nd ORDER Z-transform PAIR GENERAL ENTRY for z-Transform TABLE

  13. 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] );

  14. Complex POLE-ZERO PLOT

  15. UNIT CIRCLE • MAPPING BETWEEN

  16. FREQUENCY RESPONSEfrom POLE-ZERO PLOT

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

  18. Complex POLE-ZERO PLOT

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

  20. Complex POLE-ZERO PLOT

  21. 3 DOMAINS MOVIE: IIR POLE MOVES H(z) H(w) h[n]

  22. THREE INPUTS • Given: • Find the output, y[n] • When

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

  24. Step Response Partial Fraction Expansion

  25. Step Response

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

  27. SINUSOID starting at n=0 • We’ll look at an example in MATLAB • cos(0.2pn) • Pole at –0.8, so an 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

  28. Cosine input

  29. STABILITY • When Does the TRANSIENT DIE OUT ?

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

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

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