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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 12 Frequency Response of FIR Filters. READING ASSIGNMENTS. This Lecture: Chapter 6, Sections 6-1, 6-2, 6-3, 6-4, & 6-5

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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 12 Frequency Response of FIR Filters

  3. READING ASSIGNMENTS • This Lecture: • Chapter 6, Sections 6-1, 6-2, 6-3, 6-4, & 6-5 • Other Reading: • Recitation: Chapter 6 • FREQUENCY RESPONSE EXAMPLES • Next Lecture: Chap. 6, Sects. 6-6, 6-7 & 6-8

  4. LECTURE OBJECTIVES MAG PHASE • SINUSOIDAL INPUT SIGNAL • DETERMINE the FIR FILTER OUTPUT • FREQUENCY RESPONSE of FIR • PLOTTING vs. Frequency • MAGNITUDE vs. Freq • PHASE vs. Freq

  5. DOMAINS: Time & Frequency • Time-Domain: “n” = time • x[n] discrete-time signal • x(t) continuous-time signal • Frequency Domain (sum of sinusoids) • Spectrum vs. f (Hz) • ANALOG vs. DIGITAL • Spectrum vs. omega-hat • Move back and forth QUICKLY

  6. DIGITAL “FILTERING” • CONCENTRATE on the SPECTRUM • SINUSOIDAL INPUT • INPUT x[n] = SUM of SINUSOIDS • Then, OUTPUT y[n] = SUM of SINUSOIDS x(t) A-to-D x[n] FILTER y[n] D-to-A y(t)

  7. FILTERING EXAMPLE • 7-point AVERAGER • Removes cosine • By making its amplitude (A) smaller • 3-point AVERAGER • Changes A slightly

  8. 3-pt AVG EXAMPLE USE PAST VALUES

  9. 7-pt FIR EXAMPLE (AVG) CAUSAL: Use Previous LONGER OUTPUT

  10. SINUSOIDAL RESPONSE • INPUT: x[n] = SINUSOID • OUTPUT: y[n] will also be a SINUSOID • Different Amplitude and Phase • SAME Frequency • AMPLITUDE & PHASE CHANGE • Called the FREQUENCY RESPONSE

  11. DCONVDEMO: MATLAB GUI

  12. COMPLEX EXPONENTIAL x[n] is the input signal—a complex exponential FIR DIFFERENCE EQUATION

  13. COMPLEX EXP OUTPUT • Use the FIR “Difference Equation”

  14. FREQUENCY RESPONSE FREQUENCY RESPONSE • Complex-valued formula • Has MAGNITUDE vs. frequency • And PHASE vs. frequency • Notation: • At each frequency, we can DEFINE

  15. EXAMPLE 6.1 EXPLOIT SYMMETRY

  16. PLOT of FREQ RESPONSE RESPONSE at p/3

  17. EXAMPLE 6.2 x[n] y[n]

  18. EXAMPLE 6.2 (answer)

  19. EXAMPLE: COSINE INPUT x[n] y[n]

  20. EX: COSINE INPUT Use Linearity

  21. EX: COSINE INPUT (ans-2)

  22. MATLAB: FREQUENCY RESPONSE • HH = freqz(bb,1,ww) • VECTOR bb contains Filter Coefficients • SP-First: HH = freekz(bb,1,ww) • FILTER COEFFICIENTS {bk}

  23. Time & Frequency Relation IMPULSE RESPONSE • Get Frequency Response from h[n] • Here is the FIR case:

  24. BLOCK DIAGRAMS x[n] y[n] x[n] y[n] • Equivalent Representations

  25. UNIT-DELAY SYSTEM x[n] y[n] x[n] y[n]

  26. FIRST DIFFERENCE SYSTEM x[n] y[n] y[n] x[n]

  27. DLTI Demo with Sinusoids x[n] y[n] FILTER

  28. CASCADE SYSTEMS • Does the order of S1 & S2 matter? • NO, LTI SYSTEMS can be rearranged !!! • WHAT ARE THE FILTER COEFFS? {bk} • WHAT is the overall FREQUENCY RESPONSE ? S1 S2

  29. CASCADE EQUIVALENT • MULTIPLY the Frequency Responses x[n] y[n] x[n] y[n] EQUIVALENT SYSTEM

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