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Lesson 5

Lesson 5. Discrete Filters. Lesson 2 Recap. Lesson 3 and 4 Recap. I nput. Transfer function. O utput. Transfer function. y [n] = 0.5 y[n-1] + x[n] y [n] = (x[n-1]+x[n]+x[n+1])/3. Transfer function. y[n] = 0.5 y[n-1] + x[n]. Transfer function. y[n] = (x[n-1]+x[n]+x[n+1])/3.

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Lesson 5

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  1. Lesson 5 Discrete Filters

  2. Lesson 2 Recap

  3. Lesson 3 and 4 Recap Input Transfer function Output

  4. Transfer function • y[n] = 0.5 y[n-1] + x[n] • y[n] = (x[n-1]+x[n]+x[n+1])/3

  5. Transfer function • y[n] = 0.5 y[n-1] + x[n]

  6. Transfer function • y[n] = (x[n-1]+x[n]+x[n+1])/3

  7. IIR and FIR • Infinite Impulse Response (IIR) • Finite Impulse Response (FIR) recursive

  8. Linear Phase • Time shifting

  9. Linear Phase • Example

  10. Inverse Filter • How to remove the effect of a filter with transfer function H(z)?

  11. Frequency Scales fp fst 0.5 fs f(Hz) 0 0.5Ωs Ω=2πf (rad/sec) Ωp Ωst 0 ωst ωp π 0 ω=ΩTs (rad) ωp/π ωst/π 1 0 ω/π

  12. Loss Function

  13. Loss Function • Example

  14. Advantages of FIR • Stability (BIBO: Bounded Input Bounded Output) • Possible linear phase • Efficient implementation (convolution sum via FFT) • Minor disadvantage compared with IIR: a larger number of coefficients results in slightly larger storage

  15. FIR Filter Design

  16. FIR Filter Design • Windowing

  17. FIR Filter Design • The effect of windowing

  18. FIR Filter Design • Shifting

  19. FIR Filter Design • The effect of shifting Linear Phase

  20. FIR Filter Design • Different Window Functions • Rectangular • Triangular or Bartlett • Hamming • Kaiser • Etc.

  21. FIR Filter Design • Example

  22. FIR Filter Design • High-pass filter

  23. Properties of DTFT • Frequency shifting

  24. FIR Filter Design • High-pass filter

  25. FIR Filter Design • Example Design a high-pass filter of order 14 and a cut-off frequency 0.2π using the Kaiser window.

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