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Digital Signal Processing 2 Les 4: Elementair filterontwerp

Digital Signal Processing 2 Les 4: Elementair filterontwerp. Prof. dr. ir. Toon van Waterschoot Faculteit Industriële Ingenieurswetenschappen ESAT – Departement Elektrotechniek KU Leuven, Belgium. Digital Signal Processing 2: Vakinhoud. Les 1: Inleiding 1 (Discrete signalen en systemen)

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Digital Signal Processing 2 Les 4: Elementair filterontwerp

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  1. Digital Signal Processing 2Les 4: Elementair filterontwerp Prof. dr. ir. Toon van WaterschootFaculteit Industriële IngenieurswetenschappenESAT – Departement ElektrotechniekKU Leuven, Belgium

  2. Digital Signal Processing 2: Vakinhoud • Les 1: Inleiding 1 (Discrete signalen en systemen) • Les 2: Inleiding 2 (Wiskundige concepten) • Les 3: Spectrale analyse • Les 4: Elementair filterontwerp • Les 5: Schattingsproblemen • Les 6: Lineaire predictie • Les 7: Optimale filtering • Les 8: Adaptieve filtering • Les 9: Detectieproblemen • Les 10: Classificatieproblemen • Les 11: Codering • Les 12: Herhalingsles

  3. Les 4: Elementair filterontwerp • Graphical method for filter analysis magnitude response, phase response • Elementary filter design shelving filters, presence filters, all-pass filters + homework

  4. Les 4: Literatuur • Graphical method for filter analysis J. O. Smith III, Introduction to Digital Filters • Ch. 8, “Pole-Zero Analysis” • Section 8.2, “Graphical Amplitude Response” • Section 8.3, “Graphical Phase Response” • Elementary filter design J. O. Smith III, Introduction to Digital Filters • App. B, “Elementary Audio Digital Filters” Research paper: T. van Waterschoot and M. Moonen, “A pole-zero placement technique for designing second-order IIR parametric equalizer filters”, IEEE Trans. Audio Speech Language Process., vol. 15, no. 8, Nov. 2007, pp. 2561-2565.

  5. Les 4: Elementair filterontwerp • Graphical method for filter analysis magnitude response, phase response • Elementary filter design shelving filters, presence filters, all-pass filters + homework

  6. Graphical method for filter analysis • Graphical magnitude response • Graphical phase response

  7. Graphical magnitude response (1) • Factored form of filter frequency response: • zeros: q1, q2, … qM • poles: p1, p2, … pN • sampling period: T • radial frequency: ω • DC gain: g • Polar representation of factored form:

  8. Graphical magnitude response (2) • Magnitude response (= amplitude response) • Magnitude response at frequency ω = product of lengths of vectors drawn from zeros to point ejωT product of lengths of vectors drawn from poles to point ejωT

  9. Graphical magnitude response (3) • Example: pole-zero diagram 2nd order filter

  10. Graphical magnitude response (4) • Example: magnitude response 2nd order filter

  11. Graphical method for filter analysis • Graphical magnitude response • Graphical phase response

  12. Graphical phase response (1) • Factored form of filter frequency response: • zeros: q1, q2, … qM • poles: p1, p2, … pN • sampling period: T • radial frequency: ω • DC gain: g • Polar representation of factored form:

  13. Graphical phase response (2) • Phase response • Phase response at frequency ω = offset + sum of angles of vectors drawn from zeros to point ejωT sum of angles of vectors drawn from poles to point ejωT

  14. Graphical phase response (3) • Note that phase response offset can be interpreted by • assuming that every digital filter has equal number of poles and zeros • when N > M, assuming N−M zeros at z = 0 • when N< M, assuming M−N poles at z = 0

  15. Graphical phase response (4) • Example: pole-zero diagram 2nd order filter (note: offset = 0)

  16. Graphical phase response (5) • Example: phase response 2nd order filter

  17. Les 4: Elementair filterontwerp • Graphical method for filter analysis magnitude response, phase response • Elementary filter design shelving filters, presence filters, all-pass filters + homework

  18. Elementary digital filters: overview • Shelving filters: • definition • one-zero • one-pole • Presence filters: • definition • two-zero • two-pole • biquadratic • All-pass filters: • definition • biquadratic Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  19. Elementary digital filters: shelving filters • Definition: • a shelving filter is a filter that amplifies a signal in the frequency range Hz (boost), while attenuating it in the range Hz (cut), or vice versa • Low-pass filter: • low-frequency boost, high-frequency cut • High-pass filter: • low-frequency cut, high-frequency boost • Cut-off frequency: • the cut-off frequency is usually defined as the frequency at which the filter gain is 3dB less than the gain at Hz (low-pass) or Hz (high-pass) Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  20. Elementary digital filters: shelving filters • One-zero shelving filter: • difference equation: • transfer function: • signal flow graph:  Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  21. Elementary digital filters: shelving filters • One-zero shelving filter: • 1 real zero: • highpass if • lowpass if Im Im highpass lowpass Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  22. Elementary digital filters: shelving filters • One-zero shelving filter: • frequency response • frequency magnitude response: • frequency phase response: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  23. Elementary digital filters: shelving filters • One-zero shelving filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  24. Elementary digital filters: shelving filters • One-pole shelving filter: • difference equation: • transfer function: • signal flow graph:  Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  25. Elementary digital filters: shelving filters • One-pole shelving filter: • 1 real pole: • highpass if • lowpass if Im Im highpass lowpass Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  26. Elementary digital filters: shelving filters • One-pole shelving filter: • frequency response • frequency magnitude response: • frequency phase response: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  27. Elementary digital filters: shelving filters • One-pole shelving filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  28. Elementary digital filters: presence filters • Definition: • a presence filter is a filter that amplifies a signal in the frequency range around a center frequency Hz (boost), while attenuating elsewhere (cut), or vice versa • Resonance filter: • boost at center frequency (band-pass) • Notch filter: • cut at center frequency (band-stop) • Bandwidth: • the bandwidth is defined as the frequency difference between the frequencies at which the filter gain is 3dB lower/higher than the resonance/notch gain Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  29. Elementary digital filters: presence filters • Two-zero presence filter: • diff. eq.: • transfer function: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  30. Elementary digital filters: presence filters • Two-zero presence filter: • 2 zeros: • if : real zeros  cascade shelving filters • if : complex conj. zero pair  notch filter Im Im cascade shelving filters notch filter Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  31. Elementary digital filters: presence filters • Two-zero notch filter: • transfer function in radial representation: • radial center frequency • zero radius Im notch filter Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  32. Elementary digital filters: presence filters • Two-zero notch filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  33. Elementary digital filters: presence filters • Two-pole presence filter: • diff. eq.: • transfer function: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  34. Elementary digital filters: presence filters • Two-pole presence filter: • 2 poles: • if : real poles  cascade shelving filters • if : comp. conj. pole pair  resonance filter Im Im cascade shelving filters resonance filter Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  35. Elementary digital filters: presence filters • Two-pole resonance filter: • transfer function in radial representation: • radial center frequency • pole radius Im resonance filter Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  36. Elementary digital filters: presence filters • Two-pole resonance filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  37. Elementary digital filters: presence filters • Biquadratic presence filter: • difference equation: • transfer function: • 2 poles: • 2 zeros: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  38. Elementary digital filters: presence filters • Biquadratic presence filter: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  39. Elementary digital filters: presence filters • Constrained biquadratic presence filter: constrained biquadratic resonance filter constrained biquadratic notch filter Im Im Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  40. Elementary digital filters: presence filters • Design procedure biquadratic presence filter • five specifications: • five filter coefficients:

  41. Elementary digital filters: presence filters • Design procedureconstrained biquadratic presence filter • four specifications: • four filter coefficients:

  42. Elementary digital filters: presence filters • Design equations resulting from graphical analysis resonance filter notch filter

  43. Elementary digital filters: all-pass filters • Definition: • a (unity-gain) all-pass filter is a filter that passes all input signal frequencies without gain or attenuation • hence a (unity-gain) all-pass filter preserves signal energy • an all-pass filter may have any phase response Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  44. Elementary digital filters: all-pass filters • Biquadratic all-pass filter: • it can be shown that for the unity-gain constraint to hold, the denominator coefficients must equal the numerator coefficients in reverse order, e.g., • the poles and zeros are moreover related as follows Toon van Waterschoot & Marc Moonen INTRODUCTION-2

  45. Homework • Read research paper T. van Waterschoot and M. Moonen, “A pole-zero placement technique for designing second-order IIR parametric equalizer filters”, IEEE Trans. Audio Speech Language Process., vol. 15, no. 8, Nov. 2007, pp. 2561-2565. • Play around with Matlab code ftp://ftp.esat.kuleuven.be/pub/SISTA/vanwaterschoot/abstracts/06-177.html

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