Part 3: IIR Filters – Bilinear Transformation                                Method                        Tutorial ISCAS ...
Introduction            t A procedure for the design of IIR filters that would satisfy               arbitrary prescribed s...
Introduction            t A procedure for the design of IIR filters that would satisfy               arbitrary prescribed s...
Introduction Cont’d     Given an analog filter with a continuous-time transfer function     HA (s ), a digital filter with a...
Introduction Cont’d     The bilinear transformation method has the following important     features:            t A stable...
Introduction Cont’d     The bilinear transformation method has the following important     features:            t A stable...
Introduction Cont’dFrame # 5   Slide # 7   A. Antoniou   Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’dFrame # 6   Slide # 8   A. Antoniou   Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect     If we let ω and represent the frequency variable in the analog     filter and the derived digital fil...
The Warping Effect Cont’d      ···                                 2     T                                  ω=        tan ...
The Warping Effect Cont’d      ···                                 2     T                                  ω=        tan ...
The Warping Effect Cont’d                                  6.0                                                T=2         ...
The Warping Effect Cont’d      The warping effect changes the band edges of the digital filter      relative to those of th...
Prewarping          t From Eq. (B), i.e.,                                                2     T                          ...
Prewarping          t From Eq. (B), i.e.,                                                    2     T                      ...
Prewarping Cont’d          t If prescribed passband and stopband edges ˜ 1 ,              ˜ 2 , . . . , ˜ i , . . . are to...
Prewarping Cont’d          t If prescribed passband and stopband edges ˜ 1 ,              ˜ 2 , . . . , ˜ i , . . . are to...
Design Procedure      Consider a normalized analog LP filter characterized by HN (s )      with an attenuation             ...
Design Procedure Cont’d           A(ω)                                                                  Aa                ...
Design Procedure      A denormalized LP, HP, BP, or BS filter that has the same      passband ripple and minimum stopband a...
Design Procedure      A denormalized LP, HP, BP, or BS filter that has the same      passband ripple and minimum stopband a...
Design Procedure Cont’d                                              ¯                        Standard forms of fX (s )   ...
Design Procedure Cont’d          t The digital filter designed by this method will have the             required passband a...
Design Procedure Cont’d          t The digital filter designed by this method will have the             required passband a...
General Design Procedure for LP Filters      An outline of the methodology for the derivation of general      solutions fo...
Design of LP Filters Cont’d           A(ω)                                                                  Aa            ...
Design of LP Filters Cont’d        2. Apply the LP-to-LP analog-filter transformation to HN (s ) to           obtain a deno...
Design of LP Filters Cont’d        2. Apply the LP-to-LP analog-filter transformation to HN (s ) to           obtain a deno...
Design of LP Filters Cont’d        2. Apply the LP-to-LP analog-filter transformation to HN (s ) to           obtain a deno...
Design of LP Filters Cont’d           A(Ω)                                                               Aa               ...
Design of LP Filters Cont’d        5. Solve for λ, the parameter of the LP-to-LP analog-filter           transformation.Fra...
Design of LP Filters Cont’d        5. Solve for λ, the parameter of the LP-to-LP analog-filter           transformation.   ...
Design of LP Filters Cont’d        7. The same methodology is applicable for HP filters, except           that the LP-HP an...
Design of LP Filters Cont’d        7. The same methodology is applicable for HP filters, except           that the LP-HP an...
Formulas for LP and HP Filters                                    ωp                                       ≥ K0           ...
Design of LP Filters Cont’d          t The table of formulas presented is applicable to all the             classical type...
Design of LP Filters Cont’d          t The table of formulas presented is applicable to all the             classical type...
General Design Procedure for BP Filters      An outline of the methodology for the derivation of general      solutions fo...
Design of BP Filters Cont’d        2. Apply the LP-to-BP analog-filter transformation to HN (s ) to           obtain a deno...
Design of BP Filters Cont’d        2. Apply the LP-to-BP analog-filter transformation to HN (s ) to           obtain a deno...
Design of BP Filters Cont’d        4. At this point, assume that the derived discrete-time transfer           function has...
Design of LP Filters Cont’d           A(Ω)                        Aa                                                      ...
Design of LP Filters Cont’d        5. Solve for B and ω0 , the parameters of the LP-to-BP           analog-filter transform...
Design of LP Filters Cont’d        5. Solve for B and ω0 , the parameters of the LP-to-BP           analog-filter transform...
Design of LP Filters Cont’d        5. Solve for B and ω0 , the parameters of the LP-to-BP           analog-filter transform...
Design of LP Filters Cont’d        5. Solve for B and ω0 , the parameters of the LP-to-BP           analog-filter transform...
Formulas for the design of BP Filters                              √                             2 KB                     ...
Formulas for the Design of BS Filters                              √                             2 KB                     ...
Formulas for ωp and n          t The formulas presented apply to any type of normalized             analog LP filter with a...
Formulas for ωp and n          t The formulas presented apply to any type of normalized             analog LP filter with a...
Formulas for ωp and n          t The formulas presented apply to any type of normalized             analog LP filter with a...
Formulas for Butterworth Filters                        LP    K = K0                                   1                  ...
Formulas for Chebyshev Filters                        LP   K = K0                                  1                      ...
Formulas for Elliptic Filters                                           k                          ωp                     ...
Example – HP Filter      An HP filter that would satisfy the following specifications is      required:                     ...
Example – HP Filter Cont’d      Solution                        Filter type       n           ωp                   λ      ...
Example Cont’d                           70                                              Elliptic                         ...
Example – BP Filter      Design an Elliptic BP filter that would satisfy the following      specifications:      Ap = 1 dB, ...
Example – BP Filter Cont’d                           70                           60                           50         ...
Example – BS Filter      Design a Chebyshev BS filter that would satisfy the following      specifications:      Ap = 0.5 dB...
Example – BS Filter Cont’d                           60                           50                           40         ...
D-Filter      A DSP software package that incorporates the design      techniques described in this presentation is D-Filt...
Summary          t A design method for IIR filters that leads to a complete             description of the transfer functio...
Summary          t A design method for IIR filters that leads to a complete             description of the transfer functio...
Summary          t A design method for IIR filters that leads to a complete             description of the transfer functio...
Summary          t A design method for IIR filters that leads to a complete             description of the transfer functio...
References          t A. Antoniou, Digital Signal Processing: Signals, Systems,             and Filters, Chap. 15, McGraw-...
This slide concludes the presentation.                Thank you for your attention.Frame # 47 Slide # 68   A. Antoniou   P...
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Warping Concept (iir filters-bilinear transformation method)

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A good description on warping effect.

Uet Peshwar Atd campus.
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Warping Concept (iir filters-bilinear transformation method)

  1. 1. Part 3: IIR Filters – Bilinear Transformation Method Tutorial ISCAS 2007 Copyright © 2007 Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org July 24, 2007Frame # 1 Slide # 1 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  2. 2. Introduction t A procedure for the design of IIR filters that would satisfy arbitrary prescribed specifications will be described.Frame # 2 Slide # 2 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  3. 3. Introduction t A procedure for the design of IIR filters that would satisfy arbitrary prescribed specifications will be described. t The method is based on the bilinear transformation and it can be used to design lowpass (LP), highpass (HP), bandpass (BP), and bandstop (BS), Butterworth, Chebyshev, Inverse-Chebyshev, and Elliptic filters. Note: The material for this module is taken from Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 12.)Frame # 2 Slide # 3 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  4. 4. Introduction Cont’d Given an analog filter with a continuous-time transfer function HA (s ), a digital filter with a discrete-time transfer function HD (z ) can be readily deduced by applying the bilinear transformation as follows: H D (z ) = H A (s ) (A) 2 z −1 s= T z +1Frame # 3 Slide # 4 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  5. 5. Introduction Cont’d The bilinear transformation method has the following important features: t A stable analog filter gives a stable digital filter.Frame # 4 Slide # 5 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  6. 6. Introduction Cont’d The bilinear transformation method has the following important features: t A stable analog filter gives a stable digital filter. t The maxima and minima of the amplitude response in the analog filter are preserved in the digital filter. As a consequence, – the passband ripple, and – the minimum stopband attenuation of the analog filter are preserved in the digital filter.Frame # 4 Slide # 6 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  7. 7. Introduction Cont’dFrame # 5 Slide # 7 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  8. 8. Introduction Cont’dFrame # 6 Slide # 8 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  9. 9. The Warping Effect If we let ω and represent the frequency variable in the analog filter and the derived digital filter, respectively, then Eq. (A), i.e., H D (z ) = H A (s ) (A) 2 s= T z −1 z +1 gives the frequency response of the digital filter as a function of the frequency response of the analog filter as H D (e j T ) = HA (j ω) 2 z −1 provided that s = T z +1 2 ej T −1 2 T or jω = or ω = tan (B) T ej T +1 T 2Frame # 7 Slide # 9 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  10. 10. The Warping Effect Cont’d ··· 2 T ω= tan (B) T 2 t For < 0.3/T ω≈ and, as a result, the digital filter has the same frequency response as the analog filter over this frequency range.Frame # 8 Slide # 10 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  11. 11. The Warping Effect Cont’d ··· 2 T ω= tan (B) T 2 t For < 0.3/T ω≈ and, as a result, the digital filter has the same frequency response as the analog filter over this frequency range. t For higher frequencies, however, the relation between ω and becomes nonlinear, and distortion is introduced in the frequency scale of the digital filter relative to that of the analog filter. This is known as the warping effect.Frame # 8 Slide # 11 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  12. 12. The Warping Effect Cont’d 6.0 T=2 4.0 ω 2.0 |HA(jω)| 0.1π 0.2π 0.3π 0.4π 0.5π Ω, rad/s |HD(e jΩT)|Frame # 9 Slide # 12 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  13. 13. The Warping Effect Cont’d The warping effect changes the band edges of the digital filter relative to those of the analog filter in a nonlinear way, as illustrated for the case of a BS filter: 40 35 30 Digital Analog 25 filter filter Loss, dB 20 15 10 5 0 0 1 2 3 4 5 ω, rad/sFrame # 10 Slide # 13 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  14. 14. Prewarping t From Eq. (B), i.e., 2 T ω= tan (B) T 2 a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence 2 ωT = tan−1 T 2Frame # 11 Slide # 14 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  15. 15. Prewarping t From Eq. (B), i.e., 2 T ω= tan (B) T 2 a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence 2 ωT = tan−1 T 2 t If ω1 , ω2 , . . . , ωi , . . . are the passband and stopband edges in the analog filter, then the corresponding passband and stopband edges in the derived digital filter are given by 2 ωi T i = tan−1 i = 1, 2, . . . T 2Frame # 11 Slide # 15 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  16. 16. Prewarping Cont’d t If prescribed passband and stopband edges ˜ 1 , ˜ 2 , . . . , ˜ i , . . . are to be achieved, the analog filter must be prewarped before the application of the bilinear transformation to ensure that its band edges are given by 2 ˜ iT ωi = tan T 2Frame # 12 Slide # 16 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  17. 17. Prewarping Cont’d t If prescribed passband and stopband edges ˜ 1 , ˜ 2 , . . . , ˜ i , . . . are to be achieved, the analog filter must be prewarped before the application of the bilinear transformation to ensure that its band edges are given by 2 ˜ iT ωi = tan T 2 t Then the band edges of the digital filter would assume their prescribed values i since 2 ωi T i = tan−1 T 2 2 T 2 ˜ iT = tan−1 · tan T 2 T 2 = ˜i for i = 1, 2, . . .Frame # 12 Slide # 17 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  18. 18. Design Procedure Consider a normalized analog LP filter characterized by HN (s ) with an attenuation 1 AN (ω) = 20 log |HN (j ω)| (also known as loss) and assume that 0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞ Note: The transfer functions of analog LP filters are reported in the literature in normalized form whereby the passband edge is typically of the order of unity.Frame # 13 Slide # 18 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  19. 19. Design Procedure Cont’d A(ω) Aa Ap ω ωp ωc ωaFrame # 14 Slide # 19 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  20. 20. Design Procedure A denormalized LP, HP, BP, or BS filter that has the same passband ripple and minimum stopband attenuation as a given normalized LP filter can be derived from the normalized LP filter through the following steps: ¯ 1. Apply the transformation s = fX (s ) ¯ H X (s ) = H N (s ) ¯ s =fX (s ) ¯ where fX (s ) is one of the four standard analog-filters transformations, given by the next slide.Frame # 15 Slide # 20 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  21. 21. Design Procedure A denormalized LP, HP, BP, or BS filter that has the same passband ripple and minimum stopband attenuation as a given normalized LP filter can be derived from the normalized LP filter through the following steps: ¯ 1. Apply the transformation s = fX (s ) ¯ H X (s ) = H N (s ) ¯ s =fX (s ) ¯ where fX (s ) is one of the four standard analog-filters transformations, given by the next slide. ¯ 2. Apply the bilinear transformation to HX (s ), i.e., ¯ H D (z ) = H X (s ) z −1 ¯ 2 s= T z +1Frame # 15 Slide # 21 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  22. 22. Design Procedure Cont’d ¯ Standard forms of fX (s ) X ¯ fX (s ) LP ¯ λs HP ¯ λ/s ω02 BP 1 ¯ s+ B s¯ Bs¯ BS ¯ s 2 + ω0 2Frame # 16 Slide # 22 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  23. 23. Design Procedure Cont’d t The digital filter designed by this method will have the required passband and stopband edges only if the parameters λ, ω0 , and B of the analog-filter transformations and the order of the continuous-time normalized LP transfer function, HN (s ), are chosen appropriately.Frame # 17 Slide # 23 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  24. 24. Design Procedure Cont’d t The digital filter designed by this method will have the required passband and stopband edges only if the parameters λ, ω0 , and B of the analog-filter transformations and the order of the continuous-time normalized LP transfer function, HN (s ), are chosen appropriately. t This is obviously a difficult problem but general solutions are available for LP, HP, BP, and BS, Butterworth, Chebyshev, inverse-Chebyshev, and Elliptic filters.Frame # 17 Slide # 24 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  25. 25. General Design Procedure for LP Filters An outline of the methodology for the derivation of general solutions for LP filters is as follows: 1. Assume that a continuous-time normalized LP transfer function, HN (s ), is available that would give the required passband ripple, Ap , and minimum stopband attenuation (loss), Aa . Let the passband and stopband edges of the analog filter be ωp and ωa , respectively.Frame # 18 Slide # 25 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  26. 26. Design of LP Filters Cont’d A(ω) Aa Ap ω ωp ωc ωa Attenuation characteristic of continuous-time normalized LP filterFrame # 19 Slide # 26 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  27. 27. Design of LP Filters Cont’d 2. Apply the LP-to-LP analog-filter transformation to HN (s ) to obtain a denormalized discrete-time transfer function ¯ HLP (s ).Frame # 20 Slide # 27 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  28. 28. Design of LP Filters Cont’d 2. Apply the LP-to-LP analog-filter transformation to HN (s ) to obtain a denormalized discrete-time transfer function ¯ HLP (s ). ¯ 3. Apply the bilinear transformation to HLP (s ) to obtain a discrete-time transfer function HD (z ).Frame # 20 Slide # 28 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  29. 29. Design of LP Filters Cont’d 2. Apply the LP-to-LP analog-filter transformation to HN (s ) to obtain a denormalized discrete-time transfer function ¯ HLP (s ). ¯ 3. Apply the bilinear transformation to HLP (s ) to obtain a discrete-time transfer function HD (z ). 4. At this point, assume that the derived discrete-time transfer function has passband and stopband edges that satisfy the relations ˜ p ≤ p and a ≤ ˜a where ˜ p and ˜ a are the prescribed passband and stopband edges, respectively. In effect, we assume that the digital filter has passband and stopband edges that satisfy or oversatisfy the required specifications.Frame # 20 Slide # 29 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  30. 30. Design of LP Filters Cont’d A(Ω) Aa Ap Ω ~ ~ Ωp Ωp Ωa Ωa Attenuation characteristic of required LP digital filterFrame # 21 Slide # 30 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  31. 31. Design of LP Filters Cont’d 5. Solve for λ, the parameter of the LP-to-LP analog-filter transformation.Frame # 22 Slide # 31 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  32. 32. Design of LP Filters Cont’d 5. Solve for λ, the parameter of the LP-to-LP analog-filter transformation. 6. Find the minimum value of the ratio ωp /ωa for the continuous-time normalized LP transfer function. The ratio ωp /ωa is a fraction less than unity and it is a measure of the steepness of the transition characteristic. It is often referred to as the selectivity of the filter. The selectivity of a filter dictates the minimum order to achieve the required specifications. Note: As the selectivity approaches unity, the filter-order tends to infinity!Frame # 22 Slide # 32 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  33. 33. Design of LP Filters Cont’d 7. The same methodology is applicable for HP filters, except that the LP-HP analog-filter transformation is used in Step 2.Frame # 23 Slide # 33 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  34. 34. Design of LP Filters Cont’d 7. The same methodology is applicable for HP filters, except that the LP-HP analog-filter transformation is used in Step 2. 8. The application of this methodology yields the formulas summarized by the table shown in the next slide.Frame # 23 Slide # 34 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  35. 35. Formulas for LP and HP Filters ωp ≥ K0 ωa LP ωp T λ= 2 tan( ˜ p T /2) ωp 1 ≥ ωa K0 HP 2ωp tan( ˜ p T /2) λ= T tan( ˜ p T /2) where K0 = tan( ˜ a T /2)Frame # 24 Slide # 35 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  36. 36. Design of LP Filters Cont’d t The table of formulas presented is applicable to all the classical types of analog filters, namely, – Butterworth – Chebyshev – Inverse-Chebyshev – EllipticFrame # 25 Slide # 36 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  37. 37. Design of LP Filters Cont’d t The table of formulas presented is applicable to all the classical types of analog filters, namely, – Butterworth – Chebyshev – Inverse-Chebyshev – Elliptic t Formulas that can be used to design digital versions of these filters will be presented later.Frame # 25 Slide # 37 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  38. 38. General Design Procedure for BP Filters An outline of the methodology for the derivation of general solutions for BP filters is as follows: 1. Assume that a continuous-time normalized LP transfer function, HN (s ), is available that would give the required passband ripple, Ap , and minimum stopband attenuation, Aa . Let the passband and stopband edges of the analog filter be ωp and ωa , respectively.Frame # 26 Slide # 38 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  39. 39. Design of BP Filters Cont’d 2. Apply the LP-to-BP analog-filter transformation to HN (s ) to obtain a denormalized discrete-time transfer function ¯ HBP (s ).Frame # 27 Slide # 39 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  40. 40. Design of BP Filters Cont’d 2. Apply the LP-to-BP analog-filter transformation to HN (s ) to obtain a denormalized discrete-time transfer function ¯ HBP (s ). ¯ 3. Apply the bilinear transformation to HBP (s ) to obtain a discrete-time transfer function HD (z ).Frame # 27 Slide # 40 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  41. 41. Design of BP Filters Cont’d 4. At this point, assume that the derived discrete-time transfer function has passband and stopband edges that satisfy the relations ˜ p1 ≤ p1 ˜ p2 ≥ p2 and a1 ≥ ˜ a1 a2 ≤ ˜ a2 where – p 1 and p 2 are the actual lower and upper passband edges, – ˜ p 1 and ˜ p 2 are the prescribed lower and upper passband edges, – a1 and a2 are the actual lower and upper stopband edges, – ˜ p 1 and ˜ p 2 are the prescribed lower and upper stopband edges, respectively.Frame # 28 Slide # 41 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  42. 42. Design of LP Filters Cont’d A(Ω) Aa Aa Ap Ω ~ ~ Ωa1 Ωp1 Ωp2 Ωa2 Ωa1 ~ ~ Ωa2 Ωp1 Ωp2 Attenuation characteristic of required BP digital filterFrame # 29 Slide # 42 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  43. 43. Design of LP Filters Cont’d 5. Solve for B and ω0 , the parameters of the LP-to-BP analog-filter transformation.Frame # 30 Slide # 43 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  44. 44. Design of LP Filters Cont’d 5. Solve for B and ω0 , the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ωp /ωa , for the continuous-time normalized LP transfer function.Frame # 30 Slide # 44 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  45. 45. Design of LP Filters Cont’d 5. Solve for B and ω0 , the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ωp /ωa , for the continuous-time normalized LP transfer function. 7. The same methodology can be used for the design of BS filters except that the LP-to-BS transformation is used in Step 2.Frame # 30 Slide # 45 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  46. 46. Design of LP Filters Cont’d 5. Solve for B and ω0 , the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ωp /ωa , for the continuous-time normalized LP transfer function. 7. The same methodology can be used for the design of BS filters except that the LP-to-BS transformation is used in Step 2. 8. The application of this methodology yields the formulas summarized in the next two slides.Frame # 30 Slide # 46 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  47. 47. Formulas for the design of BP Filters √ 2 KB ω0 = T ωp K1 if KC ≥ KB BP ≥ ωa K2 if KC < KB 2KA B= T ωp ˜ p2T ˜ p 1T ˜ p 1T ˜ p2T where KA = tan − tan KB = tan tan 2 2 2 2 ˜ a1 T ˜ a2 T KA tan( ˜ a1 T /2) KC = tan tan K1 = 2 2 KB − tan2 ( ˜ a1 T /2) KA tan( ˜ a2 T /2) K2 = tan2 ( ˜ a2 T /2) − KBFrame # 31 Slide # 47 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  48. 48. Formulas for the Design of BS Filters √ 2 KB ω0 = T ⎧ ⎪1 ⎪ ⎨ if KC ≥ KB ωp K2 BS ≥ ωa ⎪ 1 ⎪ ⎩ if KC < KB K1 2KA ωp B= T ˜ p2T ˜ p 1T ˜ p 1T ˜ p2T where KA = tan − tan KB = tan tan 2 2 2 2 ˜ a1 T ˜ a2 T KA tan( ˜ a1 T /2) KC = tan tan K1 = 2 2 KB − tan2 ( ˜ a1 T /2) KA tan( ˜ a2 T /2) K2 = tan2 ( ˜ a2 T /2) − KBFrame # 32 Slide # 48 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  49. 49. Formulas for ωp and n t The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞Frame # 33 Slide # 49 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  50. 50. Formulas for ωp and n t The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞ t However, the values of the normalized passband edge, ωp , and the required filter order, n , depend on the type of filter.Frame # 33 Slide # 50 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  51. 51. Formulas for ωp and n t The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞ t However, the values of the normalized passband edge, ωp , and the required filter order, n , depend on the type of filter. t Formulas for these parameters for Butterworth, Chebyshev, and Elliptic filters are presented in the next three slides.Frame # 33 Slide # 51 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  52. 52. Formulas for Butterworth Filters LP K = K0 1 HP K = K0 K1 if KC ≥ KB BP K = K2 if KC < KB ⎧ ⎪1 ⎪ if KC ≥ KB ⎨ K2 BS K = ⎪1 ⎪ ⎩ if KC < KB K1 log D 100.1Aa − 1 n≥ , D= 2 log(1/K ) 100.1Ap − 1 ωp = (100.1Ap − 1)1/2nFrame # 34 Slide # 52 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  53. 53. Formulas for Chebyshev Filters LP K = K0 1 HP K = K0 K1 if KC ≥ KB BP K = K2 if KC < KB ⎧ ⎪ 1 if K ≥ K ⎪ ⎨ C B K2 BS K = ⎪1 ⎪ ⎩ if KC < KB K1 √ cosh−1 D 100.1Aa − 1 n≥ , D= cosh−1 (1/K ) 100.1Ap − 1 ωp = 1Frame # 35 Slide # 53 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  54. 54. Formulas for Elliptic Filters k ωp √ LP K = K0 K0 1 1 HP K = √ K0 K0 √ K1 if KC ≥ KB K BP K = √ 1 K2 if KC < KB K2 ⎧ ⎪1 ⎪ 1 ⎨ if KC ≥ KB √ K2 K2 BS K = ⎪1 ⎪ 1 ⎩ if KC < KB √ K1 K1 √ cosh−1 D 100.1Aa − 1 n≥ , D= cosh−1 (1/K ) 100.1Ap − 1Frame # 36 Slide # 54 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  55. 55. Example – HP Filter An HP filter that would satisfy the following specifications is required: Ap = 1 dB, Aa = 45 dB, ˜ p = 3.5 rad/s, ˜ a = 1.5 rad/s, ωs = 10 rad/s. Design a Butterworth, a Chebyshev, and then an Elliptic digital filter.Frame # 37 Slide # 55 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  56. 56. Example – HP Filter Cont’d Solution Filter type n ωp λ Butterworth 5 0.873610 5.457600 Chebyshev 4 1.0 6.247183 Elliptic 3 0.509526 3.183099Frame # 38 Slide # 56 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  57. 57. Example Cont’d 70 Elliptic (n=3) 60 50 Passband loss, dB Chebyshev (n=4) 40 Loss, dB 30 20 Butterworth (n=5) 10 1.0 0 0 1 2 3 4 5 Ω, rad/s 1.5 3.5Frame # 39 Slide # 57 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  58. 58. Example – BP Filter Design an Elliptic BP filter that would satisfy the following specifications: Ap = 1 dB, Aa = 45 dB, ˜ p 1 = 900 rad/s, ˜ p 2 = 1100 rad/s, ˜ a1 = 800 rad/s, ˜ a2 = 1200 rad/s, ωs = 6000 rad/s. Solution k = 0.515957 ωp = 0.718302 n =4 ω0 = 1, 098.609 B = 371.9263Frame # 40 Slide # 58 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  59. 59. Example – BP Filter Cont’d 70 60 50 Passband loss, dB 40 Loss, dB 30 20 10 1.0 0 0 500 1000 1500 2000 Ω, rad/s ψ 800 900 1100 1200Frame # 41 Slide # 59 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  60. 60. Example – BS Filter Design a Chebyshev BS filter that would satisfy the following specifications: Ap = 0.5 dB, Aa = 40 dB, ˜ p 1 = 350 rad/s, ˜ p 2 = 700 rad/s, ˜ a1 = 430 rad/s, ˜ a2 = 600 rad/s, ωs = 3000 rad/s. Solution ωp = 1.0 n =5 ω0 = 561, 4083 B = 493, 2594Frame # 42 Slide # 60 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  61. 61. Example – BS Filter Cont’d 60 50 40 Passband loss, dB Loss, dB 30 20 10 0.5 0 0 200 400 800 1000 Ω, rad/s 350 430 600 700Frame # 43 Slide # 61 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  62. 62. D-Filter A DSP software package that incorporates the design techniques described in this presentation is D-Filter. Please see http://www.d-filter.ece.uvic.ca for more information.Frame # 44 Slide # 62 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  63. 63. Summary t A design method for IIR filters that leads to a complete description of the transfer function in closed form either in terms of its zeros and poles or its coefficients has been described.Frame # 45 Slide # 63 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  64. 64. Summary t A design method for IIR filters that leads to a complete description of the transfer function in closed form either in terms of its zeros and poles or its coefficients has been described. t The method requires very little computation and leads to very precise optimal designs.Frame # 45 Slide # 64 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  65. 65. Summary t A design method for IIR filters that leads to a complete description of the transfer function in closed form either in terms of its zeros and poles or its coefficients has been described. t The method requires very little computation and leads to very precise optimal designs. t It can be used to design LP, HP, BP, and BS filters of the Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.Frame # 45 Slide # 65 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  66. 66. Summary t A design method for IIR filters that leads to a complete description of the transfer function in closed form either in terms of its zeros and poles or its coefficients has been described. t The method requires very little computation and leads to very precise optimal designs. t It can be used to design LP, HP, BP, and BS filters of the Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types. t All these designs can be carried out by using DSP software package D-Filter.Frame # 45 Slide # 66 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  67. 67. References t A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 15, McGraw-Hill, 2005. t A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 15, McGraw-Hill, 2005.Frame # 46 Slide # 67 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
  68. 68. This slide concludes the presentation. Thank you for your attention.Frame # 47 Slide # 68 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method

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