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- 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. 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. 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. 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. 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. 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. Introduction Cont’dFrame # 5 Slide # 7 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
- 8. Introduction Cont’dFrame # 6 Slide # 8 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Design Procedure Cont’d A(ω) Aa Ap ω ωp ωc ωaFrame # 14 Slide # 19 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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