Part 3: IIR Filters Bilinear Transformation Method. Tutorial ISCAS 2007

Transcription

1 Part 3: IIR Filters Bilinear Transformation Method Tutorial ISCAS 2007 Copyright 2007 Andreas Antoniou Victoria, BC, Canada July 24, 2007 Frame # 1 Slide # 1 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

2 Introduction 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 A procedure for the design of IIR filters that would satisfy arbitrary prescribed specifications will be described. 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 H A (s), a digital filter with a discrete-time transfer function H D (z) can be readily deduced by applying the bilinear transformation as follows: H D (z) = H A (s) s= 2 T ( ) (A) z 1 z+1 Frame # 3 Slide # 4 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

5 Introduction Cont d The bilinear transformation method has the following important features: 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: A stable analog filter gives a stable digital filter. 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 d Frame # 5 Slide # 7 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

8 Introduction Cont d Frame # 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) z 1 z+1 s= 2 T gives the frequency response of the digital filter as a function of the frequency response of the analog filter as provided that s = 2 T or jω = 2 T H D (e j T ) = H A (jω) ( ) z 1 z + 1 ( e j T ) 1 e j T + 1 or ω = 2 T tan T 2 (B) Frame # 7 Slide # 9 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

10 The Warping Effect Cont d ω = 2 T tan T 2 (B) 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 ω = 2 T tan T 2 (B) 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. 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 = ω 2.0 H A (jω) 0.1π 0.2 π 0.3 π 0.4 π 0.5π Ω, rad/s H D (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: Loss, db Digital filter Analog filter ω, rad/s Frame # 10 Slide # 13 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

14 Prewarping From Eq. (B), i.e., ω = 2 T tan T 2 (B) a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence = 2 T tan 1 ωt 2 Frame # 11 Slide # 14 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

15 Prewarping From Eq. (B), i.e., ω = 2 T tan T 2 (B) a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence = 2 T tan 1 ωt 2 If ω 1,ω 2,...,ω i,...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 i = 2 T tan 1 ω it 2 i = 1, 2,... Frame # 11 Slide # 15 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

16 Prewarping Cont d 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 ω i = 2 T tan i T 2 Frame # 12 Slide # 16 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

17 Prewarping Cont d 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 ω i = 2 T tan i T 2 Then the band edges of the digital filter would assume their prescribed values i since i = 2 ω it T tan 1 ( 2 = 2 T T tan T tan ) i 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 H N (s) with an attenuation 1 A N (ω) = 20 log H N (jω) (also known as loss) and assume that 0 A N (ω) A p for 0 ω ω p A N (ω) A a for ω a 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) A p A a ω ω p ω c ω a Frame # 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 = f X ( s) H X ( s) = H N (s) s=fx ( s) where f X ( 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 = f X ( s) H X ( s) = H N (s) s=fx ( s) where f X ( s) is one of the four standard analog-filters transformations, given by the next slide. 2. Apply the bilinear transformation to H X ( s), i.e., H D (z) = H X ( s) s= 2 T ( z 1 z+1 ) Frame # 15 Slide # 21 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

22 Design Procedure Cont d Standard forms of f X ( s) X LP HP BP BS 1 B f X ( s) λ s λ/ s ( s + ω2 0 s B s s 2 + ω 2 0 ) Frame # 16 Slide # 22 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

23 Design Procedure Cont d The digital filter designed by this method will have the required passband and stopband edges only if the parameters λ, ω 0, and B of the analog-filter transformations and the order of the continuous-time normalized LP transfer function, H N (s), are chosen appropriately. Frame # 17 Slide # 23 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

24 Design Procedure Cont d The digital filter designed by this method will have the required passband and stopband edges only if the parameters λ, ω 0, and B of the analog-filter transformations and the order of the continuous-time normalized LP transfer function, H N (s), are chosen appropriately. 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, H N (s), is available that would give the required passband ripple, A p, and minimum stopband attenuation (loss), A a. Let the passband and stopband edges of the analog filter be ω p and ω a, respectively. Frame # 18 Slide # 25 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

26 Design of LP Filters Cont d A(ω) A p A a ω ω p ω c ω a Attenuation characteristic of continuous-time normalized LP filter Frame # 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 H N (s) to obtain a denormalized discrete-time transfer function H LP ( 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 H N (s) to obtain a denormalized discrete-time transfer function H LP ( s). 3. Apply the bilinear transformation to H LP ( s) to obtain a discrete-time transfer function H D (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 H N (s) to obtain a denormalized discrete-time transfer function H LP ( s). 3. Apply the bilinear transformation to H LP ( s) to obtain a discrete-time transfer function H D (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) A p A a ~ Ω p Ω p Attenuation characteristic of required LP digital filter Ω a ~ Ω a Ω Frame # 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 ω p /ω a for the continuous-time normalized LP transfer function. The ratio ω p /ω a 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 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 LP ω p ω a K 0 ω p T λ = 2tan( p T /2) HP ω p ω a 1 K 0 λ = 2ω p tan( p T /2) T where K 0 = tan( p T /2) tan( a T /2) Frame # 24 Slide # 35 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

36 Design of LP Filters Cont d The table of formulas presented is applicable to all the classical types of analog filters, namely, Butterworth Chebyshev Inverse-Chebyshev Elliptic Frame # 25 Slide # 36 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

37 Design of LP Filters Cont d The table of formulas presented is applicable to all the classical types of analog filters, namely, Butterworth Chebyshev Inverse-Chebyshev Elliptic 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, H N (s), is available that would give the required passband ripple, A p, and minimum stopband attenuation, A a. Let the passband and stopband edges of the analog filter be ω p and ω a, 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 H N (s) to obtain a denormalized discrete-time transfer function H BP ( 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 H N (s) to obtain a denormalized discrete-time transfer function H BP ( s). 3. Apply the bilinear transformation to H BP ( s) to obtain a discrete-time transfer function H D (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 where a1 a1 a2 a2 p1 and p2 are the actual lower and upper passband edges, p1 and p2 are the prescribed lower and upper passband edges, a1 and a2 are the actual lower and upper stopband edges, p1 and p2 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) A a A p A a ~ ~ Ωa1 Ω p1 Ω p2 Ωa2 Ωa1 ~ ~ Ω Ω a2 p1 Ω p2 Attenuation characteristic of required BP digital filter Frame # 29 Slide # 42 A. Antoniou Part3: IIR Filters Bilinear Transformation Method Ω

43 Design of LP Filters Cont d 5. Solve for B and ω 0, 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 ω 0, the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ω p /ω a, 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 ω 0, the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ω p /ω a, 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 ω 0, the parameters of the LP-to-BP analog-filter transformation. 6. Find the minimum value for the selectivity, i.e., the ratio ω p /ω a, 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 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 ω 0 = 2 K B T BP ω p ω a { K1 if K C K B K 2 if K C < K B where B = 2K A T ω p K A = tan p2 T 2 K C = tan a1 T 2 tan p1 T 2 tan a2 T 2 K 2 = K A tan( a2 T /2) tan 2 ( a2 T /2) K B K B = tan p1 T 2 tan p2 T 2 K 1 = K A tan( a1 T /2) K B tan 2 ( a1 T /2) Frame # 31 Slide # 47 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

48 Formulas for the Design of BS Filters BS ω 0 = 2 K B T 1 ω if K C K B p K 2 ω a 1 if K C < K B K 1 where B = 2K Aω p T K A = tan p2 T 2 K C = tan a1 T 2 tan p1 T 2 tan a2 T 2 K 2 = K A tan( a2 T /2) tan 2 ( a2 T /2) K B K B = tan p1 T 2 tan p2 T 2 K 1 = K A tan( a1 T /2) K B tan 2 ( a1 T /2) Frame # 32 Slide # 48 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

49 Formulas for ω p and n The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 A N (ω) A p for 0 ω ω p A N (ω) A a for ω a ω Frame # 33 Slide # 49 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

50 Formulas for ω p and n The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 A N (ω) A p for 0 ω ω p A N (ω) A a for ω a ω However, the values of the normalized passband edge, ω p, 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 ω p and n The formulas presented apply to any type of normalized analog LP filter with an attenuation that would satisfy the following conditions: 0 A N (ω) A p for 0 ω ω p A N (ω) A a for ω a ω However, the values of the normalized passband edge, ω p, and the required filter order, n, depend on the type of filter. 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 = K 0 HP K = 1 K 0 BP K = { K1 if K C K B K 2 1 BS K = 1 K 2 K 1 if K C < K B if K C K B if K C < K B log D n 2log(1/K ), D = 100.1Aa A p 1 ω p = (10 0.1A p 1) 1/2n Frame # 34 Slide # 52 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

53 Formulas for Chebyshev Filters LP K = K 0 HP K = 1 K 0 BP K = { K1 if K C K B K 2 1 BS K = 1 K 2 K 1 if K C < K B if K C K B if K C < K B n cosh 1 D cosh 1 (1/K ), D = 100.1Aa A p 1 ω p = 1 Frame # 35 Slide # 53 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

54 Formulas for Elliptic Filters k ω p LP K = K 0 K0 HP K = 1 K 0 1 K0 BP K = { K1 if K C K B K 2 1 BS K = 1 K 2 K 1 if K C < K B if K C K B if K C < K B K1 K2 1 K2 1 K1 n cosh 1 D cosh 1 (1/K ), D = 100.1Aa A p 1 Frame # 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: A p = 1dB, A a = 45 db, p = 3.5 rad/s, a = 1.5 rad/s, ω s = 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 ω p λ Butterworth Chebyshev Elliptic Frame # 38 Slide # 56 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

57 Example Cont d Elliptic (n=3) Loss, db Butterworth (n=5) Chebyshev (n=4) Passband loss, db Ω, rad/s Frame # 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: A p = 1dB, A a = 45 db, p1 = 900 rad/s, p2 = 1100 rad/s, a1 = 800 rad/s, a2 = 1200 rad/s, ω s = 6000 rad/s. Solution k = ω p = n = 4 ω 0 = 1, B = Frame # 40 Slide # 58 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

59 Example BP Filter Cont d Loss, db Passband loss, db Ω, rad/s ψ Frame # 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: A p = 0.5 db, A a = 40 db, p1 = 350 rad/s, p2 = 700 rad/s, Solution a1 = 430 rad/s, a2 = 600 rad/s, ω s = 3000 rad/s. ω p = 1.0 n = 5 ω 0 = 561, 4083 B = 493, 2594 Frame # 42 Slide # 60 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

61 Example BS Filter Cont d Loss, db Passband loss, db Ω, rad/s Frame # 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 for more information. Frame # 44 Slide # 62 A. Antoniou Part3: IIR Filters Bilinear Transformation Method

63 Summary 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 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. 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 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. The method requires very little computation and leads to very precise optimal designs. 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 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. The method requires very little computation and leads to very precise optimal designs. It can be used to design LP, HP, BP, and BS filters of the Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types. 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 A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 15, McGraw-Hill, A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 15, McGraw-Hill, 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