1. 1 Teaching Innovation - Entrepreneurial - Global The Centre for Technology enabled Teaching & Learning, MGI,India DTEL DTEL (Department for Technology Enhanced Learning)

2. DEPARTMENT OF ELECTRONICS ENGINEERING B. E. VI-SEMESTER DIGITAL SIGNAL PROCESSING 2 CHAPTER NO.4 IIR Filter Design & Realization

3. CHAPTER 4:- SYLLABUSDTEL. 1234 3 5 Topic 1 : Filter design methods Topic 2 : Designing of Butterworth filters Topic 3 : Designing of Chebyshev filters Topic 4 : Frequency Transformations Topic 5 : IIR filter structures

4. CHAPTER-4 SPECIFIC OBJECTIVE / COURSE OUTCOMEDTEL 12 4 The student will be able to: Design digital IIR filters Characterize & design of Chebyshev filters Characterize & design of Butterworth filters 34 Understand Frequency Transformations 5 Realize various IIR filter structures

5. DTEL 5 Filter design methods Magnitude response specifications in the passband and the stopband are given with some acceptable. A transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly   p - passband edge frequency.   s - stopband edge frequency.   p - peak ripple in the passband.   s - peak ripple in the stopband. LECTURE 1:- INTRODUCTION

6. DTEL 6 Digital Filter Design Transform it into the desired digital filter transfer function G(z). Convert the digital filter specifications into analog low pass prototype one. Determine the analog lowpass filter H a (s) meeting these specifications. Basic Approaches to Digital Filter Design LECTURE 1:- BASIC APPROACH

7. DTEL 7 IIR Filter Design Why this approach has been widely used: (1) Analog approximation techniques are highly advanced. (2) They usually yield closed-form solutions. (3) Extensive tables are available for analog filter design. (4) Many applications require digital simulation of analog systems.

8. DTEL 8 LECTURE 1:- TRANSFORMATION Order of the Filter After the type of the digital filter has been selected, the next step in the filter design process is to estimate the filter order N. For the design of IIR LPF, the order of H a (s) is estimated from its specifications using the appropriate formula depending on which approximation desired. Then the order of G(z) can be determined automatically from the transformation being used to convert H a (s) into G(z). LECTURE 1:- ESTIMATION

9. DTEL IIR filter structures 9 The IIR filter difference equation is recursive in nature: the current output depends upon the previous output. Since the current output depends upon the previous output and the previous output depends upon its previous output, the output depends upon the infinite past. Most analog filters have an impulse response which is infinite in duration. IIR filters are generally designed by emulating an analog prototype filter. LECTURE 1:- CLASSIFICATION

10. DTEL IIR filter structures 10 LECTURE 1:- CLASSIFICATION THANK YOU

11. DTEL 11 IIR Filter Design Most analog filters have an impulse response which is infinite in duration. IIR filters are generally designed by emulating an analog prototype filter. There are two methods for doing this analog filter emulation: The method involves designing an analogue filter and then transforming it to a digital filter. (1) the matched z-transform or impulse invariant transform (2) the bilinear transformation. In both cases, we are given an analog transfer function H(s), and we transform this function into a digital transfer function H(z). LECTURE 2:- METHODS

12. DTEL 12 The Matched z-Transform In the matched z-transform digital filter design method we try to “match” the impulse response of the analog filter with that of the digital filter being designed. To match the impulse responses, we take the inverse Laplace transform of the analog filter H(s)  h(t), then sample the impulse response h(t)  h[n], then take the z- transform of the sampled impulse response to get the z- transform transfer function h[n]  H(z). IIR Filter Design LECTURE 2:- MATCHED Z-TRANSFORM

13. DTEL 13 Once we have our z-transform transfer function H(z), we apply the definition of the transfer function to write our digital filter equations: IIR Filter Design LECTURE 2:- MATCHED Z-TRANSFORM

14. DTEL 14 IIR Filter Design LECTURE 2:- MATCHED Z-TRANSFORM THANK YOU

15. DTEL 15 Fig: Frequency Warping IIR Filter Design LECTURE 3:- FREQUENCY WARPING

16. DTEL 16 As we can see from the graph, the following function does perform the necessary mapping. We know the relationship between and w; what is the relationship between s and z? IIR Filter Design LECTURE 3:- FREQUENCY WARPING

17. DTEL 17 Since s=jW and z=e jw, we have IIR Filter Design LECTURE 3:- FREQUENCY WARPING

18. DTEL 18 IIR Filter Design LECTURE 3:- FREQUENCY WARPING THANK YOU

19. DTEL 19 To obtain G(z) replace s by f(z) in H(s) Start with requirements on G(z) G(z) G(z) Available H(s) Stable Real and Rational in zReal and Rational in s Order n L.P. (lowpass) cutoffL.P. cutoff LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design

20. DTEL 20 Mapping of s-plane into the z-plane LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design

21. DTEL 21 For with unity scalar we have or LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design

22. DTEL 22 Mapping is highly nonlinear Complete negative imaginary axis in the s-plane from to is mapped into the lower half of the unit circle in the z-plane from to Complete positive imaginary axis in the s-plane from to is mapped into the upper half of the unit circle in the z-plane from to LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design

23. DTEL 23 Nonlinear mapping introduces a distortion in the frequency axis called frequency warping effect of warping shown below LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design

24. DTEL 24 LECTURE 4:-BILINEAR TRANSFORMATION IIR Filter Design THANK YOU

25. DTEL 25 The magnitude-square response of an N-th order analog low pass Butterworth filter is given by: The Butterworth low pass filter thus is said to have a maximally-flat magnitude at  = 0. Two parameters completely characterizing a Butterworth low pass filter are  c and N. Where;  c = cutoff frequency, N = filter order LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design

26. DTEL 26 Gain in dB is: G(  )=10log 10 |H a (j  )| 2 As G(0)=0 G(  c )=10log 10 (0.5)  -3 dB  c is called 3-dB cutoff frequency. Typical magnitude responses with  c =1 are shown in right. LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design

27. DTEL 27 These are determined from the specified band edges  p and  s, and minimum pass band magnitude 1/  (1 +  2), and maximum stop band ripple 1/A. Ω c and N are thus determined from: LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design

28. DTEL 28 Solving the above, we get: Transfer function of an analog Butterworth low pass filter is given by: Where: Denominator D N (s) is known as the Butterworth polynomial of order N. LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design

29. DTEL 29 Example: Determine the lowest order of a Butterworth lowpass filter with a 1-dB cutoff frequency at 1 kHz and minimum attenuation of 40 dB at 5 kHz. Now, 10log 10 [1/(1+ε 2 )]= -1, Which yields ε 2 =0.25895; And 10log 10 (1/A 2 )=-40, Which yields A 2 =10000; Therefore 1/k1=√(A 2 -1)/ε=196.51334 And 1/k=Ωs/Ωp=5 Hence N=log10(1/k1)/log10(1/k)=3.2811 Choose N=4. LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design

30. DTEL 30 LECTURE 5:-BUTTERWORTH FILTER IIR Filter Design THANK YOU

31. DTEL 31 The magnitude-square response of an N-th order analog low pass Type 1 Chebyshev filter is given by: where T N (  ) is the Chebyshev polynomial of order N: Order N is chosen as the nearest integer greater than or equal to the above value. LECTURE 6:-CHEBYSHEV FILTER IIR Filter Design

32. DTEL 32 Typical magnitude response plots of the analog low pass Type 1 Chebyshev filter are shown below: LECTURE 6:-CHEBYSHEV FILTER IIR Filter Design

33. DTEL 33 If at  =  s the magnitude is equal to 1/A, then Solving the above we get: Order N is chosen as the nearest integer greater than or equal to the above value. LECTURE 6:-CHEBYSHEV FILTER IIR Filter Design

34. DTEL 34 LECTURE 6:-CHEBYSHEV FILTER IIR Filter Design THANK YOU

35. DTEL 35 The magnitude-square response of an N-th order analog low pass Type 2 Chebyshev (also called inverse Chebyshev) filter is given by: where T N (  ) is the Chebyshev polynomial of order N. LECTURE 7:-CHEBYSHEV FILTER IIR Filter Design

36. DTEL 36 Typical magnitude response plots of the analog low pass Type 2 Chebyshev filter are shown below: LECTURE 7:-CHEBYSHEV FILTER IIR Filter Design

37. DTEL 37 The order N of the Type 2 Chebyshev filter is determined from given ,  s, and A using: Example - Determine the lowest order of a Chebyshev lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz. LECTURE 7:-CHEBYSHEV FILTER IIR Filter Design

38. DTEL 38 LECTURE 7:-CHEBYSHEV FILTER IIR Filter Design THANK YOU

39. DTEL 39 Frequency Transformation We need to apply a suitable frequency transformation, if we wish to design band pass, band stop and high-pass filters, using the low-pass approximating function analysis previously covered. The block diagram, shown in below figure, illustrates the procedure, which produces, from the specification supplied, the required high-pass approximating function. Fig: System transform approach LECTURE 8:- INTRODUCTION

40. DTEL 40 Before we consider frequency transform techniques, lets consider the second order series-tuned LCR circuit shown in figure. Here we obtain expressions for the standard second-order transfer functions for low-pass, high-pass, and band pass filter configurations. Fig: Second - order band pass tuned circuit Frequency Transformation LECTURE 8:- INTRODUCTION

41. DTEL 41 The standard low-pass transfer function: Consider the tuned circuit configured with the output voltage measured across the capacitor C. The transfer function is: Reworking the equation, multiply above and below by sC and divide both sides by the input voltage. Frequency Transformation LECTURE 8:- TRANSFER FUNCTION

42. DTEL 42 As usual, making the coefficient of unity by dividing above and below by LC yields: This is a second-order system. Consider the generalized second-order transfer function as: We can, by comparing both equations, define the pole frequency and the Q – factor (=1/damping) as: Frequency Transformation LECTURE 8:- TRANSFER FUNCTION

43. DTEL 43 Frequency Transformation LECTURE 8:- TRANSFER FUNCTION THANK YOU

44. DTEL 44 The second-order bandpass transfer function The transfer function for the voltage measured across R is: In standard form Frequency Transformation LECTURE 9:- TRANSFER FUNCTION

45. DTEL 45 The standard high-pass transfer function Consider the output voltage across the inductance L. The transfer function is: Comparing this second-order high-pass function to the standard high-pass transfer function form: If we apply the simple frequency transformation to the low–pass second order circuit, we see the transfer function has the form of a high-pass function. Frequency Transformation LECTURE 9:- TRANSFER FUNCTION

46. DTEL 46 Low-pass to high-pass frequency transformation The specification for a high-pass filter includes the pass band edge frequency, ω hp, and the stop band edge frequency, ω hs The maximum pass band attenuation is Amax and the minimum stop band attenuation is Amin. The transformation of the high-pass specification to an equivalent normalized low-pass specification is achieved by applying the frequency transform S L = 1/s, where S L is the low-pass normalized complex frequency variable. However to account for the process of normalization we must replace S L by 1/(s/ω p) =ωp /s. Frequency Transformation LECTURE 9:- PROCEDURE

47. DTEL 47 Frequency Transformation LECTURE 9:- PROCEDURE THANK YOU

48. DTEL 48 In terms of real frequencies: Fig: Normalized HPF response Frequency Transformation LECTURE 10:- PROCEDURE

49. DTEL 49 We define values for the transformed frequency Ω as: Fig: Equivalent LPF response Frequency Transformation LECTURE 10:- PROCEDURE

50. DTEL 50 We define values for the transformed frequency Ω as: We calculate the value for ε and n for the filter type chosen using the equivalent normalized low-pass obtained previously. From these two quantities, we can then obtain the normalized attenuation function. Frequency Transformation LECTURE 10:- PROCEDURE

51. DTEL 51 This must be frequency de-normalized and the, low-pass to high-pass frequency transformation, performed. To demoralize a Butterworth approximation loss function we used but with frequency transformation, we substitute for ω/ωp with In the Chebychev case, we apply the substitution Frequency Transformation LECTURE 10:- PROCEDURE

52. DTEL 52 Frequency Transformation LECTURE 10:- PROCEDURE THANK YOU

53. DTEL 53 Basic IIR Digital Filter Basic IIR Digital Filter Structures The causal IIR digital filters we are concerned with in this course are characterized by a real rational transfer function of or, equivalently by a constant coefficient difference equation From the difference equation representation, it can be seen that the realization of the causal IIR digital filters requires some form of feedback An N-th order IIR digital transfer function is characterized by 2N+1 unique coefficients, and in general, requires 2N+1 multipliers and 2N two-input adders for implementation LECTURE 11:-REALIZATION IIR Filter Design

54. DTEL 54 Direct form IIR filters Filter structures in which the multiplier coefficients are precisely the coefficients of the transfer function Consider for simplicity a 3rd-order IIR filter with a transfer function We can implement H(z) as a cascade of two filter sections as shown on the next slide LECTURE 11:- CASCADE REALIZATION IIR Filter Design

55. DTEL 55 where The filter section can be seen to be an FIR filter and can be realized as shown in next slide LECTURE 11:- CASCADE REALIZATION IIR Filter Design

56. DTEL 56 Fig: Direct Form IIR Digital Filter Direct Form IIR Digital Filter Structures LECTURE 11:-DIRECT FORM REALIZATION IIR Filter Design

57. DTEL 57 The time-domain representation Fig: Realization H 2 (z) LECTURE 11:- DIRECT FORM REALIZATION IIR Filter Design

58. DTEL 58 LECTURE 11:- DIRECT FORM REALIZATION IIR Filter Design THANK YOU

59. DTEL 59 A cascade of the two structures realizing and leads to the realization of shown below and is known as the direct form I structure Fig: direct form I structure LECTURE 12:- DF-I REALIZATION IIR Filter Design

60. DTEL 60 The direct form I structure is noncanonic as it employs 6 delays to realize a 3rd-order transfer function A transpose of the direct form I structure is shown on the right and is called the direct form I t structure Fig: direct form I t structure LECTURE 12:-DF-I(T)REALIZATION IIR Filter Design

61. DTEL 61 Various other noncanonic direct form structures can be derived by simple block diagram manipulations as shown below LECTURE 12:- DF-II REALIZATION IIR Filter Design

62. DTEL 62 Observe in the direct form structure shown below, the signal variable at nodes 1 and 1’ and are the same, and hence the two top delays can be shared LECTURE 12:- DF-II REALIZATION IIR Filter Design

63. DTEL 63 Likewise, the signal variables at nodes 2 and 2’ and are the same, permitting the sharing of the middle two delays Following the same argument, the bottom two delays can be shared Sharing of all delays reduces the total number of delays to 3 resulting in a canonic realization shown on the next slide along with its transpose structure LECTURE 12:-DF-II(T) REALIZATION IIR Filter Design

64. DTEL 64 Fig: Direct Form IIFig: Direct Form II(Transpose) LECTURE 12:- DF-II(T) REALIZATION IIR Filter Design

65. DTEL 65 LECTURE 12:- DF-II(T) REALIZATION IIR Filter Design THANK YOU

66. DTEL 66 References Books:  PROAKIS, J. G., AND MANOLAKIS, D. G., “Digital Signal Processing: Principles, Algorithms & Application”(Prentice Hall, 2011).  RAMESH BABU, P., “Digital Signal Processing: Fourth Edition”, (Scitech Publication, 2009).  STEVE WHITE, “Digital Signal Processing”, (Centage Learning, 2008).  SALIVAHANAN, S., VALLAVARAJ, A., GNANAPRIYA, C., “Digital Signal processing” (Tata McGraw-Hills, 2008).  HAYES, M. H., “Digital Signal Processing” (Tata McGraw-Hill, 2008).  AMBARKAR, A., “Digital Signal Processing: A Modern Introduction”,(Thomson Edition, 2007).  MITRA, S. K., “Digital Signal Processing”,(Wiley, 1998).  RABINER, L. R., AND GOLD, B., “Theory and Application of Digital Signal Processing”, (Prentice Hall, 1974).