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# Digital signal processing By Er. Swapnil Kaware

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### Digital signal processing By Er. Swapnil Kaware

1. 1. A Seminar On, “Design of Digital IIR filter” Presented By, Mr. Swapnil V. Kaware, svkaware@yaoo.co.in svkaware@yahoo.co.in
2. 2. Introduction (i). Infinite Impulse Response (IIR) filters are the first choice when: Speed is paramount. Phase non-linearity is acceptable. (ii). IIR filters are computationally more efficient than FIR filters as they require fewer coefficients due to the fact that they use feedback or poles. (ii). However feedback can result in the filter becoming unstable if the coefficients deviate from their true values. svkaware@yahoo.co.in
3. 3. Filter Design Methods Butterworth :- Maximally Flat Amplitude. Chepyshev type I :- Equiriple in the passband. Chepyshev type II :- Equiriple in the stopband. Elliptic :- Equiripple in both the passband and stopband. svkaware@yahoo.co.in
4. 4. Design Procedure To fully design and implement a filter five steps are required: (1) (2) (3) (4) (5) Filter specification. Coefficient calculation. Structure selection. Simulation (optional). Implementation. svkaware@yahoo.co.in
5. 5. Filter Specification - Step 1 |H(f)| pass-band stop-band 1 f c : cut-off frequency f(norm) f s /2 (a) |H(f)| (dB) pass-band transition band |H(f)| (linear) stop-band ∆ p 1+ p δ 0 1 1− p δ pass-band ripple -3 stop-band ripple ∆ s f sb : stop-band frequency f c : cut-off frequency f pb : pass-band frequency (b) svkaware@yahoo.co.in δs f s /2 f(norm)
6. 6. IIR Filters  Better magnitude response (sharper transition and/or lower stopband attenuation than FIR with the same number of parameters: HW efficient)  Established filter types and design methods. IIR filter design procedure:1) Set up digital filter specification, 2) Determine the corresponding analog filter specification,  (frequency translation involved) 3) Design the analog filter, 4) Transform the analog filter to digital filter using various transformation methods,  Impulse invariant method  Bilinear transformation svkaware@yahoo.co.in
7. 7. IIR Filters Important parameters  Passband ripple :  Stopband attenuation :  Discrimination factor :  Selectivity factor : 1  (-3dB) cutoff frequency : 1 − p δ δ s ω p svkaware@yahoo.co.in ω s
8. 8. IIR Filters Frequency response  Transfer function : Rational  Asymptotic attenuation at high frequency  Attenuation function: (rational or polynomial function) : Square magnitude frequency response : reference frequency  If  If is monotone, so is is oscillatory, exhibits ripple. svkaware@yahoo.co.in
9. 9. IIR Filters Frequency response  For real rational transfer function   Stability requirement must include all poles of on the left half of the s plane and only those. Analog filter types  Butterworth  Chebyshev  Elliptic svkaware@yahoo.co.in
10. 10. Butterworth Filters The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and Physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers" svkaware@yahoo.co.in
11. 11. Butterworth Filters The frequency response of the Butterworth filter is maximally flat (i.e. has no ripples) in the passband and rolls off towards zero in the stopband. When viewed on a logarithmic Bode plot the response slopes off linearly towards negative infinity. A first-order filter's response rolls off at −6 dB per octave (−20 dB per decade) (all first-order lowpass filters have the same normalized frequency response). A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on. Butterworth filters have a monotonically changing magnitude function with ω, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband. svkaware@yahoo.co.in
12. 12. Butterworth Lowpass Filters • Passband is designed to be maximally flat. • The magnitude-squared function is of the form Hc ( jΩ) Hc ( s ) 2 2 = 1 1 + ( jΩ / jΩc ) Ωc 2N N The order of the filter 1 = 1 + ( s / jΩc ) 2N ΩcN ∏ (s − s k ) H a (s) = LHP poles s k = Ω ce j π ( 2 k + N +1 ) 2N , The Cutoff frequency ⋅ k = 0,1, ⋅ ⋅ 2 N − 1 svkaware@yahoo.co.in
13. 13. Butterworth filters Magnitude Squared Response | H (ω ) | 2 1 (1 − δ p ) 2 0.5 δ 2 s Properties of a LP Butterworth filter  Magnitude response : monotonically decreasing  Maximum gain : 0 at  : -3 dB point  Asymptotic attenuation at high frequency :  Maximally flat at DC (maximally flat filter) svkaware@yahoo.co.in ω p ω 0 ω s
14. 14. Butterworth filters Transfer function  2N poles:  : N poles are on the left side of the complex plane  All pole filter Im{s} Im{s} N= 3 Re{s}  Normalized transfer function : Nth-order LP Butterworth filter svkaware@yahoo.co.in N= 4 Re{s}
15. 15. Butterworth filters LP Butterworth filter design procedure 1. Set up filter spec : 2. Compute N, using 3. Choose using 4. Compute the poles 5. Compute , using , using svkaware@yahoo.co.in
16. 16. Chebyshev Filters Chebyshev filters are analog or digital filters having a steeper roll-offand more passbandripple(type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshv because its mathematical characteristics are derived from Chebyshev polynomials. Because of the passband ripple inherent in Chebyshev filters, the ones that have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications. svkaware@yahoo.co.in
17. 17. Chebyshev Filters 2 1 • Equiripple in the Hc (jΩ) = 2 1 + ε2 VN (Ω / Ωc ) passband and monotonic in the stopband. 1 • Or equiripple in the | H c ( jΩ ) | 2 = 2 1 + [ε 2VN (Ω / Ω c )]−1 stopband and monotonic in the passband. ( VN (x ) = cos N cos−1 x Type II Type I svkaware@yahoo.co.in )
18. 18. Chebyshev filters TN (x) Chebyshev polynomial of degree N = 1,2,3,4 Recursive formula: If N is even(odd), so is Monotone only in one band  Chebyshev Type I : equiripple in the passband  Chebyshev Type II : equiripple in the stopband Sharper than Butterworth due to the ripples ! Why ?  Sharpest if equiripple in both bands, pass- and stop-bands.  Phase response : Better for maximally flat or monotonic mag response filters svkaware@yahoo.co.in
19. 19. Chebyshev-I : Chebyshev Filter of the first kind Properties  All-pole filter  For  For  Monotonically decreasing because  asymptotic attenuation : svkaware@yahoo.co.in
20. 20. Chebyshev-I : Chebyshev Filter of the first kind Poles of a Nth-order LP Chebyshev-I filter π /N bω 0 aω 0 Transfer function (N=3) case svkaware@yahoo.co.in
21. 21. Chebyshev-II : Chebyshev Filter of the second kind Inverse Chebyshev filter or Chebyshev-II Properties  Passband : monotonic Stopband : equi-ripple  Contains both the poles and zeros for all : monotonically decreasing svkaware@yahoo.co.in
22. 22.  Elliptic filter (i). An elliptic filter (also known as a Cauer filter, named after  Wilhelm Cauer) is a signal processing filter with  equalized ripple (equiripple) behavior in both thepassband and  the stopband.  (ii). The amount of ripple in each band is independently adjustable,  and no other filter of equal order can have a faster transition  ingain between the passbandand the stopband, for the given  values of ripple (whether the ripple is equalized or not).  (iii). Alternatively, one may give up the ability to independently  adjust the passband and stopband ripple, and instead design a  filter which is maximally insensitive to componenAn elliptic filter (also known as a Cauer filter,  svkaware@yahoo.co.in
23. 23. Elliptic Filters Overview  Equiripple in both the passband and the stop band  Minimum possible order for a given spec : Sharpest (optimum) Magnitude Squared Response: LP elliptic filter    : Jacobian elliptic function of degree N Even(odd) function of         for even(odd)  For                              ,              oscillates between -1 and +1  oscillates between 1 and  for For oscillates between and svkaware@yahoo.co.in for
24. 24. Elliptic Filters Example N= 2 N= 3 svkaware@yahoo.co.in
25. 25. Elliptic Filters Equiripple both in stopband and in the passband | H c ( jΩ) |2 = 1 2 1 + ε 2U N (Ω) Jacobian Elliptic function
26. 26. Frequency transformation Analog filter design 1. Design a LPF (Butterworth, Chebyshev, elliptic) 2. Frequency transformation to obtain HPF, BPF, BRF Definitions  : rational function (  Transfer function of a LP filter :  Transformed filter , )            : rational function of Class and stability of the filter              is preserved after transformation.  Design domain :  Target domain :   svkaware@yahoo.co.in
27. 27. Frequency transformation LP to LP transformation LP to HP transformation LP to BP transformation LP to BS transformation svkaware@yahoo.co.in
28. 28. Digital IIR filter design Digital IIR filter design 1. Digital filter spec -> analog filter spec 2. Design analog filter 3. Transformation : Analog filter to digital filter Transformation Goal  Requirements for  Real, causal, stable, rational  The order of           should not be greater than that of           if possible.                  should be close to             where   •transform should be simple, convenient to implement and applicable  to all analog filter types and classes svkaware@yahoo.co.in
29. 29. Digital IIR filter design Impulse Invariant Transformation  Definition  Procedure 1.  2.  3.   High-pass filter cannot be transformed !!  Filter orders are not changed After transformation Example) svkaware@yahoo.co.in
30. 30. Digital IIR filter design Bilinear Transform  Definition and Properties (Approximation of continuous-time integration by discrete-time trapezoidal integration)  For 1) # of poles are preserved. => Preserve the filter order 2) # of zeros increase from q to p if p > q (p-1 zeros at z=-1) svkaware@yahoo.co.in
31. 31. Digital IIR filter design Bilinear Transform  Definition and Properties If => Preserve the stability θ Im{s} Re{s} π ω 1 z-plane svkaware@yahoo.co.in −π
32. 32. Digital IIR filter design Bilinear Transform  Definition and Properties For 1) Frequency warping : One-to-one mapping, 2) 3) Can be used for all filter types θ Im{s} Re{s} π ω 1 z-plane svkaware@yahoo.co.in −π
33. 33. Digital IIR filter design Bilinear Transform  Prewarping  Prewarp the analog frequencies -> Bilinear transform -> desired digital frequencies (A) For convenience, set => Prewarping -> BLT gives the same result.  IIR filter Design procedure using BLT 1. Convert each specified band-edge frequency of the digital filter to a corresponding band-edge freq of an analog filter, using (A) - Leave the ripple values unchanged. 2. Design 3. using BLT  svkaware@yahoo.co.in
34. 34. Bilinear Transformation Transformation is unaffected by scaling. Consider inverse transformation with scale factor equal to unity For z =1+ s 1− s s = σo + jΩo 2 (1 + σ o ) + jΩo (1 + σ o ) 2 + Ωo 2 z= ⇒z = and so 2 (1 − σ o ) − jΩo (1 − σ o ) 2 + Ωo σo = 0 → z =1 σo < 0 → z <1 σo > 0 → z >1 svkaware@yahoo.co.in
35. 35. Bilinear Transformation Mapping of s-plane into the z-plane svkaware@yahoo.co.in
36. 36. Bilinear Transformation For or z =e jω with unity scalar we have 1 − e − jω = j tan(ω / 2) jΩ = − jω 1+ e Ω = tan(ω / 2) svkaware@yahoo.co.in
37. 37. Bilinear Transformation Mapping is highly nonlinear Complete negative imaginary axis in the s-plane from Ω = −∞ toΩ = 0 is mapped into the lower half of the unit circle in the z-plane from = −1 to = 1 z z Complete positive imaginary axis in the s-plane from Ω = 0 toΩ = ∞ is mapped into the upper half of the unit z to −1 = circle in the z-plane from 1 z= svkaware@yahoo.co.in
38. 38. Bilinear Transformation Nonlinear mapping introduces a distortion in the frequency axis called frequency warping Effect of warping shown below, svkaware@yahoo.co.in
39. 39. References (1). J.G. Proakis and D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Prentice Hall, 3rd Edition, 1996, ISBN 013373762- 4. (2). S.S. Soliman and M.D. Srinath, Continuous and Discrete Signals and Systems, Prentice Hall, 1998, ISBN 013518473-8. (3). A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice Hall, 1975, ISBN 013214635-5. (4). L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing, Prentice Hall, 1975, ISBN 013914101-4. (5). E.O. Brigham, The Fast Fourier Transform and Its Applications, Prentice Hall, 1988, ISBN 013307505-2. (6). M.H. Hayes, Digital Signal Processing , Schaum’s Outline Series, McGraw Hill, 1999, ISBN 0-07027389-8 svkaware@yahoo.co.in
40. 40. Thank You!! svkaware@yahoo.co.in