1. 1 BIEN425 – Lecture 13 By the end of the lecture, you should be able to: –Outline the general framework of designing an IIR filter using frequency transform and bilinear transform –Describe the differences between various classical analog filter (Butterworth, Chebyshev-I, Chebyshev-II and Elliptic) characteristics –Design classical analog filters (Butterworth, Chebyshev-I, Chebyshev-II and Elliptic)

2. 2 Design IIR filters by prototype filters Most widely used design procedure Filter design parameters obtained from filter design specifications Recall: F p, F s,  p,  s

3. 3 Selectivity and Discrimination Selectivity factor (r) Discrimination factor (d) Ideal filter (r = 1, d = 0)

4. 4 Analog filter 1 - Butterworth

5. 5 Butterworth Magnitude response – A a (f) –F c is called 3-dB cut-off frequency The poles of Ha(s) are:

6. 6 Butterworth Laplace transform H a (s) The passband and stopband constraints are:

7. 7 Butterworth Selecting the order (n) and the cutoff frequency (F c )

8. 8 Analog filter 2 - Chebyshev-I

9. 9 Chebyshev-I Magnitude response – A a (f) –Where Tk+1(x) is called Chebyshev polynomial which is expressed recursively Because Tn(1)=1, we can define the ripple factor 

10. 10 Chebyshev-I The poles are on a ellipse Laplace transform H a (s) –Where  is defined as (-1) n p 0 p 1 p 2 …p n-1 –A a (0) is the DC gain Order (n) is determined by

11. 11 Analog filter 3 - Chebyshev-II

12. 12 Chebyshev-II Magnitude response – A a (f) Ripple factor

13. 13 Chebyshev-II Laplace transform H a (s) –Where b = sum of poles / sum of zeros –Poles are located at the reciprocals of the poles of Chebyshev-I –Zeros are located along the imaginary axis –Order (n) is computed the same way as Chebyshev-I

14. 14 Analog filter 4 - Elliptic

15. 15 Elliptic Magnitude response – A a (f) –U n is n-th order Jacobian elliptic function

16. 16 Elliptic Finding the poles and zeros of elliptic filter requires iterative solution of nonlinear algebraic equations Order (n)

17. 17 Comparison Analog Filter PassbandStopbandTransition Band Specificatio n Butterworth Monotonic BroadPass/Stop- band Chebyshev-I EquirippleMonotonicNarrowPassband Chebyshev- II MonotonicEquirippleNarrowStopband Elliptic Equiripple Very Narrow Passband

18. 18 General method 1

19. 19 General method 2

20. 20 Using frequency + bilinear transform We will cover this in the next lecture Method 1: –Normalized lowpass (analog) –Frequency transformation to LP,HP,BP,BS (analog) –Bilinear transformation (digital) Method 2: –Normalized lowpass (analog) –Bilinear transformation lowpass (digital) –Frequency transform to LP,HP,BP,BS (digital)