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

4 Analog filter 1 - Butterworth

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

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

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

8 Analog filter 2 - Chebyshev-I

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 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 Analog filter 3 - Chebyshev-II

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

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 Analog filter 4 - Elliptic

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

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

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 General method 1

19 General method 2

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)