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)
1
Design IIR filters by prototype filters
• Most widely used design procedure
• Filter design parameters obtained from filter design
specifications
• Recall: Fp, Fs, dp, ds
2
Selectivity and Discrimination
• Selectivity factor (r)
• Discrimination factor (d)
• Ideal filter (r = 1, d = 0)
3
Analog filter 1 - Butterworth
4
Butterworth
• Magnitude response – Aa(f)
– Fc is called 3-dB cut-off frequency
• The poles of Ha(s) are:
5
Butterworth
• Laplace transform Ha(s)
• The passband and stopband constraints are:
6
Butterworth
• Selecting the order (n) and the cutoff frequency (Fc)
7
Analog filter 2 - Chebyshev-I
8
Chebyshev-I
• Magnitude response – Aa(f)
– Where Tk+1(x) is called Chebyshev polynomial which is
expressed recursively
• Because Tn(1)=1, we can define the ripple factor e
9
Chebyshev-I
• The poles are on a ellipse
• Laplace transform Ha(s)
– Where b is defined as (-1)np0p1p2…pn-1
– Aa(0) is the DC gain
• Order (n) is determined by
10
Analog filter 3 - Chebyshev-II
11
Chebyshev-II
• Magnitude response – Aa(f)
• Ripple factor
12
Chebyshev-II
• Laplace transform Ha(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
13
Analog filter 4 - Elliptic
14
Elliptic
• Magnitude response – Aa(f)
– Un is n-th order Jacobian elliptic function
15
Elliptic
• Finding the poles and zeros of elliptic filter requires
iterative solution of nonlinear algebraic equations
• Order (n)
16
Comparison
Analog
Filter
Passband
Stopband
Transition
Band
Specificatio
n
Butterworth
Monotonic
Monotonic
Broad
Pass/Stopband
Chebyshev-I
Equiripple
Monotonic
Narrow
Passband
ChebyshevII
Monotonic
Equiripple
Narrow
Stopband
Elliptic
Equiripple
Equiripple
Very
Narrow
Passband
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General method 1
18
General method 2
19
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)
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