Design of iir filters

1. Digital Signal Processing
(EE-384)
1
Ammar Naseer
Lecturer
University of Engineering & Technology, Lahore
New Campus Electrical Engineering
email: ammar@uet.edu.pk
Topic 09
Digital IIR Filter Design

2. Analog Filter
Based on the frequency responses analog filter are named as
• Butterworth filter
• Chebyshev filter
• Elliptic filter
• Bessel filter
2

3. Analog filter
3

4. Analog filter
4

5. Analog filter
• Important parameter for designing filter
5

6. Butterworth filter
|𝐻 Ω |2
=
1
1 +
Ω
Ω𝑐
2𝑁
6

7. Butterworth filter
The analog filter would be evaluated as
N=even
𝐻𝑎 𝑠 =
Ω𝑐
𝑁
𝑘=1
𝑁
2
(𝑠2+𝑏𝑘Ω𝑐𝑠 + Ω𝑐
2
)
N=odd
𝐻𝑎 𝑠 =
Ω𝑐
𝑁
(𝑠 + Ω𝑐) 𝑘=1
𝑁−1
2
(𝑠2+𝑏𝑘Ω𝑐𝑠 + Ω𝑐
2
)
where
𝑏𝑘 = 2 sin
2𝑘 − 1 𝜋
2𝑁
7

8. Butterworth filter
All these equations are evaluated considering Ω𝑝 ≠ Ω𝑐
𝑁 ≥
1
2
log[
𝛿𝑠
−2 − 1
𝛿𝑝
−2
− 1
]
log[Ω𝑠/Ω𝑝]
𝑠𝑘 = Ω𝑐𝑒
𝑗 𝑁+2𝑘+1 𝜋
2𝑁 𝑘 = 0, 1, … , 𝑁 − 1
If 𝛿𝑝 =-3dB
Ω𝑐 = Ω𝑝
𝛿𝑝 ≠-3dB
Ω𝑐 =
Ω𝑝
(
1
𝛿𝑝
2 − 1)
1
2𝑁
8

9. Example
Design a low pass butterworth filter that has passband -3dB bandwidth
at cutoff frequency Ω𝑐 = 1000𝜋 and stopband attenuation of 40dB at
Ω𝑠 = 2000𝜋.
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10. Chebyshev filter
|𝐻 Ω |2
=
1
1+∈2 𝐶𝑁
2 Ω
Ω𝑝
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11. Chebyshev filter
To design chebyshev filter we need to follow the steps
Step 1
𝑁 =
cosh−1(
1
𝛿𝑠
2 − 1
1
𝛿𝑝
2 − 1
)
cosh−1(
Ω𝑠
Ω𝑝
)
11

12. Chebyshev filter
To design chebyshev filter we need to follow the steps
Step 2: Finding values of major and minor axis
𝛿𝑝 =
1
1 + 𝜖2
𝜇 = 𝜖−1
+ 1 + 𝜖−2
Minor axis
𝑎 = Ω𝑝
(𝜇
1
𝑁 − 𝜇−
1
𝑁)
2
Major axis
𝑏 = Ω𝑝
(𝜇
1
𝑁 + 𝜇−
1
𝑁)
2
12

13. Chebyshev filter
To design chebyshev filter we need to follow the steps
Step 3: Find poles of chebyshev filter
𝑠𝑘 = a cos 𝜙𝑘 + 𝑗𝑏 sin 𝜙𝑘
where
𝜙𝑘 =
𝜋
2
+
2𝑘 + 1 𝜋
2𝑁
𝑘 = 0, 1, … . , 𝑁 − 1
13

14. Chebyshev filter
To design chebyshev filter we need to follow the steps
Step 4: Calculating the transfer function
The denominator is calculated as
𝑘=0
𝑁−1
𝑠 − 𝑠𝑘 𝑘 = 0, 1, … . , 𝑁 − 1
The numerator is calculated
N= odd
Substitute 𝑠 = 0
N=Even
Substitute 𝑠 = 0
and divide by 1 + 𝜖2
14

15. Example
Design a type 1 low pass butterworth filter that has 1dB ripple in the
passband, a cutoff frequency Ω𝑝 = 1000𝜋 and stopband attenuation of
40dB at Ω𝑠 = 2000𝜋.
15

16. IIR Filter Design
IIR filter are designed using Analog filter s → z transformation
• Impulse Invariant Design
• Bilinear transformation design
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17. Impulse Invariant Design
• Filter specification are given in discrete frequency 𝜔 but the filter is
design using analog frequency Ω
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• Hence the transformation is
performed using
𝜔 = Ω𝑇
where 𝑇𝑑 is the design sampling
period

18. Transformation Table
19

19. Example
We have an analog filter defined as
𝐻 𝑠 =
0.5 𝑠 + 4
(𝑠 + 1)(𝑠 + 2)
What would be its digital filter using impulse invariant method
How would a complex function be reduced
20

20. Steps
Butterworth Filter
Step 1: Mark 𝛿𝑠, 𝛿𝑝, 𝜔𝑠 and 𝜔𝑝
Step 2: Calculate Ω𝑠 and Ω𝑝 using
IID
Step 3: Calculate the order of filter
Step 4: Calculate Ω𝑐 (if required)
Step 5: Calculate the poles
Step 6: Find the transfer function
in s domain
Step 7: Compute transformation to
HP BP or BS filter (if required)
Step 8: Compute s to z
transformation using table
Chebyshev Filter
Step 1: Mark 𝛿𝑠, 𝛿𝑝, 𝜔𝑠 and 𝜔𝑝
Step 2: Calculate Ω𝑠 and Ω𝑝 using
IID
Step 3: Calculate the order of filter
Step 4: Finding value of minor and
major axis
Step 5: Calculate the poles
Step 6: calculate the numerator
Step 7: Find the transfer function
in s domain
Step 8: Compute transformation to
HP BP or BS filter (if required)
Step 9: Compute s to z
transformation using table
21

21. Aliasing
22
Need to set small
T to avoid aliasing

22. Bilinear Transformation
Design
The impulse invariant technique suffers from aliasing due to the many-
to-one mapping. To correct this problem employ a one-to-one mapping
which compresses the entire s-plane into a strip.
23

23. Bilinear Transformation
Design
• Bilinear transform equation can be defined as
𝑠 =
2
𝑇𝑑
1−𝑧−1
1+𝑧−1 or 𝑧 =
1+
𝑇𝑑
2
𝑠
1−
𝑇𝑑
2
𝑠
• In a practical design in order to preserve the desired discrete time
critical frequencies 𝜔𝑠 and 𝜔𝑝 the corresponding continuous-time
frequencies Ω𝑠 and Ω𝑝 must be pre-warped using
Ω =
2
𝑇
tan
𝜔
2
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24. Example
Design a single pole low pass digital filter with a 3dB bandwidth of 0.2π
using the bilinear transformation applied to the analog filter
𝐻 𝑠 =
Ω𝑐
𝑠 + Ω𝑐
25

25. Steps
Butterworth Filter
Step 1: Mark 𝛿𝑠, 𝛿𝑝, 𝜔𝑠 and 𝜔𝑝
Step 2: Calculate Ω𝑠 and Ω𝑝 using
BTD
Step 3: Calculate the order of filter
Step 4: Calculate Ω𝑐 (if required)
Step 5: Calculate the poles
Step 6: Find the transfer function
in s domain
Step 7: Compute transformation to
HP BP or BS filter (if required)
Step 8: Compute s to z
transformation using bilinear
transformation equation
Chebyshev Filter
Step 1: Mark 𝛿𝑠, 𝛿𝑝, 𝜔𝑠 and 𝜔𝑝
Step 2: Calculate Ω𝑠 and Ω𝑝 using
BTD
Step 3: Calculate the order of filter
Step 4: Finding value of minor and
major axis
Step 5: Calculate the poles
Step 6: calculate the numerator
Step 7: Find the transfer function
in s domain
Step 8: Compute transformation to
HP BP or BS filter (if required)
Step 9: Compute s to z
transformation using bilinear
transformation equation
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26. End of Lecture 09
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