1. Chapter 8 Design of Infinite Impulse Response (IIR) Digital Filter

2. Basic properties of IIR filters
Recursive equation of IIR filter
Transfer function of IIR filter
(8-1)
where is impulse response of the filter,
and are coefficients of the filter
(8-2)

3. Factorized form of transfer function for IIR filter
Features of IIR filter
IIR filter is a feedback system of some sort
IIR filter normally requires fewer coefficients than FIR filter when sharp cutoff and high throughput are the important requirements
IIR filter can become unstable or its performance significantly degraded
(8-3)
where are the zeros of , and
are the poles of , and
is constant

4. Design of IIR filter using Analog Filter
Most common methods of converting analog filters into equivalent digital filter
Impulse invariant method
has equal impulse responses between equivalent digital filter and analog filter at sampling point
Bilinear z-transform
Analog transfer function in s-plane is mapped into the discrete transfer function in z-plane
jw axis in the s-plane is mapped onto the unit circle
brings frequency warping

5. Impulse Response Series
Impulse invariant method
Analog Filter
Transfer Function
Digital Filter
Impulse Response
Impulse Response Series
Fig. 8-1.

6. Design a LPF using Impulse invariant method
(8-4)
Fig. 8-2.

7. Impulse response using inverse Laplace transform
Impulse response of the equivalent digital filter
Sampling with period T (substituting t=nT)
Transfer function by using z-transform
(8-5)
(8-6)
(8-7)

8. Analog filter with single poles
Impulse response
Inverse Laplace transform
Impulse response of the equivalent digital filter
Sampling with period T(substituting t=nT)
(8-8)
where and are the real and imaginary part of
(8-9)
(8-10)

9. Transfer function of digital filter by using z-transform
Using commutative law
(8-11)
(8-12)

10. Using an infinite series
Designed digital filter by using impulse invariant method
(8-13)
(8-14)

11. Analog filter with repeated poles
Transfer function with repeated poles of l-th order
Using z-transform
(8-15)
(8-16)

12. Analog filter with complex conjugate poles (1)
Transfer function
Using z-transform
(8-17)
(8-18)

13. Analog filter with complex conjugate poles (2)
Transfer function
Using z-transform
(8-19)
(8-20)

14. Example 8-1 Transfer function for second-order Butterworth filter
Using partial fraction
Impulse response of the equivalent digital filter (T=1)

15. Magnitude response
Fig. 8-3.

16. Summary of impulse invariance method
The analog filter is bandlimited before the impulse invariant method is applied
To have the same filter gain in digital and analog filters, multiply H(z) by T
(8-21)

17. Bilinear z-transform
Replacing s to convert an analog filter into an equivalent digital filter
Imaginary axis in the s-plane is mapped onto the unit circle
The left-half s-plane is mapped inside the unit circle
(8-22)
(8-23)

18. Considering frequency scaling Substituting ,
(8-24)
(8-25)
where is analog frequency, and
is digital frequency

19. Relationship between analog and digital frequency Frequency warping
(8-26)
(8-27)

20. Frequency warping
Fig. 8-4.

21. Example 8-2
Using bilinear z transform
For

22. Impulse response of analog filter :
Magnitude and phase responses of analog filter :
Impulse response of digital filter :
Magnitude response =
Phase response =

23. Frequency response of digital filter :
Relationship between analog and digital frequencies
Magnitude response =
Phase response =

24. Example 8-3 Specification of the desired filter
Filter response
Power gain at 1000Hz : -3dB
Power gain at 3000Hz : -10dB
Sampling frequency : 10kHz
Monotonic decrease in transition region :1000 ~ 3000Hz
Digital parameter from specification

25. Considering Frequency warp
Prewarping
Determining order of Butterworth filter
Desired analog Butterworth filter

26. Using bilinear z-transform
Fig. 8-5.

27. Comparing the two transform
Converting analog filter into equivalent digital filter
Impulse invariant method
Preserving the impulse response of analog filter
Bilinear z-transform
Frequency warping
(8-28)
(8-29)

28. Example 8-4
Transfer function of analog filter
Frequency response

29. Using impulse invariant method
Frequency response
substituting

30. Using bilinear z-transform
Frequency response

31. Analog
Impulse Response
Bilinear z Transform
Fig. 8-6.

32. Example 8-5
Using partial fraction
Inverse Laplace transform

33. Cutoff frequency(-3dB) of Analog filter :
Frequency response
Cutoff frequency(-3dB) of Analog filter :
(8-30)

34. Impulse invariant method
Using partial fraction
Substituting
(8-31)

35. Bilinear z-transform
(8-32)

36. Frequency transformation
Design of various filters using frequency transformation
Analog low pass filter
(normalization filter)
Analog
frequency
transform
Low pass
Low pass High pass
Band pass
Band reject
Desired digital filter
Bilinear z transform or
Impulse invariant method
Fig. 8-7.

37. Analog frequency transform Digital frequency transform
low pass filter
(normalization)
Digital
Low pass
High pass
Band pass
Band reject
Bilinear transform
Analog frequency transform
Digital frequency transform
Fig. 8-8.

38. Low pass filter
Cutoff frequency :

39. High pass filter Cutoff frequency : Cutoff frequency :

40. Band pass filter Cutoff frequency : where is passband width
is upper passband edge frequency
is lower passband edge frequency

41. Band reject filter Cutoff frequency : where is reject band width
is upper band reject frequency
is lower band reject frequency

42. Low pass filter (cutoff frequency )
Table. 8-1 Analog Frequency Transform
Low pass filter
(cutoff frequency )
Low pass filter (cutoff frequency : )
High pass filter (cutoff frequency : )
Band pass filter
(Upper cufoff frequency : , Lower cufoff frequency: , Band pass frequency : )
Band reject filter
(Upper cufoff frequency : , Lower cufoff frequency : , Band reject frequency : )

43. Example 8-6 Specification of filter design Using bilinear z-transform
Cutoff frequency : 10Hz
Sampling frequency ( )
Transfer function :
Using bilinear z-transform
Prewarping

44. Transfer function of analog LPF
Substituting

45. For computational efficiency
For accurate frequency

46. Fig. 8-9.

47. Example 8-7 Specification of filter design Using bilinear z-transform
cutoff frequency :
Sampling frequency :
Transfer function of low pass filter:
Using bilinear z-transform
Prewarping

48. Transfer function of analog HPF from table 8-1
Substituting

49. Fig

50. Example 8-8 Specification of filter design Using bilinear z-transform
Passband :
Sampling frequency :
Order of filter : 2
Using bilinear z-transform
From table 8-2
For second order bandpass filter
Using transfer function of LPF

51. Calculation of lower and upper passband frequencies

52. Analog bandpass filter
Substituting

53. Fig