1. Chapter 7 Finite Impulse Response(FIR) Filter Design

2. 1. Features of FIR filter Characteristic of FIR filter
FIR filter is always stable
FIR filter can have an exactly linear phase response
FIR filter are very simple to implement. Nonrecursive FIR filters suffer less from the effect of finite wordlength than IIR filters

3. 2. Linear phase response Phase response of FIR filter
Phase delay and group delay
(1)
where
(2)

4. Condition of linear phase response
(3)
(4)
Where and is constant
Constant group delay and phase delay response

5. If a filter satisfies the condition given in equation (3)
From equation (1) and (2)
thus

6. It is represented in Fig 7.1 (a),(b)

7. When the condition given in equation (4) only
The filter will have a constant group delay only
It is represented in Fig 7.1 (c),(d)

8. Center of symmetry
Fig. 7-1.

9. Table 7.1 A summary of the key point about the four types of linear phase FIR filters

10. Example 7-1
Symmetric impulse response for linear phase response. No phase distortion

11. Frequency response
where

12. (3)
where

13. 3. Zero distribution of FIR filters
Transfer function for FIR filter

14. Four types of linear phase FIR filters
have zero at
is real and is imaginary

15. If zero on unit circle
If zero not exist on the unit circle
If zeros on

16. Necessary zero
Necessary zero
Necessary zero
Necessary zero
Fig. 7-2.

17. 4. FIR filter specifications
peak passband deviation (or ripples)
stopband deviation
passband edge frequency
stopband edge frequency
sampling frequency

18. ILPF
Satisfies spec’s
Fig. 7-3.

19. Characterization of FIR filter
Most commonly methods for obtaining
Window, optimal and frequency sampling methods

20. 5. Window method FIR filter Frequency response of filter
Corresponding impulse response
Ideal lowpass response

21. Fig. 7-4.

22. Truncation to FIR
Rectangular Window

23. Fig. 7-5.

24. Fig. 7-6.

25. Fig. 7-7.

26. Table 7.2 summary of ideal impulse responses for standard frequency selective filters
and are the normalized passband or stopband edge frequencies; N is the length of filter

27. Common window function types
Hamming window
where N is filter length and
is normalized transition width

28. Characteristics of common window functions
Fig. 7-8.

29. Table 7.3 summary of important features of common window functions

30. Kaiser window
where is the zero-order modified Bessel function of the first kind
where typically

31. Kaiser Formulas – for LPF design

32. Example 7-2 Obtain coefficients of FIR lowpass using hamming window
Lowpass filter
Passband cutoff frequency
Transition width
Stopband attenuation
Sampling frequency

33. Using Hamming window

37. Fig. 7-9.

38. Example 7-3 Obtain coefficients using Kaiser or Blackman window
Stopband attenuation
passband attenuation
Transition region
Sampling frequency
Passband cutoff frequency

39. Using Kaiser window

44. Fig

45. Summary of window method
1. Specify the ‘ideal’ or desired frequency response of filter,
2. Obtain the impulse response, , of the desired filter by
evaluating the inverse Fourier transform
3. Select a window function that satisfies the passband or
attenuation specifications and then determine the number of filter coefficients
4. Obtain values of for the chosen window function and the values of the actual FIR coefficients, , by multiplying by

46. Advantages and disadvantages
Simplicity
Lack of flexibility
The passband and stopband edge frequencies cannot be precisely specified
For a given window(except the Kaiser), the maximum ripple amplitude in filter response is fixed regardless of how large we make N

47. 6. The optimal method Basic concepts Equiripple passband and stopband
For linear phase lowpass filters
m+1 or m+2 extrema(minima and maxima)
Weighted
Approx. error
Weighting
function
Ideal desired
response
Practical
response
where m=(N+1)/2 (for type1 filters) or m =N/2 (for type2 filters)

48. Practical response
Ideal response
Fig

49. Fig

50. Optimal method involves the following steps
Use the Remez exchange algorithm to find the optimum set of extremal frequencies
Determine the frequency response using the extremal frequencies
Obtain the impulse response coefficients

51. Optimal FIR filer design
where
where and ,
Let
This weighting function permits different peak error in the two band

52. where are and
Find

53. Alternation theorem Let
If has equiripple inside bands and more than m+2 extremal point
then
where

54. From equation (7-33) and (7-34)
Equation (7-35) is substituted equation (7-32)
Matrix form

55. Summary Step 1. Select filter length as 2m+1
Step 2. Select m point in F
Step 3. Calculate and e using equation (7)
Step 4. Calculate using equation (5). If in some
of f , go to step 5, otherwise go to step 6
Step 5. Determine m local minma or maxma points
Step 6. Calculate when
where

56. Example 7-4 Specification of desired filter Ideal low pass filter
Filter length : 3
Normalized frequency

57. From
Cutoff frequency
: not the optimal filter

58. : has the minimum
(N=3)

59. Fig

60. Optimization using MATLAB
Park-McClellan
Remez
where N is the filter order (N+1 is the filter length)
F is the normalized frequency of border of pass band
M is the magnitude of frequency response
WT is the weight between ripples

61. Example 7-5 Specification of desired filter
Band pass region : 0 – 1000Hz
Transition region : 500Hz
Filter length : 45
Sampling frequency : 10,000Hz
Normalized frequency of border of passband
Magnitude of frequency response

62. Table 7-4.

63. Fig.7-14.

64. Example 7-6 Specification of desired filter
Band pass region : 3kHz – 4kHz
Transition region : 500Hz
Pass band ripple : 1dB
Rejection region : 25dB
Sampling frequency : 20kHz
Frequency of border of passband

65. Transform dB to normal value
Filter length
Remezord (MATLAB command)
where and ripple value(dB) of pass band and rejection band

66. Table 7-5.

67. Fig

68. 7. Frequency sampling method
Frequency sampling filters
Taking N samples of the frequency response at intervals of
Filter coefficients
where are samples of the ideal or target frequency response

69. For linear phase filters (for N even)
For N odd
Upper limit in summation is
where

70. Fig

71. Example 7-7
(1) Show the
Expanding the equation
is real value

72. Sampling frequency : 18kHz Filter length : 9
(2) Design of FIR filter
Band pass region : 0 – 5kHz
Sampling frequency : 18kHz
Filter length : 9
Fig

73. Samples of magnitude in frequency
Table 7-6.

74. 8. Comparison of most commonly method
Window method
The easiest, but lacks flexibility especially when passband and stopband ripples are different
Frequency sampling method
Well suited to recursive implementation of FIR filters
Optimal method
Most powerful and flexible