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

2. Properties of FIR filter
FIR filter has characteristics:
Always stable
Exactly linear phase response
Simple to implement
The effect of finite wordlength in FIR is less than in IIR filters
(7-1)
(7-2)

3. Linear phase response Phase response of FIR filter
Phase delay and group delay
(7-3)
where
(7-4)
(7-5)
(7-6)

4. Conditions to be linear phase response
(7-7)
(7-8)
where and are constants
 Constant group and constant phase delay responses

5. Impulse response of the filter should have positive symmetry
to satisfy Eq. (7-7), thus
(7-9)
(7-10)

6. For symmetry condition
(7-11)
(7-12)
(7-13)
(7-14)

7. For the condition given in Eq. (7-8) Constant group delay only
Negative symmetric impulse response
(7-15)
(7-16)

8. Fig. 7-1.

9. Example 7-1 Symmetric impulse response for linear phase response
No phase distortion
Using the symmetry condition
For

10. Using the symmetry condition
Frequency response
Using the symmetry condition
Compact form
where

11. Using the symmetry condition
where

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

13. Zeros of FIR filters Transfer function for FIR filter
Positive symmetry (types 1 and 2)
(7-17)

14. This means that should be zero.
has zero at (i.e., ) , then should have zero at
This means that should be zero.
If each complex zero does not exist on the unit circle,
then have 4 conjugate reciprocal zeros :
If is real and is complex number, should have conjugate zero at

15. If zero exists on unit circle, then , i.e.,
becomes
If zeros are real and do not exist on the unit circle, then
If zeros exist on , then

16. Impulse response of type 3 (even N) and type 4 (odd N)
If zeros exist on , then
For symmetry and odd N,
Should have mandatory zero at
Impulse response of type 3 (even N) and type 4 (odd N)
For z = 1
Mandatory zero at
For and even N
(7-18)

17. Mandatory zero
Mandatory zero
Mandatory zero
Mandatory zero
Fig. 7-2.

18. FIR filter design Design of FIR Filter Need to decide :
Type of filter
Amplitude and/or phase responses
Tolerances
Sampling frequency
Wordlength of the input data
To calculate filter coefficients, select the method from:
Window method, Optimal method, Frequency sampling method

19. Filter specifications
Important parameters
Another important parameter
peak passband deviation (or ripples)
stopband deviation
passband edge frequency
stopband edge frequency
sampling frequency
Filter order N

20. ILPF
Fig. 7-3.

21. FIR coefficient calculation
Most commonly used methods for obtaining
Window, optimal, and frequency sampling methods

22. Window method Design of FIR filter using window methods
Frequency response of filter,
Impulse response,
Ideal lowpass response
(7-19)
(7-20)

23. Fig. 7-4.

24. Truncation for FIR
Rectangular Window

25. Fig. 7-5.

26. Fig. 7-6.

27. Fig. 7-7.

28. Table 7.2 Summary of ideal impulse responses for standard frequency selective filters
and are the normalized passband or stopband cutoff frequencies

29. Some common window functions
Hamming window
Appropriate relationship between transition width and filter length
(7-21)
(7-22)
where N is filter order and
is normalized transition width

30. Properties of common window functions
Fig. 7-8.

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

32. Kaiser window Trade-off transition width against ripple
Using a ripple control parameter
(7-23)
where is zero-order modified Bessel function of the first kind
where typically

33. Determination of parameter Using the stopband attenuation requirements
Using empirical relationships
The number of filter coefficients N
where is the stopband attenuation
, since the passband and stopband ripples are nearly equal
(7-25)
where is the normalized transition width

34. The window method of calculating FIR filter coefficients
Step 1 : specify the desired frequency response of filter,
Step 2 : obtain the impulse response, , of desired filter by evaluating
the inverse Fourier transform
Step 3 : select a window function and then determine the number of
coefficients using the appropriate relationship between the filter
length and the transition width,
Step 4 : obtain values of for chosen window function and the values
of the actual FIR coefficient, , by multiplying by
(7-26)

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

36. Using Hamming window
Considering the smearing effect of the window function

37. Symmetrical function
Calculation of
Using the symmetry property to obtain the other coefficients

39. Fig. 7-9.

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

41. Using Kaiser window
The number of filter order N
The ripple parameter
Normalized cutoff frequency

42. Calculation of FIR coefficients

43. Symmetrical function
Calculation of
Using the symmetry property to obtain the other coefficients

47. Fig

48. Advantages and disadvantages
Simplicity
Lack of flexibility
The passband and stopband edge frequencies cannot be precisely specified
For a given attenuation specification, filter designer must find a suitable window

49. 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)

50. Practical response
Ideal response
Fig

51. Fig

52. The procedure of optimal method
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

53. Optimal FIR filer design
Transfer function of lowpass filter
Symmetric property gives
(7-28)
where
where and ,

54. Let Normalized passband : Normalized stopband :
Desired magnitude response
Weighting function
(7-30)
(7-31)

55. Find with ,
(7-32)
(7-33)
where are and

56. Alternation theorem Let
If has equiripple inside and exhibit at least m+2 alternations, then
(7-34)
where

57. From equation (7-33) and (7-34)
Matrix form
(7-35)

58. Summary Step 1. Select filter length as 2m+1
Step 2. Select m+2 points of in F
Step 3. Calculate and e using equation (7-36)
Step 4. Calculate using equation (7-29). If , go to step 5,
otherwise go to step 6
Step 5. Determine m local minima or maxima points
Step 6. Calculate ,

59. Example 7-4 Specification of desired filter Normalized frequency
Filter length : 3 ,
Normalized frequency

60. From and
(7-37)

61. Transfer function of optimal filter
Selection of new
Transfer function of optimal filter
(7-38)
(7-39)

62. Fig

63. Optimal method using MATLAB
Based on Park-McClellan and Remez algorithm
Calculation of coefficient for FIR filter using Remez
where N is the filter length
F is the normalized frequency band edges
M is the magnitude response
WT is the relative weight between ripples

64. Example 7-5 Specification of desired filter
Pass band : 0 – 1000Hz
Transition band : 500Hz
Filter length : 45
Sampling frequency : 10,000Hz
Normalized frequency band edges
Magnitude response

65. Table 7-4.

66. Fig.7-14.

67. Example 7-6 Specification of desired filter
Pass band : 3kHz – 4kHz
Transition band : 500Hz
Pass band ripple : 1dB
Rejection band attenuation : 25dB
Sampling frequency : 20kHz
Frequency band edges and magnitude response

68. Estimation of filter length Pass and rejection band ripples
Using Remezord in MATLAB
where and are ripples of dB scale in pass and rejection band

69. Table 7-5.

70. Fig

71. Frequency sampling method
Design of FIR filter
Taking N samples of the frequency response at intervals of ,
Filter coefficients
(7-40)
where , are samples of desired frequency response

72. Linear phase filters with positive symmetrical impulse response
For N even
For N odd
Upper limit in the summation is
(7-41)
where

73. Fig

74. Example 7-7 (1) Show the From equation (7-40) is symmetry
is real value
(7-42)

75. (2) Design of FIR filter Specification of desired filter
Pass band : 0 – 5kHz
Sampling frequency : 18kHz
Filter length : 9
Selection of frequency samples at intervals of
Fig

76. Coefficient of FIR filter using equation (7-42)
Table 7-6.

77. Comparison of the window, optimum and frequency sampling methods
Optimal method
Easy and efficient way of computing FIR filter coefficients
Making filter with good amplitude response characteristics for reasonable values of N
Window method
In the absence of the optimal software or when the passband and stopband ripples are equal, the window method represents a good choice
Particularly simple method to apply and conceptually easy to understand
Frequency sampling method
Filters with arbitrary amplitude-phase response can be easily designed
Lack of precise control for the location of the bandedge frequencies or the passband ripples