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

2. - The design of a digital filter involves five steps:
Filter design steps
- The design of a digital filter involves five steps:
(1) Specifications of the filter requirements
(2) calculation of suitable filter coefficients
(3) Representation of the filter by a suitable structure (realization)
(4) Analysis of the effects of finite wordlength on the filter performance
(5) Implementation of filter in software and/or hardware

3. Filter design Need to decide : Type of filter
Amplitude and/or phase responses
Tolerances
Sampling frequency
Wordlength of the input data

4. 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

5. ILPF
Fig. 7-3.

6. FIR coefficient calculation
Most common methods used for calculating
Window method
Optimal method

7. 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)

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

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

10. Fig. 7-4.
(The frequency scale is normalized by T = 1)

11. Truncation for FIR
Rectangular window

12. Fig. 7-5.

13. Fig. 7-6.

14. Fig. 7-7.

15. 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

16. Properties of common window functions
Fig Window functions: (a) Rectangular, (b) Hamming, (c) Blackman

17. Table 7.3 Summary of important features of common window functions

18. 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

19. Determination of parameter
by using the stopband attenuation requirement through empirical relationships below
The number of filter coefficients N
where is the stopband attenuation value and
since the passband and stopband ripples are nearly equal
(7-25)
where is the normalized transition width

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

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

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

24. Fig. 7-9.

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

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

27. Calculation of FIR coefficients

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

32. Fig

34. The optimal method Basic concepts Equiripple passband and stopband
Mathematically expressed as
over the passbands and stopbands
Weighted
Approx. error
Weighting
function
Ideal desired
response
Practical
response

35. Practical response
Ideal response
Fig

36.  equiripple passband and stopband of the resulting filter response,
with the ripple alternating in sign between equal amplitude levels
The minima and maxima are known as extrema.
For linear lowpass filters, for example, there are either m+1 or m+2 extrema,
m=(N+1)/2, type 1 filter
m= N/2, type 2 filter

37. Fig. 7-12. Extremal frequencies

38. The procedure of optimal method
Use the alternation theorem (Parks and McClellan) to find the optimum set of extremal frequencies
Determine the frequency response using the extremal frequencies
Obtain the impulse response coefficients

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

40. Let be defined by normalized frequency, then Normalize as
-> Normalized passband :
-> Normalized stopband :
Desired magnitude response
Weighting function
(7-30)
(7-31)

41. (7-32)
(7-33)
for in and

42. Alternation theorem
Let
is the unique best approximation if and only if is equiripple at and has at least m+2 extremal points in , that is, there
exists
such that
(7-34)
where

43. From equations (7-33) and (7-34)
Equation (7-32) into equation (7-35) yields
Matrix form
(7-35)

44. then the optimal filter is given by
Summary
Step 1. Choose filter length as 2m+1
Step 2. Choose 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. Obtain , ,
then the optimal filter is given by

45. Example 7-4 Specification of desired filter
Filter length : 3 ,
Choose three frequencies and normalize them,
two of them are cutoff frequencies, the third one arbitrarily

46. The error at and is 2x0.196= 0.392, and the error at is 0.196.
From and
The error at and is 2x0.196= 0.392, and the error at is
=> not have
=> Not the characteristic of the optimal filter
(7-37)

47. For this filter for all f in
Choose a new set of
For this filter for all f in
Transfer of the optimal function
(7-38)
(7-39)

48. Fig. 7-13. Characteristics of filter H(f)

49. Optimal method using MATLAB
Based on Parks-McClellan and Remez algorithm
Calculation of coefficients 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

50. 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

51. Table 7-4. Impulse response coefficients

52. Fig.7-14. Frequency response of filter

53. 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

54. 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

55. Table 7-5. Impulse response coefficients of filter

56. Fig. 7-15. Frequency response of filter

57. 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

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

59. Fig

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

61. (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

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

63. 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