1. Chapter 7 FIR Digital Filter Design

2. Introduce
In this chapter, we consider the FIR digital filter design problem.
Unlike the IIR digital filter design problem, it is always possible to design FIR digital filters with exact linear-phase.
In the case of FIR digital filter design, the stability is not a design issue as the transfer function is a polynomial in z-1.

3. Introduce
First, we describe a popular approach to the design of FIR digital filters with linear-phase.
We then consider the computer-aided design of linear-phase FIR digital filters.
To this end, we restrict our discussion to the use of MATLAB in determining the transfer functions.

4. Advantages FIR digital filter is always guaranteed stable.
It is always possible to design FIR digital filters with exact linear-phase.

5. Disadvantages
The order of the FIR transfer function is usually much higher than that of an IIR transfer function meeting the same frequency response specification.

6. 7.1 Linear-phase FIR transfer function
In the previous section, we pointed out why it is important to have a transfer function with a linear-phase property. In this section, we develop the forms of a linear-phase FIR transfer function H(z) with real impulse response h(n).

7. 7.1 Linear-phase FIR transfer function
If H(z) is required to have a linear-phase, its frequency response must be of the form
Where Hg(ω) is called the amplitude response.
NOTE:
Hg(ω) is different from |H(ejω)|.
Θ(ω) is a linear-phase function.

8. 7.1 Linear-phase FIR transfer function
Type 1 linear-phase:
Type 2 linear-phase:
A unify representation:

9. 7.1 Linear-phase FIR transfer function
It can be proven that the transfer function have a linear-phase, if its impulse response h(n) is either symmetric
or is antisymmetric

10. 7.1 Linear-phase FIR transfer function

11. 7.1 Linear-phase FIR transfer function

12. 7.1 Linear-phase FIR transfer function

13. 7.1 Linear-phase FIR transfer function

14. 7.1 Linear-phase FIR transfer function

15. 7.1 Linear-phase FIR transfer function

16. 7.1 Linear-phase FIR transfer function
Type 1 FIR Transfer Function
The impulse response of a Type1 FIR transfer function is of odd length, that is, the degree N is even and satisfies the symmetry condition. Its amplitude response Hg(ω) is given by

17. 7.1 Linear-phase FIR transfer function
Type 2 FIR Transfer Function
The impulse response of a Type2 FIR filter is of even length, that is, the degree N is odd and satisfies the symmetry condition. Its amplitude response Hg(ω) is given by

18. 7.1 Linear-phase FIR transfer function
Type 3 FIR Transfer Function
The impulse response of a Type3 FIR filter is of even length, that is, the degree N is odd and satisfies the antisymmetry condition. Its amplitude response Hg(ω) is given by

19. 7.1 Linear-phase FIR transfer function
Type 4 FIR Transfer Function
The impulse response of a Type1 FIR filter is of even length, that is, the degree N is odd and satisfies the antisymmetry condition. Its amplitude response Hg(ω) is given by

22. 7.1 Linear-phase FIR transfer function

23. 7.1 Linear-phase FIR transfer function

24. 7.1 Linear-phase FIR transfer function

25. 7.1 Linear-phase FIR transfer function

26. 7.1 Linear-phase FIR transfer function

27. 7.1 Linear-phase FIR transfer function

28. 7.1 Linear-phase FIR transfer function

29. 7.1 Linear-phase FIR transfer function

30. 7.2 FIR Filter Design based on Windows Function Method
We now turn our attention to the design of real coefficient FIR filters.
These filters are described by a transfer function that is a polynomial in z-1and therefore require different approaches for their design.

31. 7.2 FIR Filter Design based on Windows Function Method
A direct and straightforward method is based on truncating the Fourier series representation of the prescribed frequency response is discussed in this section.

32. 7.2.1 Windows function method
Let Hd(ejω) denote the desired frequency response function.
Since Hd(ejω) is a periodic function of ω with a period 2π, it can be expressed as a Fourier series,

33. 7.2.1 Windows function method
Where the Fourier coefficients are precisely the corresponding impulse response samples and are given by

34. 7.2.1 Windows function method
Thus, given a frequency response specification Hd(ejω), we can compute hd[n] using the above equation and, hence, determine the transfer function Hd(z).
However, for most practical applications, the desired frequency response is piecewise constant with sharp transitions between bands, in which case, the corresponding impulse response sequence {hd[n]} is of infinite length and noncausal.

35. 7.2.1 Windows function method
Our objective is to find a finite-duration impulse response sequence {ht[n]} of length 2M+1 whose DTFT Ht(ejω) approximates the desired DTFT Hd(ejω) in some sense.

36. 7.2.1 Windows function method
One commonly used approximation criterion is to minimize the integral-squared error
where

37. 7.2.1 Windows function method
Using Parseval’s relation, we can further obtained
It is evident that the integral-squared error is minimum when for In other words, the best finite-length approximation to ideal infinite-length impulse response in the mean-square error sense is simply obtained by truncation.

38. 7.2.1 Windows function method
A causal FIR filter with an impulse response h[n] can be derived from ht[n] by delaying the latter sequence by M samples, that is, by forming
The causal filter h[n] has the same magnitude response as that of the noncausal filter ht[n] and its phase response has a linear phase shift of ωM ridians with respect to that of the noncausal filter.

39. 7.2.2 impulse Response of Ideal Filters
Ideal frequency selective filters are characterized by passband and stopband with sharp transitions from passband to stopband.
Four such commonly used filters are the lowpass, highpass, bandpass, and bandstop filters whose ideal frequency response have been discussed.

40. 7.2.2 impulse Response of Ideal Filters
The frequency response of the ideal lowpass filter is
The corresponding impulse response coefficients is
It is doubly infinite, not absolutely summable, and therefore unrealizable.

41. 7.2.2 impulse Response of Ideal Filters
By setting all impulse response coefficients outside the range –M<=n<=M equal to zero, we arrive at a finite-length noncausal approximation of length 2M+1, which when shifted to the right yields the coefficients of a causal FIR lowpass filter:

42. 7.2.2 impulse Response of Ideal Filters
The frequency responses of the ideal highpass, bandpass and bandstop filter is

43. 7.2.2 impulse Response of Ideal Filters
The impulse responses of the ideal highpass, bandpass and bandstop filter is

44. 7.2.2 impulse Response of Ideal Filters
They are also characterized by doubly infinite impulse response that are not absolutely summable, making them unrealizable.
They can be made realizable by truncating the impulse response sequence to finite lengths and shifting the truncated coefficients to the right appropriately.

45. 7.2.3 Gibbs Phenomenon
The causal FIR filters obtained by simply truncating the impulse response coefficients of the ideal filters given in the previous section exhibit an oscillatory behavior in their respective magnitude responses, which is more commonly referred to as the Gibbs phenomenon.

46. 7.2.3 Gibbs Phenomenon
The reason behind the Gibbs phenomenon can be explained by considering the truncation operation as multiplication by a finite-length window sequence w[n] and by examining the windowing process in the frequency domain.

47. 7.2.3 Gibbs Phenomenon
The FIR filter obtained by truncation can be alternatively expressed as
From the modulation theorem, the Fourier transform is given by

48. 7.2.3 Gibbs Phenomenon 其中 Wg(ω)称为矩形窗的幅度函数，a=(N-1)/2。则 所以
加窗后的滤波器的幅度特性等于理想低通滤波器的幅度特性与矩形窗幅度特性的卷积。 所以

49. 图 矩形窗对理想低通
幅度特性的影响

50. 7.2.3 Gibbs Phenomenon 加窗后的影响： （1）在理想特性不连续点ωc附近形成过渡带。
（2）通带内产生了波纹，最大值在ωc-2π/N处。阻带产生了余振，最大的负峰在 ωc+2π/N。
（3）通带与阻带中的波纹与窗函数的幅度谱有关，其旁瓣幅度直接影响滤波器的波纹幅度。

51. 7.2.1 Windows function method
如何减小Gibbs效应的影响
增加矩形窗的长度，是否可以减少Gibbs效应的影响？ 因为
加大N只能使过渡带变窄，但不能有效减小Gibbs效应的影响。
结论：要减少带内波动以及增大阻带衰减，只能从窗函数的形状上找解决问题的方法。

52. 7.2.4 Fixed Windows Function
The commonly used window of length N=2M+1 listed below:
Rectangular:
Barlett:
Hann:
Hamming:
Blackman:

53. 7.2.4 Fixed Windows Function
The frequency response are listed below:
Rectangular:
Barlett:
Hanning:
Hamming:
Blackman:

54. 7.2.4 Fixed Windows Function

55. 7.2.4 Fixed Windows Function

56. 7.2.4 Fixed Windows Function
The essential properties of the above widow functions, except the Bartlett window

57. 表7.2.1 凯塞窗参数对滤波器的性能影响

58. 7.2.4 Fixed Windows Function
表 六种窗函数的基本参数

59. 7.2.4 Fixed Windows Function
In the case of the window function of Rectangular, Hann and Hamming, the value of the ripple δ does not depend on the filter length, or the cutoff frequency ωc, and is essentially constant.

60. 7.2.4 Fixed Windows Function
In addition, the transition bandwidth is approximately given by
Where c is a constant for most practical purposes and is determined from the above table after the specific windows has been selected.

61. 7.2.5 Design Step Step 1: select windows function and N
Step 2: modeling the desired frequency response
根据设计需要，一般选择相应的线性相位理想滤波器。
Step 3: calculate
Step 4: obtain the impulse response of the desired FIR filter

62. Example
用窗函数法设计线性相位高通FIR DF。
通带截止频率：
阻带截止频率：
通带最大衰减：
阻带最小衰减：

63. Example Step 1: 选择窗函数。 已知阻带最小衰减为40dB,可知Hann和Hamming窗均满足要求，我们选Hamming窗。
Hamming窗的过度带宽度ΔB=8π/N,所以有 可得
对高通滤波器，N必须取奇数，所以，N=33。

64. Example
Step 2: 其中

65. Example
Step 3: 求hd(n)。
，所以

66. Example
Step 4: 加窗

67. 7.2.6 FIR Filter Design using MATLAB
hn=fir1(M,wc)
%ep721.m: 例7.2.1
wp=pi/2;
ws=pi/4;
Bt=wp-ws;
N0=ceil(6.2*pi/Bt);
N=N0+mod(N0+1,2);
wc=(wp+ws)/2/pi;
hn=fir1(N-1,wc,’high’,hanning(N))；

68. 7.3 FIR filter design using frequency sampling
Basic idea
Let the desired frequency function be Hd(ejω). We can obtain Hd(k) by sampling it in ω∈[0,2π] with interval T
Taking N-point IDFT, we can obtain

69. 7.3 FIR filter design using frequency sampling
Taking Z-transform, the transfer function H(z) is given
This can be implemented with the direct form. Or
It can be implemented with the frequency sampling structure.

70. 7.3 FIR filter design using frequency sampling
Constrain condition of Hd(k)
By the linear-phase condition:
H(n) is a real sequence

71. 7.3 FIR filter design using frequency sampling
We can obtain that the frequency response should satisfy

72. 7.3 FIR filter design using frequency sampling
The sampling points in [0,2π] are at
We can obtain
Constrain condition

73. 7.3 FIR filter design using frequency sampling
Assume the desired filter Hd(ejω) is a ideal lowpass filter with ωc, then N-point Hg(k) and θ(k) can be computed:
When N=odd

74. 7.3 FIR filter design using frequency sampling
When N=even

75. 7.3 FIR filter design using frequency sampling
Error and improvement
If the frequency response of desired FIR filter is Hd(ejω) and its impulse response is hd(n), then,
If Hd(ejω) is not a smooth continuous function, the corresponding impulse response hd(n) is infinite length. It leads to aliasing h(n).

76. 7.3 FIR filter design using frequency sampling
The increasing N leads to decreasing transition band width, high computational complexity, but αp and αs can not be obviously improved.
The improved αs can be obtained by interpolating at the discontinuous points. m 1 2 3 as
44~54dB
65~75dB
85~95dB

77. 图 理想低通滤波器增加过渡点

78. 7.3 FIR filter design using frequency sampling
Design step
根据阻带衰减确定过渡带采样点的个数m；
确定过渡带宽度，估算频域采样点数N；
构造希望逼近的频率响应函数；
进行频率采样；
取IDFT，得到h(n)；
检验结果，如未达到指标要求，修改过渡带采样值。

79. 7.3 FIR filter design using frequency sampling
Examples
See P220

80. 7.4 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
In this section, we consider the application of computer-aided optimization technique for the design of FIR filters.

81. 7.4 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
The basic idea behind the computer-based technique is to minimize iteratively an error measure that is function of the difference between the desired frequency response D(ejω) and the frequency response H(ejω) of the filter being designed.

82. 7.4 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
The design objective is to iteratively adjust the filter parameters so that ε defined by the followed equation is a minimum.
Minimax criterion
Least-squares criterion

83. 7.4 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
The linear-phase FIR filter obtained by minimizing the peak absolute value of the weighted error ε given by the above equation is usually called the equiripple FIR filter.
Since here, after ε has been minimized, the weighted error function ε(ω) exhibits an equiripple behavior in the frequency range of interest.

84. The Parks-McClellan Algorithm
A weighted-Chebyshev approximation method has been advanced by Parks and McClellan for designing equiripple liear-phase FIR filters.

85. The Parks-McClellan Algorithm
The general form of the frequency response H(ejω) of a causal linear-phase FIR filter of length N+1 is given by
Where the amplitude response
is a real
function ω.

86. The Parks-McClellan Algorithm
The weighted error function in this case involves the amplitude response and is given by
The desired amplitude response
A positive weighting function
The Parks-McClellan algorithm is based on iteratively adjusting the coefficients of the amplitude response until the peak absolute value of ε(ω) is minimized.

87. The Parks-McClellan Algorithm
If the minimum value of the peak absolute value of ε(ω) in a ωa<= ω<= ωb is ε0, then the absolute error satisfies

88. The Parks-McClellan Algorithm
In typical filter design applications, the desired amplitude response is given by
The amplitude response
is required to
satisfy the above desired response with a ripple of ±δp in the passband and a ripple δs in the stop.

89. The Parks-McClellan Algorithm
As a result, it is evident from the weighted error function that the weighting function can be chosen either as
or as

90. The Parks-McClellan Algorithm
By a clever manipulation, the expression for the amplitude response for each of the four types of linear-phase FIR filters can be expressed in the same form.
The same algorithm can be adapted to design any one of the four types of filters.
To develop this general form for the amplitude response expression, we consider each of the four types of filters separately.

91. The Parks-McClellan Algorithm
For the Type 1 linear-phase FIR filter, the amplitude response can be rewritten using the notation N=2M in the form
where

92. The Parks-McClellan Algorithm
For Type 2 linear-phase FIR filter, the amplitude response can be rewritten in the form
where

93. The Parks-McClellan Algorithm
For Type 2 linear-phase FIR filter, the amplitude response can be rewritten in the form
where

94. The Parks-McClellan Algorithm
For the Type 3 linear-phase FIR filter, the amplitude response can be rewritten in the form
where

95. The Parks-McClellan Algorithm
For the Type 4 linear-phase FIR filter, the amplitude response can be rewritten in the form
where

96. The Parks-McClellan Algorithm
The amplitude response for all four types of linear-phase FIR filters can be expressed in the form
Where the first factor Q(ω) is given by

97. The Parks-McClellan Algorithm

98. The Parks-McClellan Algorithm
We arrive at a modified form of the weighted approximation function given by
Using the notation

99. The Parks-McClellan Algorithm
The optimization problem now becomes the determination of the coefficients
Then, the corresponding coefficients of the original amplitude response are computed from which the filter coefficients are obtained.

100. Alteration Theorem
The amplitude function A(ω) is the best unique approximation of the desired amplitude response obtained by minimizing the peak absolute valueεof ε(ω) if and only if there exist at least L+2 extremal angular frequencies.

101. The Parks-McClellan Algorithm
The Remez exchange algrithm, a highly efficient iterative procedure, is used to determine the locations of the extremal frequencies and consists of the following steps at each iteration stage:

102. The Parks-McClellan Algorithm
Step 1: A set of initial value for the extremal frequencies are either chosen or are available from the completion of the previous iteration.

103. The Parks-McClellan Algorithm
Step 2: The value ε is then computed by solving the following equation

104. The Parks-McClellan Algorithm
Step 3: The values of the amplitude response A(ω) at ω= ωi are then computed using

105. The Parks-McClellan Algorithm
Step 4: The polynomial A(ω) is determine by interpolating the above values at the L+2 extremal frequencies using the Lagrange interpolation formula:

106. The Parks-McClellan Algorithm
Step 5: The new weighted error function ε(ω) is computed at a dense set S(S>=L) of frequencies. In practice, S=16L is adequate. Determine the L+2 new extremal frequencies from the values of ε(ω) evaluated at the dense set of frequencies.

107. The Parks-McClellan Algorithm
Step 6: If the peak values ε are equal in magnitude, the algorithm has converged. Otherwise, go bask to Step 2.

108. 10.5 FIR Digital Filter Design Using MATLAB
The signal Processing ToolBox of MATLAB includes a variety of M-files for the design of FIR digital filters.

109. 10.5.1 FIR Digital Filter Order Estimation Using MATLAB
As in the case of FIR digital filter design, the FIR digital filter design process also consists of two steps.
In the first step, the filter order is estimated from the given specification
In the second step, the coefficients of the transfer function are determined using the estimated order and filter specifications..

110. 10.5.1 FIR Digital Filter Order Estimation Using MATLAB
Example FIR Filter Order Estimation Using Kaiser’s Formula
N=kaiord( , , 1800, 2000, 12000)

111. 10.5.1 FIR Digital Filter Order Estimation Using MATLAB
Example FIR Filter Order Estimation Using Bellanger’s Formula.
N=kaiord( , , 1800, 2000, 12000)

112. 10.5.1 FIR Digital Filter Order Estimation Using MATLAB
Example FIR Filter Order Estimation Using Hermann’s Formula
fedge = input('Type in the bandedges = ');
mval = input('Desired magnitude values in each band = ');
dev = input('Allowable deviation in each band = ');
FT = input('Type in the sampling frequency = ');
[N, fpts, mag, wt] = remezord(fedge, mval, dev, FT);
fprintf('Filter order is %d \n',N);

113. 10.5.2 Equiripple Linear-Phase FIR Filter Design Using MATLAB
For designing an equiripple linear-phase FIR filter employing the Parks-McClellan algorithm, we can use the MATLAB M-files remez.

114. 10.5.2 Equiripple Linear-Phase FIR Filter Design Using MATLAB
Example Design of an Equiripple Linear-Phase FIR Lowpass Filter Using Remez Algorithm.

115. 10.5.2 Equiripple Linear-Phase FIR Filter Design Using MATLAB
% Program 10_2
% Design of Equiripple Linear-Phase FIR Filters %
format long
fedge = input('Band edges in Hz = ');
mval = input('Desired magnitude values in each band = ');
dev = input('Desired ripple in each band =');
FT = input('Sampling frequency in Hz = ');
[N,fpts,mag,wt] = remezord(fedge,mval,dev,FT);
b = remez(N,fpts,mag,wt);
disp('FIR Filter Coefficients'); disp(b)
[h,w] = freqz(b,1,256);
plot(w/pi,20*log10(abs(h)));grid;
xlabel('\omega/\pi'); ylabel('Gain, dB');

117. 10.5.2 Equiripple Linear-Phase FIR Filter Design Using MATLAB
Example Design of an Equiripple Linear-Phase FIR bandpass Filter Using Remez Algorithm.
N=input(‘Filter order= ‘);
Fpts=input(‘Normalized band edges = ‘);
Mag=input(‘ Desired weights for each band = ‘);
[N,fpts,mag,wt] = remezord(fedge,mval,dev,FT);
b = remez(N,fpts,mag,wt);
disp('FIR Filter Coefficients'); disp(b)
[h,w] = freqz(b,1,256);
plot(w/pi,20*log10(abs(h)));grid;
xlabel('\omega/\pi'); ylabel('Gain, dB');

118. 10.5.4 Windows-Based FIR Filter Design Using MATLAB
The windows-based FIR filter design process involves three steps:
First, the type of window to be used is determined to meet the desired stopband attenuation.
second, the order of the FIR filter to be designed is estimated.
Finally, the desired impulse response of the ideal filter is computed, which is then multiplied by the window coefficients generated in the first step to yield the coefficients of FIR filter.

119. Window Generation W=blackman(L) W=hamming(L) W=hann(L) W=chebwin(L,Rs)
W=kaiser(L,beta)

120. Window Generation Example 10.24 Kaiser window generation using MATLAB
fpts = input('Type in the bandedges = ');
mag = input('Type in the desired magnitude values = ');
dev = input('Type in the ripples in each band = ');
[N,Wn,beta,ftype] = kaiserord(fpts,mag,dev)
w = kaiser(N+1,beta); w = w/sum(w);
[h,omega] = freqz(w,1,256);
plot(omega/pi,20*log10(abs(h)));grid;
xlabel('\omega/\pi'); ylabel('Gain, dB');

121. Filter Design
The functions available in MATLAB for the design of FIR filters using the windowed Fourier series approach are fir1 and fir2.
The function fir1 is used to design conventional lowpass, highpass, bandpass, bandstop, and multiband FIR filters.
The function fir2 is employed to design FIR filters with arbitrarily shaped magnitude response.

122. Filter Design Example 10.25 Lowpass FIR Design Using the Kaiser Window
The desired specifications for the filter are
fpts = input('Type in the bandedges = ');
mag = input('Type in the desired magnitude values = ');
dev = input('Type in the ripples in each band = ');
[N,Wn,beta,ftype] = kaiserord(fpts,mag,dev)
kw = kaiser(N+1,beta);
b = fir1(N,Wn, kw);
[h,omega] = freqz(b,1,512);
plot(omega/pi,20*log10(abs(h)));grid;
xlabel('\omega/\pi'); ylabel('Gain, dB');

123. Filter Design
Example Highpass FIR Filter Design Using the Kaiser Window
A highpass FIR filter is to be designed meeting the following specifications:

124. Filter Design fpts = input('Type in the bandedges = ');
mag = input('Type in the desired magnitude values = ');
dev = input('Type in the ripples in each band = ');
[N,Wn,beta,ftype] = kaiserord(fpts,mag,dev)
kw = kaiser(N+1,beta);
b = fir1(N,Wn,’highpass’ , kw);
[h,omega] = freqz(b,1,512);
plot(omega/pi,20*log10(abs(h)));grid;
xlabel('\omega/\pi'); ylabel('Gain, dB');

125. Filter Design
Example Design of Multiband FIR Filter Using Hamming Window
Consider the design of an FIR filter of order 100 with three different constant magnitude levels: 0.3 in the frequency range 0 to 0.28, 1.0 in the frequency range 0.3 to 5.0, and 0.7 in the frequency range 0.52 to 2.0.

126. 10.5.4 Windows-Based FIR Filter Design Using MATLAB
% Program 10_6
% Design of Multiband FIR Filter Using Hamming Window %
fpts = [ ];
mval = [ ];
b = fir2(100,fpts,mval);
[h,omega] = freqz(b,1,512);
plot(omega/pi,abs(h));grid;
xlabel('\omega/\pi'); ylabel('Magnitude');