1. Biomedical Signal processing Chapter 7 Filter Design Techniques
2018年11月29日6时3分
Biomedical Signal processing Chapter 7 Filter Design Techniques
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程《生物医学信号处理(双语)》
2018/11/29 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 7 Filter Design Techniques
7.0 Introduction
7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters
7.2 Design of FIR Filters by Windowing
7.3 Examples of FIR Filters Design by the Kaiser Window Method
7.4 Optimum Approximations of FIR Filters
7.5 Examples of FIR Equiripple Approximation
7.6 Comments on IIR and FIR Discrete-Time Filters

3. Filter Design Techniques
7.0 Introduction

4. 2018年11月29日6时3分
7.0 Introduction
Frequency-selective filters pass only certain frequencies
Any discrete-time system that modifies certain frequencies is called a filter.
We concentrate on design of causal Frequency-selective filters

5. Linear time-invariant discrete-time system
If input is bandlimited and sampling frequency is high enough to avoid aliasing,
then overall system behaves as an LTI continuous-time system
effective
continuous-time specifications are converted to discrete time specifications by:

6. Stages of Filter Design
Determine the specification of the desired properties of the system.
The approximation of the specifications using(designing) a causal discrete-time system.
The realization of the system.
Our focus is on second step
Specifications are typically given in the frequency domain.

7. Frequency-Selective Filters
Ideal lowpass filter (discrete-time system)

8. Frequency-Selective Filters
Ideal highpass filter(discrete-time system)

9. Frequency-Selective Filters
Ideal bandpass filter(discrete-time system)

10. Frequency-Selective Filters
Ideal bandstop filter(discrete-time system)

11. tolerance scheme 容限图
Figure depicts the typical representation of the tolerance limits associated with approximating an ideal lowpass filter.
tolerance scheme
容限图
discrete-time
continuous -time

12. Example 7.1 Determining Specifications for a Discrete-Time Filter H(ejw) .
Known:
Specifications of the continuous-time filter:
1. passband
2. stopband
For bandlimited input, aliasing avoided when sampling frequency is high enough

13. Example 7.1 Determining Specifications for a Discrete-Time Filter
Specifications of the continuous-time filter:
1. passband
2. stopband
tolerance scheme
容限图

14. Example 7.1 Determining Specifications for a Discrete-Time Filter
Solution:
Specifications of the discrete-time filter in
Same tolerance limits:

15. Filter Design Constraints
Designing IIR filters is to find the approximation by a rational function of z.
The poles of the system function must lie inside the unit circle(stability, causality).
Designing FIR filters is to find the polynomial approximation.
FIR filters are often required to be linear-phase.

16. Filter Design Techniques
7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters

17. 7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters
The traditional approach to the design of discrete-time IIR filters involves the
transformation of a continuous-time filter into a discrete filter meeting prescribed specification:
Impulse Invariance,
Bilinear transformation

18. Three Reasons
The art of continuous-time IIR filter design is highly advanced, and useful results can be achieved,
it is advantageous to use the design procedures already developed for continuous-time filters.

19. Three Reasons
Many useful continuous-time IIR design method have relatively simple closed form design formulas.
Therefore, discrete-time IIR filter design methods based on standard continuous-time design formulas are rather simple to carry out.

20. Three Reasons
The standard approximation methods that work well for continuous-time IIR filters do not lead to simple closed-form design formulas when these methods are applied directly to the discrete-time IIR case,
because the frequency response of a discrete-time filter is periodic, and that of a continuous-time filter is not.

21. Steps of DT filter design by transforming a prototype continuous-time filter
① Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
② Find the system function of the continuous-time filter.
③ Transform the continuous-time filter to derive the system function of the discrete-time filter.

22. H(s)极点分布与原函数的对应关系(稳定性)
几种典型情况
p211

23. Constraints of Transformation
In transformations, to preserve the essential properties of the frequency response,
the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane.

24. Constraints of Transformation
In order to preserve the property of stability, If the continuous system has poles only in the left half of the s-plane,
then the discrete- time filter must have poles only inside the unit circle.

25. 7.1.1 Filter Design by Impulse Invariance
The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system.
Relationship between two systems:
i.e. the continuous-time filter is bandlimited, and Ωs >2Ωmax
in one period

26. relation between frequencies
Relationship between two systems:
S plane
Z plane -
No Aliasing
one period

27. No Aliasing in the Impulse Invariance

28. Aliasing in the Impulse Invariance

29. Steps of DT filter design by transforming a prototype continuous-time filter
① Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
② Find the system function of the continuous- time filter.
③ Transform the continuous-time filter to derive the system function of the discrete-time filter.

30. ① Transformation of specifications from discrete to continuous
In impulse invariance design procedure, assuming the aliasing in transformation is neglected,
relation of specifications in transformation is

31. Steps of DT filter design by transforming a prototype continuous-time filter
① Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
② Find the system function of the continuous-time filter.
③ Transform the continuous-time filter to derive the system function of the discrete-time filter.

32. ② Find Continuous-time IIR filters
Butterworth filters
Chebyshev Type I filters
Chebyshev Type II filters
Elliptic filters

33. Steps of DT filter design by transforming a prototype continuous-time filter
① Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
② Find the system function of the continuous-time filter.
③ Transform the continuous-time filter to derive the system function of the discrete-time filter.

34. ③ Transformation from continuous to discrete
Laplace Transform
impulse invariance
z-Transform

35. Ex. 7.2 design DT filter by Impulse Invariance with a Butterworth Filter
Specifications for the discrete-time filter:
Solution: ①
Assume the effect of aliasing is negligible

36. Dependence of Butterworth magnitude characteristics on the order N.
② Butterworth Filter
monotonic function of frequency
magnitude-squared function
Dependence of Butterworth
magnitude characteristics on the order N.

37. Example 7.2 Impulse Invariance with a Butterworth Filter
monotonic function of frequency

38. Example 7.2 Impulse Invariance with a Butterworth Filter
round N up to
passband specifications met exactly
stopband specifications exceeded.
allows some margin for aliasing

39. Example 7.2 Impulse Invariance with a Butterworth Filter
Plole pairs:
Left half of the s-plane

40. Example 7.2 Impulse Invariance with a Butterworth Filter
Plole pairs:
Left half of the s-plane

41. Example 7.2 Impulse Invariance with a Butterworth Filter
Plole pairs:

42. Example 7.2 ③
Ak, Sk same as Td=1
same as Td=1

43. Example 7.2 Impulse Invariance with a Butterworth Filter
Frequency response of 6th-order Butterworth filter
Log magnitude in dB
Magnitude
Group delay

44. Program of Ex.7.2 using Impulse Invariance
clear all;
wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( ); Fs=1;
wap=wp*Fs; was=ws*Fs;
[n,wn]=buttord(wap,was,rp,rs,'s') % Note: 's'!
[z,p,k]=buttap(n);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
%[bs,as] = butter(n,wn,'s');
[bz,az]=impinvar(bs,as,Fs)
[Hz,Wz] = freqz(bz,az,512);
subplot(311);plot(Wz,20*log10(abs(Hz)),'r');
Butterworth analog lowpass filter prototype

45. Program of Ex.7.2 using Impulse Invariance
[Hz,Wz] = freqz(bz,az,512);
subplot(311)
plot(Wz,20*log10(abs(Hz)),'r');grid on
axis ([0 pi ])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',-100:20:20)
set(gca,'YTickLabel',{'-100','-80','-60','-40','-20','0','20'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi 0 1.2])
set(gca,'YTick',0:0.2:1.2)
set(gca,'YTickLabel',{'0','0.2','0.4','0.6','0.8','1.0','1.2'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 12])
set(gca,'YTick',0:2:12)
set(gca,'YTickLabel',{'0','2','4','6','8','10','12'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

46. Program of Ex.7.2 using Impulse Invariance
[Hz,Wz] = freqz(bz,az,512);
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',0:0.2:1.2)
set(gca,'YTickLabel',{'0','0.2','0.4','0.6','0.8','1.0','1.2'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 12])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'YTick',0:2:12)
set(gca,'YTickLabel',{'0','2','4','6','8','10','12'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

47. Program of Ex.7.2 using Impulse Invariance
Log magnitude in dB
Magnitude
Group delay

48. summary for Impulse Invariance
The basis for Impulse Invariance is to choose an impulse response for the DT-filter
that is similar in some sense to the impulse response of the CT-filter.
If the CT filter is bandlimited, then the DT filter frequency response will closely approximate the CT filter frequency response.

49. summary for Impulse Invariance
The relationship between CT and DT frequency is linear;
consequently, except for aliasing, the shape of the frequency response is preserved.

50. summary for Impulse Invariance
The relationship between CT and DT frequency is linear;
consequently, except for aliasing, the shape of the frequency response is preserved.

51. 7.1.2 Bilinear Transformation
Bilinear transformation can avoid the problem of aliasing.
Bilinear transformation maps onto
Bilinear transformation:

52. 7.1.2 Bilinear Transformation

53. 7.1.2 Bilinear Transformation

54. 7.1.2 Bilinear Transformation
Derivation (textbook): relationship between w and Ω

55. relation between frequency response for CT, DT filter
2018年11月29日6时3分
relation between frequency response for CT, DT filter
Same tolerance limit for
Passband, stopband

56. Comments on the Bilinear Transformation
It avoids the problem of aliasing encountered with the use of impulse invariance.
It is nonlinear compression of frequency axis.
S plane
Z plane -
S plane
Z plane

57. Comments on the Bilinear Transformation
bilinear transformation is useful only when this compression can be tolerated or compensated for,
as the case of filters that approximate ideal piecewise-constant magnitude- response characteristics.

58. Bilinear Transformation of
distortion in the frequency axis also manifests warping of the phase response of the filter

59. Comparisons of Impulse Invariance and Bilinear Transformation
The use of bilinear transformation is restricted to the design of approximations to filters with piecewise-constant magnitude-response characteristics,
such as highpass, lowpass and bandpass filters.
Impulse invariance can be used to design lowpass filters.
However, it cannot be used to map highpass CT designs to highpass DT designs because they are not bandlimited.

60. Comparisons of Impulse Invariance and Bilinear Transformation
Bilinear transformation cannot design filter whose magnitude-response isn’t piecewise constant, like differentiator.
However, Impulse invariance can design an bandlimited differentiator.

61. 7.1.3 Examples of Bilinear Transformation
Example 7.3 is to illustrate the steps in the design of a Butterworth filter using the bilinear transformation.
Example 7.4, 7.5 , 7.6 illustrate design of:
Butterworth Filter;
Chebyshev (I, II) Approximation;
Elliptic Approximation

62. Bilinear Transformation of a Butterworth Filter
Ex.7.3
Design DT filter, given specification:
Solution: (1) CT fiter spec,
(prewarp)

63. Ex.7.3 Bilinear Transformation: Butterworth Filter
passband specifications exceeded.
stopband specifications met exactly
(2) Butterworth Filter

64. Example 7.2 Impulse Invariance with a Butterworth Filter
Plole pairs:
Left half of the s-plane

65. Example 7.3 Bilinear Transformation of a Butterworth Filter
Plole pairs:
(3)
same as

66. Ex. 7.3 frequency response of DT filter
Log magnitude in dB
Magnitude
Group delay

67. Program of Ex.7.3 by Bilinear Transformation
clear all;
wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( ); Fs=1;
wap=2*Fs*tan(wp/2);was=2*Fs*tan(ws/2);
[n,wn]=buttord(wap,was,rp,rs,'s') % Note: 's'!
[z,p,k]=buttap(n);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
%[bs,as] = butter(n,wn,'s');
[bz,az]=bilinear(bs,as,Fs)
[Hz,Wz] = freqz(bz,az,512);
subplot(311);plot(Wz,20*log10(abs(Hz)),'r');
Butterworth analog lowpass filter prototype
[n,Wn]=buttord(.2*pi/pi,0.3*pi/pi,rp,rs) % no 's‘!
[bs,as] = butter(n,wn);

68. Program of Ex.7.3 by Bilinear Transformation
[Hz,Wz] = freqz(bz,az,512);
subplot(311)
plot(Wz,20*log10(abs(Hz)),'r');grid on
axis ([0 pi ])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',-100:20:20)
set(gca,'YTickLabel',{'-100','-80','-60','-40','-20','0','20'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi 0 1.2])
set(gca,'YTick',0:0.2:1.2)
set(gca,'YTickLabel',{'0','0.2','0.4','0.6','0.8','1.0','1.2'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 12])
set(gca,'YTick',0:2:12)
set(gca,'YTickLabel',{'0','2','4','6','8','10','12'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

69. Program of Ex.7.3 by Bilinear Transformation
Log magnitude in dB
Magnitude
Group delay

70. Program of Ex.7.3直接用数字滤波器设计指令
clear all; Fs=1
wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( );
[n,Wn]=buttord(.2*pi/pi,0.3*pi/pi,rp,rs) % no 's'!
[bz,az]=butter(n,Wn);
freqz(bz,az,512, Fs)
axis ([ ])
figure
zplane(bz,az)
[z,p,k]=tf2zp(bz,az)
zplane(z,p,k)

71. 7.1.3 Examples of Bilinear Transformation
Butterworth Filter,
Chebyshev Approximation, I, II
Elliptic Approximation
Example7.4, 7.5, 7.6
Design DT filter, given specification:

72. Example 7.4 Butterworth Approximation
CT filter specification

73. Example 7.4 frequency response
Log magnitude in dB
Magnitude in passband only
Group delay

74. Example 7.4 frequency response
1.007
Magnitude in passband only
passband edge frequency
0.99
passband
stopband
0.983

75. Program of Ex.7.4 by Bilinear Transformation
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
wap=2*Fs*tan(wp/2);was=2*Fs*tan(ws/2);
[n,wn]=buttord(wap,was,rp,rs,'s') % Note: 's'!
[z,p,k]=buttap(n);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
% [bs,as] = butter(n,wn,'s');
[bz,az]=bilinear(bs,as,Fs)
bz=bz *(1+e);
[Hz,Wz] = freqz(bz,az,512);
% Hz=Hz*(1+e);

76. Program of Ex.7.4 by Bilinear Transformation
[Hz,Wz] = freqz(bz,az,512);
% Hz=Hz*(1+e);
subplot(311)
plot(Wz,20*log10(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',-100:20:20)
set(gca,'YTickLabel',{'-100','-80','-60','-40','-20','0','20'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.99:0.005:1.01)
set(gca,'YTickLabel',{'0.99','0.995','1.0','1.005','1.01'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 25])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

77. Program of Ex.7.4 by Bilinear Transformation
Log magnitude in dB
Frequency response
Magnitude
Group delay

78. Program of Ex.7.4直接用数字滤波器设计指令
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
[n,Wn]=buttord(.4*pi/pi, .6*pi/pi,rp,rs) % no's'!
[z,p,k]=butter(n,Wn);
[sos,g] = zp2sos(z,p,k);
Hd = dfilt.df2tsos(sos,g);
h = fvtool(Hd)
figure
zplane(z,p,k)
[bz,az]=zp2tf (z,p,k)

79. Chebyshev filter (type I)
Chebyshev filters
Chebyshev filter (type I)
双曲余弦
Chebyshev polynomial
N=0, V0(x)=1;
N=1, V1(x)=cos(cos-1x)= x;
N=2, V2(x)=cos(2cos-1x) =2x2-1;
N=3, V3(x)=4x3-3x
VN+1(x)=2xVN(x) - VN-l(x)

80. Chebyshev filter (type I)
Chebyshev filters
Chebyshev filter (type I)
双曲余弦
Chebyshev polynomial

81. Chebyshev filter (type I)
Chebyshev filters
Chebyshev filter (type I)
Chebyshev polynomial x

82. Ws is stopband cut-off frequency
Three parameters of Chebyshev filter：
Wc ：given passband cut-off frequency
e ：is specified by the allowable passband ripple
由通带衰减(passband attenuation)决定
N：filter order，
等于通带内最大最小值的总数
Ws is stopband cut-off frequency
阻带衰减越大所需阶数越高

83. Chebyshev低通滤波器幅度平方函数的极点分布
以上Ωc,ε和N确定后，可以求出滤波器的极点，并确定Ha(p)，p=s/Ωc。
有用的结果:设Ha(s)的极点为si=σi+jΩi，可证明：
上式是一个椭圆方程，因为ch(x)大于sh(x)，长半轴为Ωcchξ(在虚轴上)，短半轴为Ωcshξ(在实轴上)。

84. 令bΩc和aΩc分别表示长半轴和短半轴，可推导出：
设N=3，平方幅度函数的极点分布如图所示(极点用X表示)。为稳定，用左半平面的极点构成Ha(p)，即

85. c是待定系数。根据幅度平方函数式可导出：c=ε·2N-1，代入上式，得到归一化的传输函数为
去归一化后的传输函数为
三阶切比雪夫滤波器的极点分布图

87. Chebyshev filters Chebyshev filter (type II) Chebyshev filter (type I)1
Chebyshev filter (type I)

88. Example 7.5 Chebyshev Type I , II Approximation

89. Example 7.5 frequency response of Chebyshev
Type I
Type II
Magnitude in passband only

90. Program of Ex.7.5 by Bilinear Transformation
% Chebyshev I Matlab program:
clear all; Fs=1; e=0.007(书图中最大幅值大约值1+e)
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
wap=2*Fs*tan(wp/2);was=2*Fs*tan(ws/2);
[n,wn]=cheb1ord(wap,was,rp,rs,'s') % 's'!
[z,p,k]=cheb1ap(n,rp);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
% [bs,as] = cheby1(n,rp,wn,'s');
[bz,az]=bilinear(bs,as,Fs)
bz=bz *(1+e);

91. Program of Ex.7.5 by Bilinear Transformation
% Chebyshev II Matlab program
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
wap=2*Fs*tan(wp/2);was=2*Fs*tan(ws/2);
[n,wn]=cheb2ord(wap,was,rp,rs,'s')
[z,p,k]=cheb2ap(n,rs);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
%[bs,as] = cheby2(n,rs,wn,'s');
[bz,az]=bilinear(bs,as,Fs)
bz=bz *(1+e);
be careful !

92. Program of Ex.7.5 by Bilinear Transformation
[Hz,Wz] = freqz(bz,az,512);
%Hz=Hz*(1+e);
subplot(311)
plot(Wz,20*log10(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',-100:20:20)
set(gca,'YTickLabel',{'-100','-80','-60','-40','-20','0','20'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r'); axis ([0 pi ])
set(gca,'YTick',0.99:0.005:1.01)
set(gca,'YTickLabel',{'0.99','0.995','1.0','1.005','1.01'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 25])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

93. Program of Ex.7.5 by Bilinear Transformation
Frequency response
Chebyshev I型

94. Program of Ex.7.5 by Bilinear Transformation
Log magnitude in dB
Frequency response
Chebyshev II型
Magnitude in passband
Group delay

95. Program of Ex.7.5直接用数字滤波器设计指令
% Chebyshev I型M程序如下
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
[n,wn]=cheb1ord (.4*pi/pi,0.6*pi/pi,rp,rs)
[z,p,k]= cheby1(n,rp,wn);
k=k*(1+e);
[sos,g] = zp2sos(z,p,k);
Hd = dfilt.df2tsos(sos,g);
h = fvtool(Hd)
figure
zplane(z,p,k)
[bz,az]=zp2tf (z,p,k)

96. Program of Ex.7.5直接用数字滤波器设计指令
% Chebyshev II型M程序如下
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
[n,wn]=cheb2ord (.4*pi/pi,0.6*pi/pi,rp,rs)
[z,p,k]= cheby2(n,rs,wn);
k=k*(1+e);
[sos,g] = zp2sos(z,p,k);
Hd = dfilt.df2tsos(sos,g);
h = fvtool(Hd)
figure
zplane(z,p,k)
[bz,az]=zp2tf (z,p,k)
zplane(bz,az)

97. elliptic filters 1
Elliptic filter
Jacobian elliptic function

98. Example 7.6 Elliptic Approximation

99. Example 7.6 frequency response of Elliptic
Log magnitude in dB
Magnitude in passband
Group delay

100. Program of Ex.7.6 by Bilinear Transformation
% Elliptic filter M程序如下
clear all; Fs=1; e=0.005
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
wap=2*Fs*tan(wp/2);was=2*Fs*tan(ws/2);
[n,wn]=ellipord(wap,was,rp,rs,'s')
[z,p,k]=ellipap(n,rp,rs);
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
%[bs,as] =ellip(n,rp,rs,wn,'s');
[bz,az]=bilinear(bs,as,Fs);
bz=bz *(1+e);

101. Program of Ex.7.5 by Bilinear Transformation
[Hz,Wz] = freqz(bz,az,512);
%Hz=Hz*(1+e);
subplot(311)
plot(Wz,20*log10(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
set(gca,'YTick',-100:20:20)
set(gca,'YTickLabel',{'-100','-80','-60','-40','-20','0','20'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r'); axis ([0 pi ])
set(gca,'YTick',0.99:0.005:1.01)
set(gca,'YTickLabel',{'0.99','0.995','1.0','1.005','1.01'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 25])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

102. Program of Ex.7.5 by Bilinear Transformation
Log magnitude in dB
Frequency response
Elliptic filter
Magnitude in passband
Group delay

103. Program of Ex.7.4直接用数字滤波器设计指令
% Elliptic filter M程序如下
clear all; Fs=1; e=0.007
wp=.4*pi; rp=-20*log10(0.99*(1/(1+e)));
ws=.6*pi ; rs=-20*log10(0.001*(1/(1+e)));
[n,wn]=ellipord (.4*pi/pi,0.6*pi/pi,rp,rs)
[z,p,k]= ellip(n,rp,rs,wn);
k=k*(1+e);
[sos,g] = zp2sos(z,p,k);
Hd = dfilt.df2tsos(sos,g);
h = fvtool(Hd)
figure
zplane(z,p,k)
[bz,az]=zp2tf (z,p,k)
zplane(bz,az)

104. Butterworth Filter : N=14. ( max flat)
*Comparison of Butterworth, Chebyshev, elliptic filters: Example
-Given specification
-Order
Butterworth Filter : N=14. ( max flat)
Chebyshev Filter : N= ( Cheby 1, Cheby 2)
Elliptic Filter : N= ( equiripple) B C E

105. -Pole-zero plot (analog)B C1 C2 E
-Pole-zero plot (digital) B C1 C2 E
(14)
(8)

106. -Group delay
-Magnitude C1 B B E C1 E C2 C2

107. 7.2 Design of FIR Filters by Windowing
FIR filters are designed based on directly approximating the desired frequency response of the discrete-time system.
Most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint.

108. Window Method An ideal desired frequency response
Many idealized systems have discontinuous piecewise-constant frequency response, and their impulse responses are noncausal and infinitely long.

109. Window Method
The most straightforward approach for obtaining a causal FIR approximation is to truncate the ideal impulse response.

110. Windowing in Frequency Domain
Windowed frequency response
The windowed version is smeared version of desired response
Gibbs phenomenon

111. Window Method If

112. Choice of Window
is as short as possible in duration. This minimizes computation in the implementation of the filter.
approximates an impulse.

113. Window Method
If is chosen so that is concentrated in a narrow band of frequencies around
then would look like , except where changes very abruptly.

114. Rectangular Window
for the rectangular window has a generalized linear phase.
As M increases, the width of the “main lobe” decreases.

115. will oscillate at the discontinuity.
Rectangular Window
will oscillate at the discontinuity.
The oscillations occur more rapidly, but do not decrease in magnitude as M increases.
Gibbs phenomenon

116. Rectangular Window
In theory of Fourier series, this nonuniform convergence, the Gibbs phenomenon can be moderated through the use of less abrupt truncation of the Fourier series.

117. Rectangular Window
By tapering the window smoothly to zero at each end, the height of the side lobes can be diminished.
The expense is a wider main lobe and thus a wider transition at the discontinuity. 主瓣

118. 7.2.1 Properties of Commonly Used Windows
Rectangular
Bartlett (triangular)
M-1 non zero values

119. 7.2.1 Properties of Commonly Used Windows
Hanning
Hamming

120. 7.2.1 Properties of Commonly Used Windows
Blackman

121. 7.2.1 Properties of Commonly Used Windows

122. 主瓣 Hamming Blackman Rectangular Bartlett (triangular) Hanning
Frequency Spectrum of Windows
(a)-(e) attenuation of sidelobe increases, width of mainlobe increases.

123. 7.2.1 Properties of Commonly Used Windows
Table 7.1
边瓣幅值 主瓣 宽度
biggest，high oscillations at discontinuity
smallest，the sharpest transition

124. 7.2.2 Incorporation of Generalized Linear Phase
In designing FIR filters, it is desirable to obtain causal systems with a generalized linear phase response.
The above five windows are all symmetric about the point ,i.e.,

125. 7.2.2 Incorporation of Generalized Linear Phase
Their Fourier transforms are of the form

126. 7.2.2 Incorporation of Generalized Linear Phase
2018年11月29日6时3分
7.2.2 Incorporation of Generalized Linear Phase
Hilbert filter hd[n]

127. Frequency Domain Representation
Linear Phase

128. Ex. 7.7 design Linear-Phase Lowpass Filter
The desired frequency response is
verification
Solution:

129. magnitude frequency response
Example:design a FIR filter M

130. 例7.7 matlab program for FIR Filter disign，specification：
Solution:
clc;clear all; close all
wp=0.15*pi;
ws=0.25*pi;
wdelta=ws-wp;
N=ceil(8*pi/wdelta);
Wn=( )*pi/2;
b=fir1(N,Wn/pi,hanning(N+1));
figure
% freqz(b,1,512)
[H,w]=freqz(b,1,512) ;
plot(w,abs(H))
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
axis([0,pi,0.97,1.03]);
Hanning窗

131. 7.2 Design of FIR Filters by Windowing Method
Review
To design an lowpass FIR Filters

132. 7.2.1 Properties of Commonly Used Windows
smallest，the sharpest transition
biggest，high oscillations at discontinuity

133. 7.2.3 The Kaiser ['kaizə] Window Filter Design Method
Caesar[ˈsi:zə] 凯撒
Trade side-lobe amplitude for main-lobe width

134. continuous envelopes of Kaiser windows
(b)
M=20
As  increases, attenuation of sidelobe increases, width of mainlobe increases.
 =6
(c)
As M increases, attenuation of sidelobe is preserved, width of mainlobe decreases.
(a) Window shape, M=20
(b) Frequency spectrum, M=20,
(c) beta=6
Figure 7.24

135. 2018年11月29日6时3分
Comparison
If the window is tapered more, the side lobe of the Fourier transform become smaller，
but the main lobe become wider.
Increasing M wile holding  constant causes the main lobe to decrease in width,
M=20
but does not affect the amplitude of the side lobe.
 =6
M=20

136. Transition width is a little less than mainlobe width
Table 7.1 Comparison Of Commonly Used Windows- with Kaiser windows
Transition width is a little less than mainlobe width

137. Filter Design by Kaiser Window
(1) the specifications must be established :
(2)
(3)
Peak approximation error
Transition width
To predict the values of M and β

138. Filter Design by Kaiser Window
(3)
(4)
magnitude response validation，
(5) linear phase, group delay

139. Example 7.8 Kaiser Window Design of a Lowpass Filter
analysis: to get
need
need

140. Ex. 7.8 Kaiser Window Design of a Lowpass Filter
Solution:

141. Example 7.8 Kaiser Window Design of a Lowpass Filter

142. Ex. 7.8 Kaiser Window Design of a Lowpass Filter
magnitude response validation，
peak approximation
error is δ= instead of the desired value of
h[n]=[ ]
In this case it is necessary
to increase M to 38 in order to meet the specifications.
phase response

143. Ex. 7.8 Kaiser Window Design of a Lowpass Filter
clear all; clc;
wp=0.4*pi;
ws=0.6*pi;
Wn=(wp+ws)/2;
b=fir1(37,Wn/pi, kaiser(38,5.653));
fcuts = [wp/pi ws/pi];
mags = [1 0];
devs = [ ];
[n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs);
b= fir1(n,Wn,ftype,kaiser(n+1,beta));
[H,w]=freqz(b,1,2048) ;
A(1:ceil(0.4*2048))=1-abs(H(1:ceil(0.4*2048)));
A(floor(0.6*2048)-1:2048)=-abs(H(floor(0.6*2048)-1:2048));
A(round(0.4*2048)+1:floor(0.6*2048)-1)=nan;
subplot(311)
stem(b)
subplot(312)
plot(w,20*log10(abs(H)))
axis([0,pi,-100,20]);
subplot(313)
plot(w,A)
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
axis([0,pi,-0.001,0.001]);

144. 7.3 Examples of FIR Filters Design by the Kaiser Window Method
The ideal highpass filter with generalized linear phase

145. Example 7.9 Kaiser Window Design of a Highpass Filter
Specifications:
By Kaiser window method

146. Ex. 7.8 Kaiser Window Design of a Lowpass Filter
clear all; clc;
wp=0.5*pi;
ws=0.35*pi;
fcuts = [ws/pi wp/pi];
mags = [0 1];
devs = [ ];
[n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs);
b=fir1(n,Wn,ftype,kaiser(n+1,beta));
[H,w]=freqz(b,1,2048) ;
A(1:ceil(ws/pi*2048))=-abs(H(1:ceil(ws/pi*2048)));
A(floor(wp/pi*2048)-1:2048)=1-abs(H(floor(wp/pi*2048)-1:2048));
A(round(ws/pi*2048)+1:floor(wp/pi*2048)-1)=nan;
subplot(311)
stem(b)
subplot(312)
plot(w,20*log10(abs(H)))
axis([0,pi,-100,20]);
subplot(313)
plot(w,A)
set(gca,'XTick',0:0.2*pi:pi)
set(gca,'XTickLabel',{'0','0.2π','0.4π','0.6π','0.8π','π'})
axis([0,pi,-0.025,0.025]);
n =24, beta =2.5974

147. 7.3.2 Discrete-Time Differentiator
M=10,even
Type III
M=5,odd
Type IV

148. Example 7.10 Kaiser Window Design of a Differentiator
Suppose
Ideal Impulse response
Kaiser window
type III FIR differentiator.
Impulse response

149. Example 7.10 Kaiser Window Design of a Differentiator
Suppose
Magnitude response
Approximation error for A0(ejW ).

150. Example 7.10 Kaiser Window Design of a Differentiator
Suppose
Ideal Impulse response
Kaiser window
type IV FIR differentiator.
Impulse response

151. Example 7.10 Kaiser Window Design of a Differentiator
Suppose
Magnitude response
Approximation error for A0(ejW ).

152. Group Delay
Phase:
Group Delay:
Noninteger delay

153. 7.4 Optimum Approximations of FIR Filters
Goal: Design a ‘best’ filter for a given M
In designing a causal type I linear phase FIR filter, it is convenient first to consider the design of a zero phase filter.
Then insert a delay sufficient to make it causal.

154. 7.4 Optimum Approximations of FIR Filters
zero phase filter

155. 7.4 Optimum Approximations of FIR Filters
Designing a filter to meet these specifications is to find the (L+1) impulse response values
Packs-McClellan algorithm is the dominant method for optimum design of FIR filters.
In Packs-McClellan algorithm,
is fixed, and is variable.

156. 7.4 Optimum Approximations of FIR Filters

157. 7.4 Optimum Approximations of FIR Filters

158. 7.4 Optimum Approximations of FIR Filters

159. Minimax criterion
Within the frequency interval of the passband and stopband, we seek a frequency response that minimizes the maximum weighted approximation error of

160. Other criterions

161. Alternation Theorem
Let denote the closed subset consisting of the disjoint union of closed subsets of the real axis x.
is an r th-order polynomial.
denotes a given desired function of x that is continuous on
is a positive function, continuous on
The weighted error is
The maximum error is defined as

162. Alternation Theorem
A necessary and sufficient condition that be the unique rth-order polynomial that minimizes is that exhibit at least (r+2) alternations; i.e., there must exist at least (r+2) values in such that
and such that
for

163. Example 7.11 Alternation Theorem and Polynomials
suppose Each of these polynomials is of fifth order.

164. Example 7.11 Alternation Theorem and Polynomials
The closed subsets of the real axis x referred to in the theorem are the regions
P3(x) is the unique optimal 5th-order polynomial approximation for this region.

165. 7.4.1 Optimal Type I Lowpass Filters
For Type I lowpass filter
The desired lowpass frequency response
Weighting function

166. 7.4.1 Optimal Type I Lowpass Filters
The weighted approximation error is
The closed subset is or

167. 7.4.1 Optimal Type I Lowpass Filters
The alternation theorem states that a set of coefficients will correspond to the filter representing the unique best approximation to the ideal lowpass filter with the ratio fixed at K and with passband and stopband edge and if and only if exhibits at least (L+2) alternations on ;

168. 7.4.1 Optimal Type I Lowpass Filters
i.e., if and only if alternately equals plus and minus its maximum value at least (L+2) times.
Such approximations are called equiripple approximations.

169. 7.4.1 Optimal Type I Lowpass Filters
The alternation theorem states that the optimum filter must have a minimum of (L+2) alternations, but does not exclude the possibility of more than (L+2) alternations.
In fact, for a lowpass filter, the maximum possible number of alternations is (L+3).

170. 7.4.1 Optimal Type I Lowpass Filters
Because all of the filters satisfy the alternation theorem for L=7 and for the same value of , it follows that and/or must be different for each ,since the alternation theorem states that the optimum filter under the conditions of the theorem is unique.

171. Property for type I lowpass filters from the alternation theorem
The maximum possible number of alternations of the error is (L+3)
Alternations will always occur at and
All points with zero slope inside the passband and all points with zero slope inside stopband will correspond to alternations; i.e., the filter will be equiripple, except possibly at and

172. 7.4.2 Optimal Type II Lowpass Filters
For Type II causal FIR filter:
The filter length (M+1) is even, ie, M is odd
Impulse response is symmetric
The frequency response is

173. 7.4.2 Optimal Type II Lowpass Filters

174. 7.4.2 Optimal Type II Lowpass Filters
For Type II lowpass filter,

175. 7.4.3 The Park-McClellan Algorithm
From the alternation theorem, the optimum filter will satisfy the set of equation

176. 7.4.3 The Park-McClellan Algorithm
Guessing a set of alternation frequencies
and

177. 7.4.3 The Park-McClellan Algorithm

178. 7.4.3 The Park-McClellan Algorithm
For equiripple lowpass approximation
Filter length: (M+1)

179. 7.5 Examples of FIR Equiripple Approximation 7.5.1 Lowpass Filter

180. Comments
M=26, Type I filter
The minimum number of alternations is (L+2)=(M/2+2)=15
7 alternations in passband and 8 alternations in stopband
The maximum error in passband and stopband are and , which exceed the specifications.

181. M=27, Type II filter, zero at z=-1
7.5.1 Lowpass Filter
M=27, Type II filter, zero at z=-1
The maximum error in passband and stopband are and , which exceed the specifications.
The minimum number of alternations is (L+2)=(M-1)/2+2=15
7 alternations in passband and 8 alternations in stopband

182. Comparison
Kaiser window method require M=38 to meet or exceed the specifications.
Park-McClellan method require M=27
Window method produce approximately equal maximum error in passband and stopband.
Park-McClellan method can weight the error differently.

183. 7.6 Comments on IIR and FIR Discrete-Time Filters
What type of system is best, IIR or FIR?
IIR: closed-form design formulas
straightforward noniterative computer programs for IIR filter design.
一次设计，指标要求满足
FIR: can have a (generalized) linear phase
iteration may be necessary to meet a prescribed spec­ification.
一次设计，未必满足指标要求

184. 7.6 Comments on IIR and FIR Discrete-Time Filters
Why give so many different design methods?
Which method yields the best result?
Closed-Form Formulas
Generalized Linear Phase
Order
IIR
Yes No
Low
FIR
High

185. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 5 HW
7.2, 7.4, 7.15,
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
199
2018/11/29
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