1. Biomedical Signal processing Chapter 7 Filter Design Techniques
2018年11月16日7时43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)》
2018/11/16 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 7 Filter Design Techniques
7.0 Introduction
7.1 Filter Specifications
7.2 Design of Discrete-Time IIR Filters From
Continuous-Time Filters
7.3 Discrete-Time Butterworth, Chebyshev
and Elliptic Filters
7.4 Frequency Transformations of Lowpass
IIR Filters
7.5 Design of FIR Filters by Windowing
7.6 Examples of FIR Filters Design by the
Kaiser Window Method

3. Filter Design Techniques
7.0 Introduction

4. 2018年11月16日7时43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 (IIR, FIR).

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

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

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

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

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

10. 7.1 Filter Specifications
2018年11月16日7时43分
Figure depicts the typical representation of the tolerance limits associated with approximating a discrete-time ideal lowpass filter.
tolerance scheme 容限图
may vary symmetrically around 1,
δp1=δp2 or
δp1=0
edge frequency

11. 7.1 Filter Specifications
2018年11月16日7时43分
no constraints on phase response other than implicit requirements of stability and causality for IIR filter.
The constraint for FIR filters: linear phase.
tolerance scheme 容限图
may vary symmetrically around 1,
δp1=δp2 or
δp1=0
edge frequency

12. Ex. 7.1 Specifications for LTI DT 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:

13. Ex. 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

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

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

16. Ex. 7.1 Determining Specifications for a Discrete-Time Filter
ideal passband gain in decibels
maximum passband gain
passband gain at passband edge
maximum stopband gain
tolerance scheme
容限图
edge frequency

17. 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.

18. Filter Design Techniques
7.2 Design of Discrete-Time IIR Filters From Continuous-Time Filters

19. 7.2 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

20. 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.

21. 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.

22. 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.

23. Steps of DT filter design by transforming a prototype continuous-time filter
①Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
②Determine the system function Hc(s) or hc(t) of the continuous-time filter.
③System function H(z) or h[n] of discrete-time filter is obtained by applying transformation to Hc(s) or hc(t).
require that the essential properties of the continuous-time frequency response be preserved.

24. 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

25. 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.

26. 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.

27. 7.2.1 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

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

29. No Aliasing in the Impulse Invariance

30. Aliasing in the Impulse Invariance

31. Steps of DT filter design by transforming a prototype continuous-time filter
①Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
②Determine the system function Hc(s) or hc(t) of the continuous-time filter.
③System function H(z) or h[n] of discrete-time filter is obtained by applying transformation to Hc(s) or hc(t).

32. 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:

33. Steps of DT filter design by transforming a prototype continuous-time filter
①Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
②Determine the system function Hc(s) or hc(t) of the continuous-time filter.
③System function H(z) or h[n] of discrete-time filter is obtained by applying transformation to Hc(s) or hc(t).

34. ②Determine the system function Hc(s) of Continuous-time IIR filters
Butterworth filters
Chebyshev Type I filters
Chebyshev Type II filters
Elliptic filters

35. Steps of DT filter design by transforming a prototype continuous-time filter
①Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.
②Determine the system function Hc(s) or hc(t) of the continuous-time filter.
③System function H(z) or h[n] of discrete-time filter is obtained by applying transformation to Hc(s) or hc(t).

36. ③ find H(z) or h[n] by applying transformation to Hc(s) or hc(t)
Laplace Transform 因果 稳定
impulse invariance
z-Transform
Left half of the s-plane
z-plane: inside unit circle

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

38. ② find Hc(s) or hc(t) Butterworth Filter
magnitude-squared function is
monotonic function of frequency
magnitude characteristics of Butterworth depend on order N.

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

40. 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

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

42. Example 7.2 ③
Td=1

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

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

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

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

47. 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

48. 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 ');

49. 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 ');

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

51. summary for Impulse Invariance
The basis for Impulse Invariance is to choose an impulse response h[n] for the DT-filter
that is similar to h(t) of the CT-filter.
If CT filter is bandlimited, then the DT filter frequency response will closely approximate the CT filter frequency response.

52. 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.
aliasing frequency

53. 7.2.2 Bilinear Transformation
Bilinear transformation can avoid the problem of aliasing.
Bilinear transformation maps onto
Bilinear transformation:

54. 7.2.2 Bilinear Transformation

55. 7.2.2 Bilinear Transformation

56. 7.2.2 Bilinear Transformation
Derivation (textbook): relationship between w and Ω

57. relation between frequency response for CT, DT filter
2018年11月16日7时43分
relation between frequency response for CT, DT filter
Same tolerance limit for
Passband, stopband

58. 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

59. 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.

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

61. 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.

62. 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.

63. 7.3 Discrete-Time Butterworth, Chebyshev and Elliptic Filters
Ap­pendix B: characteristics of these three classes of continuous­ time filters.
Butterworth Filter
monotonic

64. 7.3 Discrete-Time Butterworth, Chebyshev and Elliptic Filters
Chebyshev I Filter.
双曲余弦
Chebyshev polynomial x x

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

66. elliptic filters
N阶雅可比(Jacobian)椭圆函数
ε为通带波纹参数，N为阶数。

67. elliptic filters N阶雅可比(Jacobian)椭圆函数
[n,wn]=ellipord(wap,was,rp,rs,'s')
MATLAB函数涉及参数
ε为通带波纹参数，N为阶数由下式决定：

68. Design Butterworth, Chebyshev and Elliptic Filters
maximum passband gain
passband gain at passband edge
(minimum passband gain)
maximum stopband gain 1
MATLAB函数涉及参数
容限图 衰减
edge
[n,wn]=ellipord(wap,was,rp,rs,'s')
[z,p,k]=ellipap(n,rp,rs);

69. Design Butterworth, Chebyshev and Elliptic Filters
minimum passband gain
maximum stopband gain
用MATLAB函数： 衰减
[n,wn]=ellipord(wap,was,rp,rs,'s') % Note: 's'!模拟滤波器
[z,p,k]=ellipap(n,rp,rs); 原型低通椭圆滤波器: 截止频率wn=1
[n,wn]=buttord(wap,was,rp,rs,‘s’) ;模拟
[z,p,k]=buttap(n); 原型巴特沃斯滤波器: 截止频率wn=1
[n,wn]=cheb1ord(wap,was,rp,rs,'s') % 's'!
[z,p,k]=cheb1ap(n,rp);模拟原型切比雪夫I型滤波器
[bp,ap]=zp2tf(z,p,k)
[bs,as]=lp2lp(bp,ap,wn)
[bz,az]=bilinear(bs,as,Fs) 或
[bz,az]=impinvar(bs,as,Fs)

70. 7.3 Discrete-Time Butterworth, Chebyshev and Elliptic Filters
7.3.1 Examples of IIR Filter Design
Example 7.3 is to illustrate the steps in the design of a Butterworth filter using the bilinear transformation.
Example 7.4, 7.5 illustrate design of:
Butterworth Filter;
Chebyshev (I, II) Approximation;
Elliptic Approximation

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

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

73. Example 7.3 Bilinear Transformation of a Butterworth Filter
Left half of the s-plane
Plole pairs:

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

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

76. 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);

77. 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 ');

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

79. 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)
6-order zeros

80. 7.3.1 Examples of IIR Filter Design
Example7.4 design Butterworth, Chebyshev I,II, Elliptic filter.
maximum passband gain
minimum passband gain
maximum stopband gain
given specification:

81. Program of Ex.7.4 by Bilinear Transformation
clear all;
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; 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);

82. Program of Ex.7.4 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',-50:10:10)
set(gca,'YTickLabel',{'-50','-40','-30','-20','-10','0','10'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.94:0.02:1.02)
set(gca,'YTickLabel',{'0.94','0.96','0.98','1','1.02'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 22])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');

83. Ex.7.4 Butterworth filter
Log magnitude in dB
Magnitude
Group delay

84. Program of Ex.7.4 by Bilinear Transformation
Butterworth filter
clear all;
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,wn]=buttord(wp/pi,ws/pi,rp,rs) % Note: no's'!
[z,p,k] = butter(n,wn,'low');
[sos,g] = zp2sos(z,p,k); % Convert to SOS form
Hd=dfilt.df2tsos(sos,g); % Create a dfilt object
h=fvtool(Hd); % Plot magnitude response
set(h,'Analysis','freq') % Display frequency response
figure
zplane(z,p,k)

85. Program of Ex.7.4直接用数字滤波器设计指令
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,Wn]=buttord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=butter(n,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=butter(n,Wn);
Butterworth filter
15 zeros
figure
zplane(bz,az)
[z,p,k]=tf2zp(bz,az)
zplane(z,p,k)

86. Program of Ex.7.4直接用数字滤波器设计指令
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,Wn]=buttord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=butter(n,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=butter(n,Wn);
15 order zeros
[z,p,k]=tf2zp(bz,az)
zplane(z,p,k)
figure
zplane(bz,az)

87. Program of Ex.7.4直接用数字滤波器设计指令
Chebyshev I filter
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,Wn]=cheb1ord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=cheby1(n,rp,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=tf2zp(bz,az)
[z,p,k]=cheby1(n,rp,Wn);
figure
zplane(z,p,k)
7 order zeros

88. Program of Ex.7.4 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',-50:10:10)
set(gca,'YTickLabel',{'-50','-40','-30','-20','-10','0','10'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.94:0.02:1.02)
set(gca,'YTickLabel',{'0.94','0.96','0.98','1','1.02'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 22])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');
Chebyshev I filter

89. Ex.7.4 Chebyshev I filter
Log magnitude in dB
Magnitude
Group delay

90. Program of Ex.7.4直接用数字滤波器设计指令
Chebyshev II filter
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,Wn]=cheb2ord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=cheby2(n,rs,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=cheby2(n,rs,Wn);
[z,p,k]=tf2zp(bz,az)
figure
zplane(z,p,k)
zplane(bz,az)
7 order zeros

91. Program of Ex.7.4 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',-50:10:10)
set(gca,'YTickLabel',{'-50','-40','-30','-20','-10','0','10'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.94:0.02:1.02)
set(gca,'YTickLabel',{'0.94','0.96','0.98','1','1.02'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 22])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');
Chebyshev II filter

92. Ex.7.4 Chebyshev II filter
Log magnitude in dB
Magnitude
Group delay

93. Program of Ex.7.4直接用数字滤波器设计指令
Elliptic filter
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=30; Fs=1;
[n,Wn]=ellipord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=ellip(n,rp,rs,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=ellip(n,rp,rs,Wn);
[z,p,k]=tf2zp(bz,az)
figure
zplane(z,p,k)
zplane(bz,az)
5 order zeros

94. Program of Ex.7.4 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',-50:10:10)
set(gca,'YTickLabel',{'-50','-40','-30','-20','-10','0','10'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.94:0.02:1.02)
set(gca,'YTickLabel',{'0.94','0.96','0.98','1','1.02'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 22])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');
Elliptic filter

95. Ex.7.4 Elliptic filter
Log magnitude in dB
Magnitude
Group delay

96. Program of Ex.7.4直接用数字滤波器设计指令
Elliptic filter
clear all; Fs=1
wp=.5*pi; rp=0.3;
ws=.6*pi; rs=43; Fs=1;
[n,Wn]=ellipord(wp/pi,ws/pi,rp,rs) % no 's'!
[bz,az]=ellip(n,rp,rs,Wn);
freqz(bz,az,512, Fs)
axis ([0 ws/2 -rs 0])
[z,p,k]=ellip(n,rp,rs,Wn);
[z,p,k]=tf2zp(bz,az)
figure
zplane(z,p,k)
zplane(bz,az)
5 order zeros

97. Program of Ex.7.4 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',-50:10:10)
set(gca,'YTickLabel',{'-50','-40','-30','-20','-10','0','10'})
xlabel('Frequency (Hz)'), ylabel('Magnitude (dB)');
title('Log Magnitude Response ');
subplot(312)
plot(Wz,(abs(Hz)),'r');
axis ([0 pi ])
set(gca,'YTick',0.94:0.02:1.02)
set(gca,'YTickLabel',{'0.94','0.96','0.98','1','1.02'})
xlabel('Frequency (Hz)'), ylabel('Magnitude ');
title('Magnitude Response ');
subplot(313)
plot(Wz,(grpdelay(bz,az,Wz)),'r');
axis ([0 pi 0 22])
set(gca,'YTick',0:5:25)
set(gca,'YTickLabel',{'0','5','10','15','20','25'})
xlabel('Frequency (Hz)'), ylabel('groupdelay ');
title('groupdelay ');
Elliptic filter

98. Ex.7.4 Elliptic filter
Log magnitude in dB
rs=43
Magnitude
Group delay

99. Ex.7.4 Elliptic filter
Log magnitude in dB
rp=0.019
Magnitude
Group delay

100. 7.3.1 Examples of IIR Filter Design
Example7.5 Design Example for Comparison with FIR Designs(Section7.8.1 ).design Butterworth, Chebyshev I,II, Elliptic filter.
given specification:
Rescaling:

101. 7.3.1 Examples of IIR Filter Design
Example7.5 Design Example for Comparison with FIR Designs(Section7.8.1 ).design Butterworth, Chebyshev I,II, Elliptic filter.
given specification:
编程设计按:
Rescaling:

102. Example 7.5 Butterworth Approximation
1.007
CT filter specification

103. Example 7.5 frequency response
Log magnitude in dB
Magnitude in passband only
Group delay

104. Example 7.5 frequency response
1.007
Magnitude in passband only
passband edge frequency
0.99
passband
stopband
0.983

105. Program of Ex.7.5 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);

106. 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 ');

107. Program of Ex.7.5 by Bilinear Transformation
Log magnitude in dB
Frequency response
Magnitude
Group delay

108. Program of Ex.7.5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)

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

110. Example 7.5 Chebyshev Type I , II Approximation

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

112. 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);

113. 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 !

114. 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 ');

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

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

117. 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)
zplane(bz,az)

118. 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)

119. elliptic filters
N阶雅可比(Jacobian)椭圆函数
ε为通带波纹参数，N为阶数。

120. Example 7.5 Elliptic Approximation

121. Example 7.5 frequency response of Elliptic
Log magnitude in dB
Magnitude in passband
Group delay

122. Program of Ex.7.5 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);

123. 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 ');

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

125. Program of Ex.7.5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)

126. 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

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

128. 7.4 Frequency Transformations of Lowpass IIR Filters
Design other kinds of filters:
High-pass filters
Band-pass filters
Band-stop filters
Multiband filters
Tolerance scheme for amultiband filter.
Using the results of Low-pass filter and Frequency Band Transformation

129. 7.4 Frequency Transformations of Lowpass IIR Filters
Let HLP(Z) be the given prototype lowpass digital filter, and let H(z) be the desired frequency-selective digital filter.
Define a mapping:
Given that HLP(Z) is a stable and causal filter, we also want H(z) to be stable and causal.
This imposes the following requirements:

130. Frequency-band Transformation
stable and causal
This imposes the following requirements:
G(z-1) must be a rational function in z-1 so that H(z) is implementable.
The unit circle of the Z-plane must map onto the unit circle of the z-plane.
For stable filters, the inside of the unit circle of the Z-plane must also map onto the inside of the unit circle of the z-plane.

131. 7.4 Frequency Transformations of Lowpass IIR Filters
Let θ and ω be the frequency variables (angles) in the Z-plane and z-plane,
from condition 3
It is clear,
That is, the mapping must be all-pass:
where ensures that a stable filter is transformed into another stable filter (i.e. satisfies condition 2).

132. 7.4 Frequency Transformations of Lowpass IIR Filters
2018年11月16日7时43分
7.4 Frequency Transformations of Lowpass IIR Filters
the one that transforms a lowpass filter into another lowpass filter with different passband and stopband edge frequencies:

133. 7.4 Frequency Transformations of Lowpass IIR Filters
Warping of the frequency scale in lowpass-to-lowpass transformation
ωp: desired cutoff frequency θp

134. 7.4 Frequency Transformations of Lowpass IIR Filters
TABLE 7.1 Transformations from a lowpass digital filter prototype
of cutoff frequency ωp to highpass, bandpass, and bandstop filter
Filter Type Transformations Associated Design Formulas

135. 7.4 Frequency Transformations of Lowpass IIR Filters
TABLE 7.1 Transformations from a lowpass digital filter prototype
of cutoff frequency ωp to highpass, bandpass, and bandstop filter
Filter Type Transformations Associated Design Formulas

136. 7.4 Frequency Transformations of Lowpass IIR Filters
Ex.7.6 Transformation of a Lowpass Filter to a Highpass Filter.
Type I Chebyshev lowpass filter
specifications:
transform the filter to a highpass filter with passband cutoff frequency ωp = 0.6π.
from TABLE 7.1

137. 7.4 Frequency Transformations of Lowpass IIR Filters
Ex.7.6 Type I Chebyshev Lowpass Filter→ Highpass Filter.
θp = 0.6π
ωp = 0.6π
from TABLE 7.1

138. 7.4 Frequency Transformations of Lowpass IIR Filters
Ex.7.6 Transformation of a Type I Chebyshev Lowpass Filter to a Highpass Filter.

139. 7.4 Frequency Transformations of Lowpass IIR Filters
Ex.7.6 Type I Chebyshev Lowpass filter to Highpass filter.

140. Program of Ex.7.6 by Bilinear Transformation
clear all;
wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( ); Fs=1;
wnh=.6*pi; wanh=2*Fs*tan(wnh/2); 高通截止频率
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]=lp2hp(bp,ap,wanh);
[bz,az]=bilinear(bs,as,Fs)
[Hz,Wz] = freqz(bz,az,512);
subplot(311);plot(Wz,20*log10(abs(Hz)),'r');

141. Program of Ex.7.3 by Bilinear Transformation
2018年11月16日7时43分
[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 ');

142. Program of Ex.7.6 by Bilinear Transformation
clear all; wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( ); Fs=1;
wnh=.6*pi; wanh=2*Fs*tan(wnh/2); 高通截止频率
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]=lp2hp(bp,ap,wanh);
[bz,az]=bilinear(bs,as,Fs)
figure
zplane(z,p,k)

143. Program of Ex.7.6 by Bilinear Transformation
clear all;
wp=.2*pi; rp=-20*log10( );
ws=.3*pi; rs=-20*log10( ); Fs=1;
wnh=.6*pi; wanh=2*Fs*tan(wnh/2); 高通截止频率
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]=lp2hp(bp,ap,wanh);
[bz,az]=bilinear(bs,as,Fs)
figure
zplane(z,p,k)

144. 7.5 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.

145. 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.

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

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

148. Window Method If

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

150. 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.

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

152. 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

153. 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.

154. Rectangular Window
使窗锥形化减小到0
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

155. 7.5.1 Properties of Commonly Used Windows
Rectangular
Bartlett (triangular)
M-1 non zero values

156. 7.5.1 Properties of Commonly Used Windows
Hanning
Hamming

157. 7.5.1 Properties of Commonly Used Windows
Blackman

158. 7.5.1 Properties of Commonly Used Windows

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

160. 7.5.1 Properties of Commonly Used Windows
Table 7.2
等效的Kaiser窗
边瓣幅值 主瓣 宽度
biggest，high oscillations at discontinuity
smallest，the sharpest transition
最小,最陡峭的过渡带

161. 7.5.2 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.,

162. 7.5.2 Incorporation of Generalized Linear Phase
Their Fourier transforms are of the form

163. 7.5.2 Incorporation of Generalized Linear Phase
2018年11月16日7时43分
7.5.2 Incorporation of Generalized Linear Phase
Hilbert filter hd[n]

164. Frequency Domain Representation
Linear Phase

165. Ex. 7.7 design Linear-Phase Lowpass Filter
The desired frequency response
(spec.)
verification
Solution:

166. magnitude frequency response
2018年11月16日7时43分
magnitude frequency response 选窗
1.选窗
2.定M
Example:design a FIR filter 定M

167. 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]);
(1)选窗
可选Hanning窗
(2)定M

168. 7.5 Design of FIR Filters by Windowing Method
Review
To design an lowpass FIR Filters

169. 7.5.1 Properties of Commonly Used Windows
等效的Kaiser窗
smallest，the sharpest transition
biggest，high oscillations at discontinuity

170. 7.5.3 The Kaiser ['kaizə] Window Filter Design Method
Caesar[ˈsi:zə] 凯撒
Trade side-lobe amplitude for main-lobe width

171. 7.5.3 The Kaiser ['kaizə] Window Filter Design Method
Transition width
Peak approximation error:

172. 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, M=10, 20, 40
Figure 7.24

173. 2018年11月16日7时43分
Comparison
窗两端减小的越多(窗更锥形化)
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

174. Relationship of the Kaiser windows to Other Windows
Table 7.2
Transition width
of Equivalent Kaiser Window
approximate
width of mainlobe
a much better predictor of the transition width for the other windows than would the main-lobe width.

175. figure 7.33 a lowpass filter (M = 32 and ωc=π/2).
Relationship of the Kaiser windows to Other Windows
Approximation error vs. Transition width[*fixed windows,o=Kaiser(β=整数)]
M=20
Bartlett, β=1.33
Hanning, β=3.86
Hamming, β=4.86
Blackman,
β=7.04
"Kaiser 6" means β=6
figure 7.33 a lowpass filter (M = 32 and ωc=π/2).

176. 7.6 Examples of FIR Filters Design by the Kaiser Window Method
Design Lowpass Filter, Highpass Filter, Discrete-Time Differentiators
7.6.1 Lowpass Filter
specifications as in Ex.7.5

177. Filter Design by Kaiser Window
2018年11月16日7时43分
Filter Design by Kaiser Window
(1) the specifications must be established :
(2)
(3)
To predict the values of M and β
Peak approximation error
Transition width

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

179. 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

180. 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]);

181. The ideal highpass filter with generalized linear phase

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

183. Ex. Kaiser Window Design of a highpass 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 =2.65

184. Ex. Kaiser Window Design of a highpass Filter
n =24, beta =2.65
n =25, beta =2.65
type II filter
zero: z=-1
δ= >0.02
n should be n=26

185. 7.6.3 Discrete-Time Differentiator
M=10,even
Type III
M=5,odd
Type IV

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

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

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

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

190. Group Delay
Phase:
Group Delay:
Noninteger delay

191. 7.9 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.
FIR设计后需验证指标, 不满足需增加阶数, 再次(迭代)设计. IIR不需再次(迭代)设计. 一次设计，即满足指标要求。
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

192. 7.9 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

193. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 5 HW
7.2, 7.4, 7.15,
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
207
2018/11/16
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