1. Chapter 8 FIR Filter Design
Content
Introduction
Properties of Linear-Phase FIR Filters
Window Design Techniques
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Introduction
The advantages of the FIR digital filter
The phase response can be exactly linear;
They are relatively easy to design since there are no stability problems;
They are efficient to implement;
The DFT can be used in their implementation.
h(n)有限长，因此H(z)只在z=0处有极点，因此系统肯定是稳定的；另外，只要经过一定的延时，任何非因果有限长序列都能变成因果的有限长序列，因而总能用因果系统来实现。
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Introduction
The advantages of a linear-phase response
Design problem contains only real arithmetic and not complex arithmetic;
Linear-phase filter provide no delay distortion and only a fixed amount of delay;
For the filter of length N (or order N-1) the number of operations are of the order of N/2.
h(n)为实数偶对称，H(ejw)为实数
h(n)为实数奇对称，H(ejw)为纯虚数
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Introduction
The basic technique of FIR filter design
Window design techniques;
Frequency sampling design techniques;
Optimal equiripple design techniques.
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5. Properties of Linear-Phase FIR Filters
The system function and frequency response
The system function of FIR filters
Let h(n), n=0,1,…,N-1 be the impulse response of length N. Then the system function is
It has (N-1) poles at the origin and N-1 zeros located anywhere in the z-plane.
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6. Properties of Linear-Phase FIR Filters
The frequency response of FIR filters
Amplitude：振幅
Magnitude：幅度
Magnitude response function
Amplitude response function
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7. Properties of Linear-Phase FIR Filters
The difference between and
is always positive and the associated phase response is a discontinuous function.
may be both positive and negative and the associated phase response is a continuous function.
Consider the following example:
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9. Properties of Linear-Phase FIR Filters
Linear-phase conditions
For
A constant group delay
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10. Properties of Linear-Phase FIR Filters
For
The phase response is through the origin.
For
The phase response is not through the origin.
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11. Properties of Linear-Phase FIR Filters
Frequency response of linear-phase FIR filters
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12. Properties of Linear-Phase FIR Filters
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Symmetric impulse response
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Antisymmetric impulse response
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15. Properties of Linear-Phase FIR Filters
The properties of amplitude function
Type 1: symmetric impulse response, N is odd
Cos((N-1)/2-n)也是以(N-1)/2为对称中心的偶对称函数。
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16. Properties of Linear-Phase FIR Filters
The middle sample
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17. Properties of Linear-Phase FIR Filters
Type 2: symmetric impulse response, N is even
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18. Properties of Linear-Phase FIR Filters
Type 3: antisymmetric impulse response, N is odd
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19. Properties of Linear-Phase FIR Filters
Type 4: antisymmetric impulse response, N is even
Type 3和Type4用来设计微分器和移相器
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20. Properties of Linear-Phase FIR Filters
Zero locations of linear-phase FIR filters
If has a zero at
Then for linear phase there must be a zero at
在黑板上讲P331的例子
For a real-valued filter, there must be zeros at
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21. Window Design Techniques
Basic window design idea
Choose a proper ideal frequency-selective filter (which always has a noncausal, infinite-length impulse response);
Then truncate (window) its impulse response to obtain a linear-phase and causal FIR filter.
The emphasis is on
Selecting an appropriate ideal filter;
Selecting an appropriate windowing function.
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22. Window Design Techniques
Denote an ideal frequency-selective filter by
由图可知，hd(n)是偶对称的，因此只能用类型1和2来设计。
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Windowing
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The effect of Window function
设用矩形窗
WR(w)：振幅函数
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26. Window Design Techniques
The conclusion
Since the window has a finite length equal to N, its response has a peaky main lobe whose width is proportional to 1/N, and has side lobes of smaller heights.
The periodic convolution produces a smeared version of the ideal response
The main lobe produces a transition band in whose width is responsible for the transition width. This width is then proportional to 1/N. The wider the main lobe, the wider will be the transition width.
Smear：涂抹扩散
The side lobes produces ripples that have similar shapes in both the passband and stopband.
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27. Window Design Techniques
Windowing functions
Rectangular window
This is the simplest window function but provides the worst performance from the viewpoint of stopband attenuation. The width of main lobe is
窗函数要满足两项要求：
1、窗谱主瓣要尽可能窄，以获得较陡的过渡带，即过渡带很窄
2、尽量减少窗谱最大旁瓣的相对幅度，也就是能量尽量集中于主瓣，这样使肩峰和波纹减小，就可以增大阻带的衰减。
但这两项要求不能同时满足，往往是增加主瓣宽度来换取对旁瓣的抑制。
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28. Window Design Techniques
Gibbs phenomenon
The truncation of the infinite length will introduce ripples in frequency response The oscillatory behavior near the band edge of the filter is called the Gibbs phenomenon.
When the N is increased:
The transition band of the filter will decrease
But the relative amplitude of the peaky values will remain constant.
---The transition band of the filter is determined by the main lobe width of the window function W(omega),
When N increases, this width will decrease.(4*pi/N)
---the relative amplitude of the peaky values of the filter in stopband is determined by the relative amplitude of the
side lobe of the window function W(omega). When N increases, this relative amplitude remain constant.
主瓣与旁瓣的相对比例由sinx/x决定，或者说只由窗函数的形状来决定，与N无关。
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Bartlett window
Since the Gibbs phenomenon results from the fact that the rectangular window has a sudden transition from 0 to 1 (or 1 to 0), Bartlett suggested a more gradual transition in the form of a triangular window. The width of main lobe is
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Hanning window
This is a raised cosine window function given by:
The width of main lobe is:
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Hamming window
This is a modified version of the raised cosine window.
The width of main lobe is:
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Blackman window
This is a 2-order raised cosine window.
The width of main lobe is:
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Kaiser window
This is one of the most useful and optimum windows.
Where is the modified zero-order Bessel function, and is a parameter that can be chosen to yield various transition widths and stopband attenuation. This window can provide different transition widths for the same N.
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The design equations for Kaiser window
Given
The norm transition width:
The filter order N:
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35. Min. stopband attenuation
Summary of window function characteristics
Window name
Window function
Filter
Peak value of side lobe
The width of main lobe
Transition width
Min. stopband attenuation
Rectangular
-13 dB
-21 dB
Bartlett
-25 dB
Hanning
-31 dB
-44 dB
Hamming
-41 dB
-53 dB
Blackman
-57 dB
-74 dB
1、同一类窗函数，其过渡带宽度随窗宽度N的增加而减小。
2、同一类窗函数，其最小阻带衰减是固定的。
3、最小阻带衰减只由窗形状决定（不同的窗形其最大旁瓣相对幅度不同，最大旁瓣相对幅度越小，最小阻带衰减越大），不受N的影响；
4、最大旁瓣相对幅度越小的窗函数，其主瓣宽度也越宽，因而其滤波器的最小阻带衰减也越大，但同时过渡带也越宽；
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36. Window Design Techniques
Design procedure
Given the ideal frequency response
Compute the impulse response of ideal filter
Determine the window shape and N from the minimum stopband attenuation and the transition width
Compute the impulse response of the designed filter
Compute the frequency response of the designed filter and verify the performance
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37. Window Design Techniques
Examples of FIR linear-phase filter design
Digital FIR lowpass filter
Example
Design a digital FIR lowpass filter:
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Solution:
Compute the digital frequencies
Derive the frequency response of ideal FIR lowpass filter
wc指的是两个肩峰之间的中心频率，因此由上述公式算出来的wc有一定的近似。
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Compute the impulse response of the ideal filter
Determine the window shape and N
Both the Hamming and Blackman window can provide attenuation of more than 50dB. Let us choose the Hamming window, which provide the smaller transition band and hence has the smaller order.
Hamming
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Compute the impulse response of the designed filter
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Compute the frequency response of the designed filter
Verify the performance of the designed filter
It is not satisfied by this design
Let N = 34 and redesign
打开(digital signal processing\chap7\FIR_filter_design\lowpass_filter.m)
看ws=0.4pi处的衰减值，小于50dB，为46dB
因此需要将N加1，即N=33+1=34，重新设计，得到As为52dB。
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42. Window Design Techniques
Digital FIR highpass filter
An ideal FIR highpass filter can be obtained from two ideal FIR lowpass filters, provided they have the same phase response.
截止频率为wc的理想高通滤波器相当于两个理想低通相减，一个是截止频率为pi的理想低通，一个是截止频率为wc的理想低通
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The frequency response of an ideal FIR highpass filters
The impulse response of an ideal FIR highpass filters
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Example
Design a digital FIR highpass filter :
Solution:
Compute the digital frequencies
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Determine the window shape and N
Blackman
Note: the N must be odd for FIR highpass filters
Derive the impulse response of ideal FIR highpass filter
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Compute the impulse response of the designed filter
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Compute the frequency response of the designed filter
Verify the performance of the designed filter
It is satisfied by this design
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48. Window Design Techniques
Digital FIR bandpass filter
An ideal FIR bandpass filter can be obtained from two ideal FIR lowpass filters, provided they have the same phase response.
截止频率为wc1和wc2的理想带通滤波器相当于两个理想低通相减，一个是截止频率为wc2的理想低通，一个是截止频率为wc1的理想低通
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The frequency response of an ideal FIR bandpass filters
The impulse response of an ideal FIR highpass filters
截止频率为wc1和wc2的理想带通滤波器相当于两个理想低通相减，一个是截止频率为wc2的理想低通，一个是截止频率为wc1的理想低通
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Example
Design a digital FIR bandpass filter :
Solution:
Compute the digital frequencies
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Determine the window shape and N
Blackman
Derive the impulse response of ideal FIR bandpass filter
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Compute the impulse response of the designed filter
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Compute the frequency response of the designed filter
Verify the performance of the designed filter
It is satisfied by this design
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Digital FIR bandstop filter
An ideal FIR bandstop filter can be obtained from three ideal FIR lowpass filters, provided they have the same phase response.
截止频率为wc1和wc2的理想带阻滤波器相当于三个理想低通相加减，一个截止频率为pi的理想低通,减去一个截止频率为wc2的理想低通，再加上一个截止频率为wc1的理想低通
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The frequency response of an ideal FIR bandpass filters
The impulse response of an ideal FIR highpass filters
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Example
Design a digital FIR bandstop filter :
Solution:
Compute the digital frequencies
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Determine the window shape and N
Hanning
Note: the N must be odd for FIR bandstop filters
Derive the impulse response of ideal FIR bandstop filter
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Compute the impulse response of the designed filter
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Compute the frequency response of the designed filter
Verify the performance of the designed filter
It is satisfied by this design
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60. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\linear_phase_FIR\magVSamp.m)
Piecewise：分段地
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61. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\linear_phase_FIR\linear_phase_condition1.m)
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62. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\linear_phase_FIR\linear_phase_condition2.m)
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63. Copyright © 2005. Shi Ping CUC
偶对称
(digital signal processing\chap7\linear_phase_FIR\example1.m)
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64. Copyright © 2005. Shi Ping CUC
奇对称
(digital signal processing\chap7\linear_phase_FIR\example1.m)
不能用来设计高通、带阻滤波器
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65. Copyright © 2005. Shi Ping CUC
奇对称
(digital signal processing\chap7\linear_phase_FIR\example1.m)
不能用来设计低通、高通、带阻滤波器
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66. Copyright © 2005. Shi Ping CUC
偶对称
(digital signal processing\chap7\linear_phase_FIR\example1.m)
不能用来设计低通滤波器
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jIm[z]
Re[z]
Quadruplet
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(digital signal processing\chap7\window_design\impulse_of_Hd.m)
wc=0.2pi, a=10
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69. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\impulse_of_Hd.m)
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70. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\windowR_amplitude.m)
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71. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\Hw.m)
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72. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\Hw.m)
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73. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\Hw.m)
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(digital signal processing\chap7\window_design\window_functions\rectangular_window.m)
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(digital signal processing\chap7\window_design\window_functions\rectangular_window.m)
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76. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\Gibbs_phenomenon.m)
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79. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\window_functions\bartlett_window.m)
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80. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\window_functions\hanning_window.m)
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81. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\window_functions\hamming_window.m)
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82. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\window_functions\blackman_window.m)
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(digital signal processing\chap7\window_design\window_functions\kaiser_window.m)
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84. Copyright © 2005. Shi Ping CUC
(digital signal processing\chap7\window_design\window_functions\kaiser_window.m)
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(digital signal processing\chap7\FIR_filter_design\lowpass_filter.m),N=33
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(digital signal processing\chap7\FIR_filter_design\lowpass_filter.m),N=34
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(digital signal processing\chap7\FIR_filter_design\highpass_filter.m),N=55
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(digital signal processing\chap7\FIR_filter_design\bandpass_filter.m),N=74
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(digital signal processing\chap7\FIR_filter_design\bandstop_filter.m),N=63
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