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1、DoubleDoubleDoubleDouble NotchNotchNotchNotch FilterFilterFilterFilter DesignDesignDesignDesignXunXunXunXun WangWangWangWang 10112746101011274610101127461010112746101.Introduction1.Introduction1.Introduction1.IntroductionA lot of signals in real situation are often associated with noise in signal pr
2、ocessing.In order to reduce or eliminate the influence of noise and extract useful signals,we use a filter to decrease the scale of noise signal.Filter can be defined as a system which can produce the response which we need by inputing the original signal to the system.The requirement of response ca
3、n be exist in time domain or frequency domain.The function of filter is mainly decided by the system function.The pole location have a significant effect on the properties of amplitude,the position of the peak of wave and extent of acuity. The zero location mainly affect the position of trough of wa
4、ve and the extent of hollow.We can qualitatively draw the shape of amplitude according to the analysis of pole and zero points.This theory provide us a method to directly design the filter:At first we can set the position of poles according to the properties of amplitude,and make sure what the syste
5、m function is.Comparing it with the one we need,if the system function can not satisfy our requirement,we can revise it by adding or deleting the number of zero points,or by move the positions of zero points.This method is called pole- zero placement filter design method.It can be used not only in d
6、esigning IIR digital filters but also in designing FIR digital filters.2. 2. 2. 2.FilterFilterFilterFilter designdesigndesigndesignIn order to obtain the system function,we must try many times to set the the poles and zeros of the system and observe if the properties can satisfy the requirement.If i
7、t does not,we must try these steps again and again to make the filter satisfy our requirement.In fact it is a process of approaching.We should obey some rules when using this method. (1) we should consider which kind of filter it belongs to.If we need to design a FIR filter,we should not set poles,a
8、nd just need to set zeros.By doing this we can design a non-recursive filter.It is possible that it becomes to recursive if we set some poles in the designing process. (2) If we want to filter a point of frequency,we can set a zero at this point of digital frequency. (3) The poles must be in the uni
9、t circle according to the requirement of a stable system. (4) Aiming at accomplish the structure of filter,the poles are zeros are both should be set symmetrically(other than they are in the real axis.It means that the a couple of zeros must be conjugate.The situation is the same as poles. (5) If we
10、 want to make the curve of frequency deeper or sharper at a certain point,we can set more than one zeros or poles near this point.Proccess:Proccess:Proccess:Proccess: Step1:analysisStep1:analysisStep1:analysisStep1:analysis ofofofof thethethethe signalsignalsignalsignal inininin frequencyfrequencyfr
11、equencyfrequency domaindomaindomaindomainBy analysing the frequency of the signal,we can make sure which frequency we should to filter.The value of an ideal notch filter should be 0 at the point of frequency which we need to eliminate and be 1 at the other frequency.It just be like this:( ) =00 10，H
12、(2-1)The sampling period is81921,so the sampling frequency is 8192Hz.The frequency f1=697Hz and f2=1209 Hz are the points of frequency which we should eliminate.Step2:settingStep2:settingStep2:settingStep2:setting thethethethe zerozerozerozero pointpointpointpoint ofofofof thethethethe filter.filter
13、.filter.filter. Because the property of digital filter in frequency domain is its Z-transfer in unit-circle of unit impulse response,we just need to set zero at the points in unit circle which correspond to the location of the frequency which you need to eliminate.17. 020851. 0281926971=(2-2(a)295.
14、02819212092=(2-2(b)So we set four zeros at )295. 0*(-exp),295. 0*(expc),0.17*(-expc),0.17*exp(j4321jcjjc= The system function is42244321)z1.2z-)(1z1.7208z-1 ()-)(-)(-)(-()(zzczczczczKzH+=4-3-2-1 -4234zz1092. 2z5206. 4z1092. 2-1z1z1092. 2-z5206. 4z1092. 2-z+=+=(2-3)K is a constant.we can suppose it i
15、s 1 now.We can use matlab to draw the curve of this system function. Input matlab code:clear all; close all; clc; BB=exp(697/8192)*2*pi*i) exp(-(697/8192)*2*pi*i) exp(1209/8192)*2*pi*i) exp(- (1209/8192)*2*pi*i); B=poly(BB); A=1; W=-3*pi:0.01:3*pi;H=freqz(B,A,W); subplot(211), plot(W/pi,abs(H),LineW
16、idth,2); xlabel(omega/pi);ylabel(|H(ejomega)|); title(amplitude-frequency response);axis(-2 2 -0.2 15); grid; subplot(212), plot(W/pi,angle(H),LineWidth,2);xlabel(omega/pi);ylabel(theta(omega);title(phase-frequency response);axis(-2 2 -4 4); grid; set(gcf,color,w);We can get the curve:Figure 1 the amplitude-frequency response and phase-frequency response curvesFigure 2 The affection of the two zeros in curveThe two fi
