信号与系统信号与系统信号与系统 (14).pdf
BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Analysis for C-T signals and systems by MATLABPartial fraction Expansion of X(s)by MATLABPoles and zeros and system properties by MATLABr,p,k=residue(num,den)num,den are coefficient vectors of numerator and denominator,respectivelyr1r2,rnare expansion coefficients,and p1p2,pn are poles.If X(s)is a proper rational fraction,then k1,k2,km equal zero.sasa saX sb sbsb sbnnnmmmm()11011101spspspkk sk srrrnmnm121212Partial fraction Expansion of X(s)by MATLABExample 6.31:Determine the x(t)by partial fraction expansion.num=1 2;den=1 4 3 0;r,p,k=residue(num,den);sssX ss43()232Answer:r=1/6,1/2,2/3 p=3,1,0k=X(s)is represented assssX s31()1/61/22/3L L x tX su tu tu ttt623()()e()e()()112 13num=2 3 0 5;den=conv(1 1,1 1 2);r,p,k=residue(num,den)magr=abs(r)%magnitude of complex number rangr=angle(r)%phase of complex number r）（sssX sss1(2)()235232Example 6.32:Determine the x(t)by partial fraction expansion.Answer:r=2.0000+1.1339i,2.0000 1.1339i,3.0000 p=0.5000+1.3229i,0.5000 1.3229i,1.0000 k=2 magr=2.2991,2.2991,3.0000 angr=2.6258,2.6258,0 sssX s0.5j1.32290.5j1.32291()22.2991e2.2991e3j2.6258j2.6258x tttu tu ttt()2()1.1495ecos(1.32292.6258)()3e()0.5）（sssX sss1(2)()235232Example 6.32:Determine the x(t)by partial fraction expansion.the poles and zeros of H(s)can be ploted by pzmap()pzmap(sys)The function pzmap()depicts the pole-zero plot according to sys.a sasa saH sb sbsb sbnnnnmmmm()11011101sys=tf(num,den)Poles and zeros and system properties by MATLABnum,den are coefficient vectors of numerator and denominator,respectivelynum=1;den=1 2 2 1;sys=tf(num,den);figure(1);pzmap(sys);t=0:0.01:10;h=impulse(num,den,t);figure(2);plot(t,h)title(Impulse Respone)w=0:0.01:10;H=freqs(num,den,w);figure(3);plot(w,abs(H)xlabel(Frequency omega)title(Magnitude Respone)sssH s221()132Example 6.33:Depict the pole-zero diagram,and determine the impulse response h(t)and frequency response H(jw)of the causal LTI system.h(t)H(jw)pam orez-eloPs i xA gamIs ixA laeR18.06.04.02.002.0-4.0-6.0-8.0-1-02.0-4.0-6.0-8.0-1-2.1-4.1-)s(emitenopseR eslupmI54.04.053.03.052.02.051.01.050.0050.0-019876543210enopseR edutingaMw ycneuqerF19.08.07.06.05.04.03.02.01.00019876543210pole-zero plotExample 6.33:Depict the pole-zero plot,and determine the impulse response h(t)and frequency response H(jw)of the causal LTI system.sssH s221()132stable system%one-order system with one pole and no zerosp1=input(p1=);num=1;den=1 p1;sys=tf(num,den);figure(1);pzmap(sys);t=0:0.0001:0.005;h=impulse(num,den,t);figure(2);plot(t,h)w=0:0.1:1000*2*pi;H=freqs(num,den,w);figure(3);plot(w/2/pi,abs(H);xlabel(Hz)Example 6.34:Analyze the impulse response h(t)and magnitude response|H(jw)|of the causal LTI systems with different poles and zeros.spH s()110wjs400pOne real pole,at the left halfone real pole:p1=400p,no zerosH ss()400400Low-pass,fc=200HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G majorThe real pole moving left0wjs600p400pH ss()600600one real pole:p1=600p,no zerosLow-pass,fc=300HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major0wjs800pOne pole and one zeroH sss()800one real pole:p1=800p,one real zero:z1=0High-pass,fc=400HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major1000p0wjsH sss()1000one real pole:p1=1000p,one real zero:z1=0The real pole moving leftHigh-pass,fc=500HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major18.06.04.02.00zH00010090080070060050040030020010应响度幅的统系0wjs600p600p40pconjugate complex poles:p140p+j600p,p2 40pj600p;z10H ssss()8080(600)22Band-pass,center frequency f0=300HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major0wjs800p800p40pImaginary part changedH ssss()8080(800)22conjugate complex poles:p140p+j800p,p2 40pj800p;z10Band-pass,center frequency f0=400HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major18.06.04.02.00zH00010090080070060050040030020010应响度幅的统系15.005.0-1-s10.0800.0600.0400.0200.00应响激冲的统系谱频的统系过通阶音调大G18.06.04.02.00zH00010090080070060050040030020010p1 40p+j1000p,p2 40pj1000p;z1=j1000p,z2=j1000p0wjs0wjs1000p1000p40pH ssss()(1000)80(1000)2222Band-stop,center frequency f0=500HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G major0wjs0wjs15.005.0-1-s10.0800.0600.0400.0200.00应响激冲的统系谱频的统系过通阶音调大G18.06.04.02.00zH00010090080070060050040030020010800p800p40pH ssss()(800)80(800)2222p1 40p+j800p,p2 40pj800p;z1=j800p,z2=j800pBand-stop,center frequency f0=400HzImpulse responseMagnitude responseSpectrum of G majorSpectrum of the output signal for G majorAcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Analysis for C-T signals and systems by MATLAB
