信号与系统信号与系统信号与系统 (9).pdf
BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Bilateral Laplace transform and its inversionDefinition of bilateral Laplace transformProperties of bilateral Laplace transformInversion of bilateral Laplace transformWhy to define the bilateral Laplace transform?Complex frequency domain analysis for describing systems,However the unilateral Laplace transform only for causal systems.To describe the causal and noncausal systems in s-domain,it is necessary to define the bilateral Laplace transform(BLT).Definition of bilateral Laplace transformBilateral Laplace transform(BLT):X sx ttst()()ed x tX ssstj2()()e d1jjInversion of bilateral Laplace transform:Definition of bilateral Laplace transformX sx ttst()()edDefinition of bilateral Laplace transformThe condition for convergence of the bilateral Laplace transform is the absolute integrability of x(t)et.sx ttx ttCsttRe()|()e|d|()e|d,The region of for which the BLT converges is termed the region of convergence(ROC).（1）finite duration signalsDetermine the BLT of signal x(t)=u(t+2)u(t2)and its ROC.Solution:sRe()sss(ee)122X su tu ttst()(2)(2)edstststede|12222Definition of bilateral Laplace transformojROC is the entire s-plane.no pole for the X(s)（2）right-sided signalsX su tttst()e()ed2sst2e|10(2)s21 sRe()2ttsteed02Determine the BLT of signal x(t)=e2tu(t)and its ROC.Solution:Definition of bilateral Laplace transform0j2real part of the pole（3）left-sided signals sRe()1X sutttst()e()ed sst1e|1(1)0s11ttsteed0Determine the BLT of signal x(t)=etu(t)and its ROC.Solution:0j1Definition of bilateral Laplace transformreal part of the poleutte()sRe()1sX s1()1u tte()2sX s2()1 sRe()2L Lsu tste(),Re()1L Lsutste(),Re()1X(s)does not correspond to x(t)uniquely.Definition of bilateral Laplace transformX(s)+ROC corresponds to x(t)uniquely.is real part of the pole（4）two-sided signalsX suttu tttsttst()e()ede()ed020ss1211 s2Re()1Definition of bilateral Laplace transformDetermine the BLT of signal x(t)=etu(t)+e2tu(t)+and its ROC.Solution:10j2Re(s)2Does any signal x(t)correspond to X(s)+ROC?x tu tuttt()=e()e()2X stttsttst()eedeed020 sRe()1Impossible！Definition of bilateral Laplace transform sRe()2andBilateral Laplace transform and its inversionDefinition of bilateral Laplace transformProperties of bilateral Laplace transformInversion of bilateral Laplace transformifL Lx tX ss()(),Re()0L Lx ttX sst()e(),00sRe()0thenTime shift propertyProperties of bilateral Laplace transformL Lx tX ss()(),Re()0L LtX sssx td (),d()Re()0 Differentiation propertyifthenProperties of bilateral Laplace transformL LtX sssx tnnn()d(),Re d()0L Lx tX ss()(),Re()0L LsxX st()d ()sRe()max(,0)0 Integration propertyifthenProperties of bilateral Laplace transformL Lx tX ss()(),Re 12R ssx;Re 12Properties of bilateral Laplace transform ntegrationI ifferentiationD differntiationdomain-s domain shift-s onvolutionC ime shiftT 0a calingS inearityL OCR)s(X)t(x iesropertP1 122()()a x ta x t1122()()a X sa Xs12xxRR()x at1()sXaa/xaR0()x tt0e()stX sxR12()()x tx t12()()X s Xs12xxRRe()tx t()X sxR()tx td()dX ssxRd()dnnx tt()ns X sxR()dtx()X ssRe 0 xsRBilateral Laplace transform and its inversionDefinition of bilateral Laplace transformProperties of bilateral Laplace transformInversion of bilateral Laplace transform x tX ssstj 2()()e d1jjInversion of bilateral Laplace transformDirect inversion of the Laplace transform is complicated.We can determine it by partial fraction expansion(PFE).Partial fraction expansionL Lssutt(,Re()Re)e()1L Ltutsst()Re()Re(e(),12Right-sided signalsLeft-sided signalsInversion of bilateral Laplace transformL Lssu tt(,)e()1Re()ReL Lsstu tt)e(e()1,Re()R2real part of the polereal part of the poleSolution:ssX s23()35（1）ssX ss(2)(3)()21 sRe()2（2）s3Re()2（3）sRe()3Example 6.19:Determine x(t)corresponding to X(s)by PFE.X(s)associated with different ROCs corresponds to different signals x(t)20j3302j230jSolution:ssX s23()35（1）ssX ss(2)(3)()21 sRe()2 x tu tu ttt()3e()5e()23（2）s3Re()2x tutu ttt()3e()5e()23（3）sRe()3x tututtt()3e()5e()23Right-sided signalTwo-sided signalLeft-sided signalExample 6.19:Determine x(t)corresponding to X(s)by PFE.X(s)associated with different ROCs corresponds to different signals x(t)x(t)=？ABCD4()(2)(3)sX sss3Re()2s 3j02提交1分单选题AcknowledgmentsMaterials 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.Bilateral Laplace transform and its inversion
