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1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Complex frequency-domain analysis for signalss-domain representation for C-T signalsUnilateral Laplace transform of typical signalsProperties of unilateral Lapla
2、ce transformInversion of unilateral Laplace transformProperties of unilateral Laplace transformLinearityTime shiftScalingConvolutionMultiplicationExponential weightingLinear weightingDifferentiationIntegrationInitial&final value theoremsifthenL Lx tX ss()(),Re()111L Lx tXss()(),Re()222L Lx tx tX sXs
3、()()()()1212 sRe()max(,)12 Linearity propertyProperties of unilateral Laplace transformL Lx tX ss()(),Re()0Lx tt u ttX sst()()e()000sRe()0Time shift propertyt(0)0Properties of unilateral Laplace transformifthen Lssu t()1)0,Re(According to Linearity propertyDue to Lsssu tu tss()(1)e,111e sRe()L，ssu t
4、s e(0 R)(1)e12)t(x101tSolution:signal x(t)is represented as x tu tu t()()(1)According to Time-shift propertyExample 6.4:Determine unilateral Laplace transform of x(t).sX sss()(12ee)122 sRe()2101)t(xtx tr tr tr t()()2(1)(2)L L sr ttu ts()(),Re()012According to Linearity and Time-shift propertyFor a f
5、inite-duration signal,its ROC includes the entire s-planeExample 6.5:Determine unilateral Laplace transform of x(t).Solution:signal x(t)is represented as Due tox1(t).4523210t)t(xx tx tnTn()()01Firstly find the unilateral LT X1(s)corresponding to signal x1(t),by properties of Linearity and Time-shift
6、,we can obtainLLx tXsnsn()()e012Re(s)0Xss1e()21x tnn(2)01Example 6.6:Determine unilateral Laplace Transform of x(t).signal x(t)is represented as x1(t)and its delay x1(tnT)Solution:x tu tu t1()2()(1)L stsxs),e()1 eR()(21ssX sX sssss1 e1 e1e()2 1 e21()221sRe()0 x1(t).4523210t)t(xExample 6.6:Determine
7、unilateral Laplace Transform of x(t).ifthenL Lx tX ss()()Re()0L Laax atXas()(),1 (0)Scaling propertysaRe()0Properties of unilateral Laplace transformL Lx tx tX s Xs()*()()()1212 sRe()max(,)12 Convolution propertyifthenL Lx tX ss()(),Re()111L Lx tXss()(),Re()222Properties of unilateral Laplace transf
8、ormConvolution in time-domain;multiplication in s-domain2101)t(xt101)t(1xtx tx tx t11()()()sX sX sX sss()()()(),Re()1e112x tu tu t()()(1)1According to the Convolution propertywhereL L sx tXsss()(),Re()1e11Example 6.7:Determine unilateral Laplace Transform of x(t).Solution:signal x(t)is represented a
9、s L Lx tX ss()(),Re()111L Lx tXss()(),Re()222L x t x tX sXsj 2()()()*()11212sRe()12 Multiplication propertyifthenProperties of unilateral Laplace transformMultiplication in time-domain;convolution in s-domainL Lx tX ste()()sRe()Re()0L Lx tX ss()(),Re()0if Exponential weighting property(s-domain shif
10、t property)thenProperties of unilateral Laplace transformAccording to exponential weighting propertyx(t)=et cos(0t)u(t),is real.L Lt u ttecos()()0L Lst u tsscos()(),Re()00220ss()022 sRe()Example 6.8:Determine unilateral Laplace Transform of x(t).Solution:the unilateral Laplace transform of cos(0t)u(
11、t)is asL Lstx tX sd()d()L Lx tX ss()(),Re()0ifthen Linear weighting property(s-domain differentiation property)sRe()0Properties of unilateral Laplace transformMultiplication by t in time-domain;Differentiation in s-domainAccording to the linear weighting propertyL L s stu td()()d 1L Lsu ts(),Re()01s
12、s)0,1Re(2Applying the property again and again,we can deriveLLLLt u tt tu tnn()()1ssnn,Re()0!1tu(t),t nu(t),te-t u(t),t ne-t u(t),n is positive number.Example 6.9:Determine unilateral Laplace Transform of x(t).Solution:L L s stu ttd(e()()d1L L su tste(),Re()Re()1 ss()Re)()(,1Re2L Ltu tnt e()snn()!1
13、sRe()Re()tu(t),t nu(t),te-t u(t),t ne-t u(t),n is positive number.Example 6.9:Determine unilateral Laplace Transform of x(t).Solution:According to the linear weighting propertyApplying the property again and again,we can deriveL Lx tX ss()(),Re()0L L txssX sx td()(0),()d()Re0 Differentiation propert
14、yifthenL Ltttx tx tstddedd()d()0 x tx tststst()e|()(e)d00 xsx ttsX sxst(0)()ed()(0)0sRe()0Properties of unilateral Laplace transformL Lts X ssxxx td ()(0)(0)d()222Applying the property repeatedly,we can deriveL Lts X ssxsxxx tnnnnnnd ()(0)(0)(0)d()1(2)(1)Properties of unilateral Laplace transform Di
15、fferentiation propertyThe property is useful to solve the differential equation Relationship of the signals tu(t),u(t),(t),(n)(t)in s-domainRamp signal tu(t)Step signal u(t)Impulse signal(t)tn()()ss(01),Re2ss01),Re(s1 Re(,)ssn),Re(differentiationL LL LL LL LdifferentiationdifferentiationL Lts X ssxx
16、x td ()(0)(0)d()222 s X s()2ssX ss()12ee22sRe()2101)t(xt12101)t(xt)1()2210)1()t(xtTherefore,we can obtain ss12ee2Example 6.10:Determine unilateral Laplace Transform of x(t).Solution:by the differentiation propertyThe mathematical model for inductors in s-domain V ssLIsLi()()(0)LLL )s(LV)s(LI Lsinduc
17、torttLi td()d()LL)t(Lv)t(Li_ _+LModel in s-domainL LtsX sxx td ()(0)d()L(0)LiL Lx tX ss()(),Re()0L LssxX sxt()d ()(0)(1)sRe()max(,0)0 Integration propertyifthenLLLLxxu t()d(0)()101(0)xsProperties of unilateral Laplace transformL Lx tX ss()(),Re()0L LssxX sxt()d ()(0)(1)sRe()max(,0)0if x(1)(0)=0,then
18、L LsxX st()d()Properties of unilateral Laplace transform Integration propertyifthen011)t(xt)t(y011tx ty ttt()()dBy the integration propertyssY ssXss(e)R0,()1 e2y tu tu t()()(1)Due toSolution:andyy tt(0)()d0(1)0L L sy tss),Re()1 eExample 6.11:Determine unilateral Laplace Transform of x(t).capacitorMo
19、del in s-domainCtit()()d1CCsCsVsIsv()()(0)11CCCL ssxX sxt()d()(0)(1)t(Cv)t(Ci_ _+L )s(CV)s(CI The mathematical model for capacitors in s-domain 1sCC1(0)vsL Lx tX ss()(),Re()0 x txsX stslim()(0)lim()0 x txsX stslim()()lim()0If the order of the numerator polynomial is less than the order of denominato
20、r polynomial in X(s),the initial value theorem states thatIf the ROC of sX(s)includes jaxis in s-plane,then the final value theorem states that Initial and final value theoremsProperties of unilateral Laplace transformExample 6.12:Determine the initial and final value of x(t).Solution:sX ss1(),Re()1
21、1xsX ss(0)lim()1 xsX ss()lim()00ROC of sX(s)includes j axis in s-plane,so we can directly apply the final value theoremX(s)is a proper rational function,so we can directly apply the initial value theorem xsXss(0)lim()11 xsX ss()lim()00X ssss()1111X1(s)Xssss1(),Re()1Example 6.13:Determine the initial
22、 and final value of x(t).Solution:X(s)is an improper rational fraction,so we can not directly apply the initial value theorem X1(s)is a proper fraction,by the initial value theorem ROC of sX(s)includes j axis,by the final value theoremAcknowledgmentsMaterials 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.Properties of unilateral Laplace transform