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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 transformUnilateral Laplace transform of typical signalsImpulse signalStep signalExponential signalSinusoidal signalRamp signal impulse signalttn(),()()L Ltttst()()ed0sRe()Impulse signalL L stsnn(),Re()()tste10L L ts()()1,ReSimilarlyfirst derivative,higher
3、 derivativeL L stsRe()(),Re(s)0Step signal step signal u(t)L Lssu t Re()0(),1L Lu tu ttst()()ed0tsted0 ssste|110Lu ttttste()e ed0Exponential signal exponential signal et u(t)，is a real or complex number.L Lssu tt(,)e()1Re()Re sss te|110()sRe()Re()L Lsu ttje()10j0L Lsu tt(j)e()10(j)0If=,real exponent
4、ial signalssRe()0sRe()sRe()L Lsu tte()1If=j0,imaginary exponential signals If=+j0,complex exponential signalsL Lssu tt(,)e()1Re()ReExponential signalt u tu ttt2cos()()()ee0jj00L Lss2jj()11100t u tu ttt2jsin()()()ee0jj00L Lss2jjj()11100Sinusoidal signal sinusoidal signal cos(0t)u(t),sin(0t)u(t)sssRe(
5、)0=,022ss0=,Re()0220Ltu tttst()ed0Ramp signal ramp signal tu(t)ssttstst=(e)ed100L Lsstu t)0(),Re(12sRe()0ststed10 ssste|11022Solution:signal x(t)FT-X(j)LT-X(s)tu t3e()tu t3e()t u tcos(2)()1j3 ss3Re 31noness3Re 31 2j(j)42(2)(2)sss2Re 022Example 6.3:unilateral Laplace transform and Fourier transform tu t3e()tu t3e()tu tcos2()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.Unilateral Laplace transform of typical signals