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1、9 The Laplace Transform 9.The Laplace Transform 9.1 The Laplace Transform(1)Definition(2)Region of Convergence(ROC)ROC:Range of for X(s)to convergeRepresentation:A.Inequality B.Region in S-plane9 The Laplace TransformExample for ROCReReS-planeS-planeImIm-a-a9 The Laplace Transform(3)Relationship bet
2、ween Fourier and Laplace transform Example 9.1 9.2 9.3 9.5 9 The Laplace Transform 9.2 The Region of Convergence for Laplace TransformProperty1:The ROC of X(s)consists of strips parallel to j-axis in the s-plane.Property2:For rational Laplace transform,the ROC does not contain any poles.Property3:If
3、 x(t)is of finite duration and is absolutely integrable,then the ROC is the entire s-plane9 The Laplace TransformProperty4:If x(t)is right sided,and if the line Res=0 is in the ROC,then all values of s for which Res0 will also in the ROC.9 The Laplace TransformProperty5:If x(t)is left sided,and if t
4、he line Res=0 is in the ROC,then all values of s for which Res0 will also in the ROC.x(t)T2te-0te-1t9 The Laplace TransformProperty6:If x(t)is two sided,and if the line Res=0 is in the ROC,then the ROC will consist of a strip in the s-plane that includes the line Res=0.9 The Laplace TransformS-plane
5、ReReReImImImRLLR9 The Laplace TransformProperty7:If the Laplace transform X(s)of x(t)is rational,then its ROC is bounded by poles or extends to infinity.In addition,nopoles of X(s)are contained in the ROC.Property8:If the Laplace transform X(s)For rational Laplace transform,the ROC does not contain
6、any poles.Property3:If x(t)is of finite duration and is absolutely integrable,then the ROC is the entire s-plane9 The Laplace TransformProperty7:If the Laplace transform X(s)of x(t)is rational,then its ROC is bounded by poles or extends to infinity.In addition,no poles of X(s)are contained in the RO
7、C.Property8:If the Laplace transform X(s)of x(t)is rational,then if x(t)is right sided,the ROC is the region in the s-plane to the right of the rightmost pole.If x(t)is left sided,the ROC is the region in the s-plane to the left of the leftmost pole.Example 9.7 9.8 9 The Laplace Transform Appendix P
8、artial Fraction ExpansionConsider a fraction polynomial:Discuss two cases of D(s)=0,for distinct root and same root.9 The Laplace Transform(1)Distinct root:thus9 The Laplace TransformCalculate A1:Multiply two sides by(s-1):Let s=1,so Generally9 The Laplace Transform(2)Same root:thusFor first order p
9、oles:9 The Laplace TransformMultiply two sides by(s-1)r:For r-order poles:So 9 The Laplace Transform 9.3 The Inverse Laplace TransformSo9 The Laplace TransformThe calculation for inverse Laplace transform:(1)Integration of complex function by equation.(2)Compute by Fraction expansion.General form of X(s):Important transform pair:Example 9.9 9.10 9.11