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Laplace Transform (LT)

 

Laplace transform of x(t) function is

1091.png 


Properties of Laplace Transform:

  1. Linearity

1096.png 

1101.png 

1106.png 

1111.png 

  1. Time-Reversal : x(t X(s) then x(–t X(–s), ROC = –R
  2. Shift in S-domain : x(t X(s) with ROC = R then 1116.png
  3. Time-shifting: It x(t) is input-signal whose ROC is R.
     
    then 1121.png then 1127.png with ROC = R
  4. Differentiation in S-domain: If x(t X(s) ROC = R, then 1132.png
  5. Differentiation in Time : If 1137.png then 1142.png
  6. Integration in Time : 1147.png for unilateral LT 1152.png
  7. Frequency Integration : 1157.png
  8. Convolution in Time : 1162.png and 1167.png with 1172.png then 1178.png 1183.png
  9. Initial and Final Value Theorem : If x(t1188.pngX(s)
     
    then initial value theorem is x(0) = 1193.png s X(s)
     
    Final value theorem is x() = 1198.pngs X(s)

 





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