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# lect11 - SOME USEFUL LAPLACE TRANSFORMS We found earlier...

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SOME USEFUL LAPLACE TRANSFORMS We found earlier that a s t U e t a - 1 ) ( . Equivalently, } Re{ } Re{ ; 1 0 a s a s dt e e t s t a - = - . (11.1) Differentiating both sides of (11.1) with respect to a yields ( 29 } Re{ } Re{ ; 1 2 0 a s a s dt e e t t s t a - = - . (11.2) Now, the integral on the left is the laplace transform of the signal ) ( t U e t t a . Thus, we have the new transform pair ( 29 2 1 ) ( a s t U e t t a - with ROC } Re{ } Re{ a s (11.3) Differentiating (11.2) now yields ( 29 } Re{ } Re{ ; 2 3 0 2 a s a s dt e e t t s t a - = - , which gives us the transform pair ( 29 3 2 2 ) ( a s t U e t t a - with ROC } Re{ } Re{ a s

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Differentiating again with respect to a yields the pair ( 29 4 3 ) 3 )( 2 ( ) ( a s t U e t t a - with ROC } Re{ } Re{ a s (11.4) Continuing this process, we find the general pair: ( 29 1 ! ) ( + - m t a m a s m t U e t with ROC } Re{ } Re{ a s (11.5) If we repeat the above process starting with the anticausal pair a s t U e t a - - - 1 ) ( , we arrive at the general transform pair ( 29 1 ! ) ( + - - - m t a m a s m t U e t with ROC } Re{ } Re{ a s (11.6) Example 11.1 a) With m = 1 (11.5) and (11.6) yield ( 29 2 1 ) ( a s t U e t t a - ( 29 2 1 ) ( a s t U e t t a - - - . b) With m = 3,
( 29 4 3 6 ) ( a s t U e t t a - ( 29 4 3 6 ) ( a s t U e t t a - - - . SOME LAPLACE TRANSFORM THEOREMS 1. The time shifting theorem t t s e s F t f - - ) ( ) ( . Proof: - - - - + - - - = = - - dt e t f e dt e t f dt e t f t f t s s t s t s ) ( ) ( ) ( ) ( ) ( t t t t ) ( s F e s t - = . Q.E.D. 2. The convolution theorem ) ( ) ( ) ( ) ( s G s F t g t f . Proof: - - - - = t t t t t t d e s G f d t g f t g t f s ) ( ) ( ) ( ) ( ) ( ) (

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where we have used the time shifting theorm under the integral sign to obtain the last expression on the right. Now taking ) ( s G out from under the integral, we have ) ( ) ( ) ( ) ( ) ( ) ( s F s G d e f s G t g t f s = - - t t t . Q.E.D. 3. A conjugate function property ( 29 * ) ( s F t f - - . Proof: - - = dt e t f s F t s ) ( ) ( - = - dt e t f s F t s ) ( ) ( - - - - = = - dt e t f dt e t f s F t s t s ) ( ) ( ) ( . Q.E.D. INVERSE LAPLACE TRANSFORMS BY PARTIAL FRACTIONS Just as with the z-transform, we can use partial fractions to find the inverse laplace transform of rational functions of s . Example 11.2
2 ) 2 )( 1 ( ) 1 3 ( ) ( + - + = s s s s s F .

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lect11 - SOME USEFUL LAPLACE TRANSFORMS We found earlier...

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