6 003 Signals and Systems Laplace Transform February 18 2010 Concept Map Continuous Time Systems Multiple representations of CT systems R X Block Diagram X R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Concept Map Continuous Time Systems Relations among representations X Block Diagram X R R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Concept Map Continuous Time Systems Two interpretations of R X Block Diagram X R R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Concept Map Continuous Time Systems Relation between System FunctionalR and System Function X Block Diagram X R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t A 1s System Function Y s 2 2 X s 2s 3s 1 Check Yourself How to determine impulse response Rfrom system functional X Block Diagram X R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Check Yourself How to determine impulse response from system functional Expand functional using partial fractions 2A2 A2 Y 2A 2A 2 1 1 X 2 3A A 1 2 A 1 A 1 2A 1 A Recognize forms of terms each corresponds to an exponential Alternatively expand each term in a series Y 1 1 1 2A 1 A A2 A3 2A 1 A A2 A3 X 2 4 8 Let X t Then 1 1 1 1 1 Y 2 1 t t2 t3 u t 2 1 t t2 t3 u 2 8 48 2 3 2 e t 2 e t u t Check Yourself How to determine impulse response Rfrom system functional X Block Diagram X R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 series partial fractions Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Concept Map Continuous Time Systems Today new relations based on Laplace transform R X Block Diagram X R R AX System Functional Y 2A2 X 2 3A A2 Y 1 2 1 Impulse Response x t R x t h t 2 e t 2 e t u t Differential Equation 2y t 3y t y t 2x t System Function Y s 2 2 X s 2s 3s 1 Laplace Transform Definition Laplace transform maps a function of time t to a function of s Z X s x t e st dt There are two important variants Unilateral 18 03 Z X s x t e st dt 0 Bilateral 6 003 Z X s x t e st dt Both share important properties will discuss differences later Laplace Transforms Example Find the Laplace transform of x1 t x1 t x1 t e t 0 if t 0 otherwise t 0 Z X1 s x1 t e st Z e dt 0 t st e e s 1 t dt s 1 0 provided Re s 1 0 which implies that Re s 1 s plane 1 Re s 1 s 1 ROC 1 1 s 1 Check Yourself x2 t x2 t e t e 2t 0 if t 0 otherwise 0 Which of the following is the Laplace transform of x2 t 1 1 X2 s s 1 s 2 Re s 1 1 Re s 2 2 X2 s s 1 s 2 s 3 X2 s s 1 s 2 Re s 1 s 4 X2 s s 1 s 2 Re s 2 5 none of the above t Check Yourself Z X2 s e t e 2t e st dt 0 Z e 0 t st e Z dt e 2t e st dt 0 1 s 2 s 1 1 1 s 1 s 2 s 1 s 2 s 1 s 2 These equations converge if Re s 1 0 and Re s 2 0 thus Re s 1 s plane 1 Re s 1 s 1 s 2 ROC 2 1 Check Yourself x2 t x2 t e t e 2t 0 if t 0 otherwise 0 Which of the following is the Laplace transform of x2 t 1 1 X2 s s 1 s 2 Re s 1 1 2 X2 s s 1 s 2 Re s 2 s 3 X2 s s 1 s 2 Re s 1 s 4 X2 s s 1 s 2 Re s 2 5 none of the above t Regions of Convergence Left sided signals have left sided Laplace transforms bilateral only Example x3 t x3 t e t 0 Z X3 s x3 t e if t 0 otherwise st t 1 Z 0 e dt t st e e s 1 t dt s 1 1 Re s 1 s 1 ROC provided Re s 1 0 which implies that Re s 1 s plane 1 0 1 s 1 Left and Right Sided ROCs Laplace transforms of left and right sided exponentials have the same form except with left and right sided ROCs respectively Laplace transform time function e t u t s plane 0 1 s 1 e t u t t 1 1 s 1 ROC 1 ROC t 1 s plane Left and Right Sided ROCs Laplace transforms of left and right sided exponentials have the same form except with left and right sided ROCs respectively Laplace transform time function e t u t s plane 0 1 s 1 e t u t t 1 1 s 1 ROC 1 ROC t 1 s plane Check Yourself Find the Laplace transform of x4 t x4 t x4 t e t 0 1 X4 s 2 X4 s 3 X4 s 4 X4 s 2 1 s2 2 1 s2 2 1 s2 2 1 s2 Re s 1 Re s 1 Re s 1 Re s 1 5 none of the above t Check Yourself Z X4 s …
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