MIT6_003S10_lec05_handout

MIT6_003S10_lec05_handout - x ( t ) x ( t ) X A X x ( t ) x...

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Unformatted text preview: x ( t ) x ( t ) X A X x ( t ) x ( t ) X A X x ( t ) x ( t ) X A X X 6.003: Signals and Systems Lecture 5 February 18, 2010 6.003: Signals and Systems Concept Map: Continuous-Time Systems Relations among representations. Laplace Transform Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 + 1 + 1 2 X Impulse Response h ( t ) = 2( e t/ 2 e t ) u ( t ) Differential Equation System Function 2 y ( t ) + 3 y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 February 18, 2010 X ( s ) = 2 s 2 + 3 s + 1 Check Yourself Concept Map: Continuous-Time Systems How to determine impulse response from system functional? Today: new relations based on Laplace transform. Block Diagram Block Diagram System Functional System Functional X Y Y 2 A 2 Y Y 2 A 2 + + = + + = 1 X 2 + 3 A + A 2 1 X 2 + 3 A + A 2 1 2 1 2 Impulse Response Impulse Response h ( t ) = 2( e t/ 2 e t ) u ( t ) h ( t ) = 2( e t/ 2 e t ) u ( t ) Differential Equation System Function Differential Equation System Function 2 y ( t ) + 3 y ( t ) + y ( t ) = 2 x ( t ) Y ( s s ) ) = 2 2 y ( t ) + 3 y ( t ) + y ( t ) = 2 x ( t ) Y ( s s ) ) = 2 X ( 2 s 2 + 3 s + 1 X ( 2 s 2 + 3 s + 1 Laplace Transform: Definition Laplace Transforms Laplace transform maps a function of time t to a function of s . Example: Find the Laplace transform of x 1 ( t ) : x 1 ( t ) X ( s ) = x ( t ) e st dt x 1 ( t ) = e t if t otherwise There are two important variants: Unilateral (18.03) e ( s +1) t 1 X 1 ( s ) = x 1 ( t ) e st dt = e t e st dt = ( s + 1) = s + 1 X ( s ) = x ( t ) e st dt provided Re ( s + 1) > which implies that Re ( s ) > 1 . Bilateral (6.003) t X ( s ) = x ( t ) e st dt Both share important properties will discuss differences later. 1 s + 1 ; Re ( s ) > 1 1 s-plane ROC 1 6.003: Signals and Systems Lecture 5 February 18, 2010 Check Yourself x 2 ( t ) x 2 ( t ) = e t e 2 t if t otherwise t Which of the following is the Laplace transform of x 2 ( t ) ?...
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This note was uploaded on 11/07/2011 for the course ELECTRICA 6.003 taught by Professor Staff during the Summer '10 term at MIT.

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MIT6_003S10_lec05_handout - x ( t ) x ( t ) X A X x ( t ) x...

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