{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT6_003S10_lec05_handout

# MIT6_003S10_lec05_handout - ˙ x t x t X A X ˙ x t x t X A...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ?...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

MIT6_003S10_lec05_handout - ˙ x t x t X A X ˙ x t x t X A...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online