lec07-handout-6up

lec07-handout-6up - 6.003: Signals and Systems Lecture 7...

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Unformatted text preview: 6.003: Signals and Systems Lecture 7 October 1, 2009 1 6.003: Signals and Systems Laplace and Z Transforms October 1, 2009 Mid-term Examination #1 Wednesday, October 7, 7:30-9:30pm, Walker Memorial. No recitations on the day of the exam. Coverage: DT Signals and Systems Lectures 15 Homeworks 14 Homework 4 will include practice problems for mid-term 1. However, it will not collected or graded. Solutions will be posted. Closed book: 1 page of notes ( 8 1 2 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open office hours. Conflict? Contact freeman@mit.edu before Friday, October 2, 5pm. Last Time Many continuous-time systems can be represented with differential equations. Example: leaky tank r ( t ) r 1 ( t ) h 1 ( t ) Differential equation representation: r 1 ( t ) = r ( t ) r 1 ( t ) Last time we considered two methods to solve differential equations: solving homogeneous and particular equations singularity matching Solving Differential Equations with Laplace Transform The Laplace transform provides a particularly powerful method of solving differential equations it transforms a differential equation into an algebraic equation. Method (where L represents the Laplace transform): differential algebraic algebraic differential equation solve differential equation algebraic equation algebraic answer solution to differential equation L solve L 1 Laplace Transform: Definition Laplace transform maps a function of time t to a function of s . X ( s ) = x ( t ) e st dt There are two important variants: Unilateral (18.03) X ( s ) = x ( t ) e st dt Bilateral (6.003) X ( s ) = x ( t ) e st dt Both share important properties will discuss differences later. Laplace Transforms Example: Find the Laplace transform of x 1 ( t ) : t x 1 ( t ) x 1 ( t ) = Ae t if t otherwise X 1 ( s ) = x 1 ( t ) e st dt = Ae t e st dt = Ae ( s + ) t ( s + ) = A s + provided Re { s + } > which implies that Re { s } > . s-plane ROC A s + ; Re { s } > 6.003: Signals and Systems Lecture 7 October 1, 2009 2 Check Yourself t x 2 ( t ) x 2 ( t ) = e t e 2 t if t otherwise Which of the following is the Laplace transform of x 2 ( t ) ? 1. X 2 ( s ) = 1 ( s +1)( s +2) ; Re { s } > 1 2. X 2 ( s ) = 1 ( s +1)( s +2) ; Re { s } > 2 3. X 2 ( s ) = s ( s +1)( s +2) ; Re { s } > 1 4. X 2 ( s ) = s ( s +1)( s +2) ; Re { s } > 2 5. none of the above Regions of Convergence Left-sided signals have left-sided Laplace transforms (bilateral only)....
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lec07-handout-6up - 6.003: Signals and Systems Lecture 7...

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