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

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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 1–5 Homeworks 1–4 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 [email protected] before Friday, October 2, 5pm. Last Time Many continuous-time systems can be represented with differential equations. Example: leaky tank r 0 ( t ) r 1 ( t ) h 1 ( t ) Differential equation representation: τ ˙ r 1 ( t ) = r 0 ( 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 −→ 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 ) = 0 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 ) : 0 t x 1 ( t ) x 1 ( t ) = Ae σt if t 0 0 otherwise X 1 ( s ) = −∞ x 1 ( t ) e st dt = 0 Ae σt e st dt = Ae ( s + σ ) t ( s + σ ) 0 = A s + σ provided Re { s + σ } > 0 which implies that Re { s } > σ . σ s -plane ROC A s + σ ; Re { s } > σ

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6.003: Signals and Systems Lecture 7 October 1, 2009 2 Check Yourself 0 t x 2 ( t ) x 2 ( t ) = e t e 2 t if t 0 0 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). Example: t x 3 ( t ) 1 x 3 ( t ) = e t if t 0 0 otherwise X 3 ( s ) = −∞ x 3 ( t ) e st dt = 0 −∞ e t e st dt = e ( s +1) t ( s + 1) 0 −∞ = 1 s + 1 provided Re { s + 1 } < 0 which implies that Re { s } < 1 .
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lec07-handout-6up - 6.003 Signals and Systems Lecture 7...

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