# lec07 - 6.003 Signals and Systems Laplace and Z Transforms...

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6.003: Signals and Systems Laplace and Z Transforms October 1, 2009

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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

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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 ) = Ú x ( t ) e st dt There are two important variants: Unilateral (18.03) X ( s ) = Ú 0 x ( t ) e st dt Bilateral (6.003) X ( s ) = Ú −∞ x ( t ) e st dt Both share important properties — will discuss diﬀerences later.

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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 ) = Ú −∞ x 1 ( t ) e st dt = Ú 0 Ae σt e st dt = Ae ( s + σ ) t ( s + σ ) ² ² ² ² ² 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
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

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Check Yourself X 2 ( s ) = Ú 0 ( e t e 2 t ) e st dt = Ú 0 e t e st dt Ú 0 e 2 t e st dt = 1 s + 1 1 s + 2 = ( s + 2) ( s + 1) ( s + 1)( s + 2) = 1 ( s + 1)( s + 2) These equations converge if Re { s + 1 } > 0 and Re { s + 2 } > 0 , thus Re { s } > 1 .
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lec07 - 6.003 Signals and Systems Laplace and Z Transforms...

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