6 003 Signals and Systems Lecture 7 6 003 Signals and Systems October 1 2009 Mid term Examination 1 Wednesday October 7 7 30 9 30pm Walker Memorial Laplace and Z Transforms 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 12 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions during open office hours October 1 2009 Conflict Contact freeman mit edu before Friday October 2 5pm Last Time Solving Differential Equations with Laplace Transform Many continuous time systems can be represented with differential equations The Laplace transform provides a particularly powerful method of solving differential equations it transforms a differential equation into an algebraic equation Example leaky tank Method where L represents the Laplace transform r0 t differential differential equation h1 t algebraic algebraic equation L solve r1 t algebraic algebraic answer Differential equation representation 1 L solution to differential equation differential equation r 1 t r0 t r1 t Last time we considered two methods to solve differential equations solving homogeneous and particular equations singularity matching Laplace Transform Definition Laplace Transforms Laplace transform maps a function of time t to a function of s Example Find the Laplace transform of x1 t Z X s x t e st dt x1 t There are two important variants Unilateral 18 03 Z X s x t e st dt X1 s 0 Z x1 t Ae t 0 if t 0 otherwise x1 t e st dt Z 0 Ae t e st dt Ae s t s provided Re s 0 which implies that Re s Bilateral 6 003 Z X s x t e st dt s plane Both share important properties will discuss differences later A Re s s 1 t 0 ROC 0 A s 6 003 Signals and Systems Lecture 7 Check Yourself Regions of Convergence Left sided signals have left sided Laplace transforms bilateral only x2 t e t e 2t 0 Example if t 0 otherwise t 0 x3 t Which of the following is the Laplace transform of x2 t 1 1 X2 s s 1 s 2 Re s 1 X3 s 1 2 X2 s s 1 s 2 Re s 2 Z e t 0 x3 t if t 0 otherwise x3 t e st dt Z 0 t 1 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 s plane s 3 X2 s s 1 s 2 Re s 1 s 4 X2 s s 1 s 2 Re s 2 1 Re s 1 s 1 5 none of the above ROC x2 t October 1 2009 1 Left and Right Sided Signals Left and Right Sided ROCs We can concisely express left and right sided signals by multiplication with step functions Laplace transforms of left and right sided exponentials have the same form except with left and right sided ROCs respectively Laplace transform time function e t u t x1 t e t if t 0 0 otherwise e t u t s plane x1 t t 0 t 0 1 s 1 ROC 1 t 1 e t u t 1 Check Yourself t 1 s 1 ROC x3 t e t if t 0 0 otherwise e t u t x3 t s plane 1 Solving Differential Equations with Laplace Transforms Solve the following differential equation Find the Laplace transform of x4 t y t y t t Take the Laplace transform of this equation x4 t L y t y t L t x4 t e t 0 1 X4 s 2 X4 s 3 X4 s 4 X4 s 2 1 s2 2 1 s2 2 1 s2 2 1 s2 The Laplace transform of a sum is the sum of the Laplace transforms prove this as an exercise t L y t L y t L t Re s 1 Re s 1 Re s 1 Re s 1 What s the Laplace transform of a derivative 5 none of the above 2 6 003 Signals and Systems Lecture 7 October 1 2009 Laplace transform of a derivative Solving Differential Equations with Laplace Transforms Assume that X s is the Laplace transform of x t Z x t e st dt X s Back to the previous problem Find the Laplace transform of y t x t Z y t e st dt Y s Z x t e st dt Z x t se st dt x t e st Let Y s represent the Laplace transform of y t L y t L y t L t Then sY s is the Laplace transform of y t sY s Y s L t What s the Laplace transform of the impulse function The first term must be zero since X s converged Thus Z Y s s x t e st dt sX s Laplace transform of the impulse function Solving Differential Equations with Laplace Transforms Let x t t Back to the previous problem X s Z Z Z sY s Y s L t 1 t e st dt This is a simple algebraic expression Solve for Y s 1 Y s s 1 We ve seen this Laplace transform previously t e st t 0 dt t 1 dt y t e t u t 1 why not y t e t u t Sifting property t sifts out the value of the integrand at t 0 Notice that we solved the differential equation y t y t t without computing homogeneous and particular solutions and without singularity matching Solving Differential Equations with Laplace Transforms Solving Differential Equations with Laplace Transforms Summary of method Recognizing the form Is there a more systematic way to take an inverse Laplace transform Start with differential equation Yes and no y t y t t Formally Take the Laplace transform of this equation Z j 1 X s est ds 2 j j but this integral is not generally easy to compute sY s Y s 1 x t Solve for Y s 1 Y s s 1 Take inverse Laplace transform by recognizing form of transform This equation can be useful to prove theorems We will find better ways e g partial fractions to compute inverse transforms for common systems y t e t u t 3 6 003 Signals and Systems Lecture 7 October 1 2009 Solving Differential Equations with Laplace Transforms Properties of Laplace Transforms Example 2 The use of Laplace Transforms to solve differential equations depends on several important properties y t 3y t 2y t t Laplace transform s2 Y s 3sY s 2Y s 1 x t X s ROC Linearity ax1 t bx2 t aX1 s bX2 s R1 R2 x t T X s e sT R Delay by T Solve 1 1 1 Y s s …
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