Ve216LectureNotesChapter6Part2-1

# - Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 6(Part II Representing Signals by Using Continuous-Time Complex Exponentials The Laplace

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Ve216 Lecture Notes Dianguang Ma Spring 2009

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Chapter 6 (Part II) Representing Signals by Using Continuous-Time Complex Exponentials: The Laplace Transform
6.3 The Unilateral Laplace Transform There are many applications of Laplace transforms in which it is reasonable to assume that the signals involved are causal. In such problems, it is advantageous to define the unilateral or one-sided Laplace transform, which is based only on the nonnegative-time portions of a signal. 0 ( ) ( ) ( ) { ( )} ( ) ( ) ( ) { ( ) ( )} u L st st x t X s X s UL x t x t e dt x t u t e dt L x t u t - - - -∞ = = = =

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6.3 The Unilateral Laplace Transform Any signal that is identically zero for t<0 has identical bilateral and unilateral Laplace transforms. The ROC for a unilateral laplace transform must always be a right-half plane. The unilateral Laplace transform is of considerable value in analyzing causal systems and particularly, systems specified by linear constant-coefficient differential equations with nonzero initial conditions.
6.4 Properties of the Unilateral Laplace Transform Lineraity ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( s bY s aY t by t ax s Y t y s X t x + +

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6.4 Properties of the Unilateral Laplace Transform Scaling 0 , 1 ) ( ) ( ) ( a a s X a at x s X t x
6.4 Properties of the Unilateral Laplace Transform Time Shift ) ( ) ( ) ( ) ( such that all for ) ( ) ( ) ( ) ( τ - - = - - - t u t x t u t x s X e t x s X t x s

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Figure 6.6 (p. 492) Time shifts for which the unilateral Laplace transform time-shift property does not apply. (a) A nonzero portion of x ( t ) that occurs at times t 0 is shifted to times t < 0. (b) A nonzero portion x ( t ) that occurs at times t < 0 is shifted to times t 0.
6.4 Properties of the Unilateral Laplace Transform s-Domain Shift ) ( ) ( ) ( ) ( 0 0 s s X t x e s X t x t s -

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6.4 Properties of the Unilateral Laplace Transform
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## This note was uploaded on 10/26/2009 for the course EECS EECS 216 taught by Professor Dianguangma during the Spring '09 term at University of Michigan-Dearborn.

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- Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 6(Part II Representing Signals by Using Continuous-Time Complex Exponentials The Laplace

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