ECE634S09_L4_LaplaceTransforms

ECE634S09_L4_LaplaceTransforms - ECE634 Signals and Systems...

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ECE634 Signals and Systems II, Spring 2009 - Lecture 4, January 30 Daniel S. Brogan 1 4.1 The Laplace Transform: The unilateral (one-sided) Laplace Transform is the only one of interest here. It is defined as ( ) ( ) 0 st X s x t e dt = where ( ) X s is a function that exists in the complex frequency domain, ( ) x t is a real (in practice) function that exists in the complex domain but is zero valued for 0 t < (causal), and s is a complex frequency variable defined as s j σ ω = + where ω relates to the periodic nature of the time domain function and σ relates to the transient response nature of the time domain function. The inverse Laplace transform has the form ( ) ( ) 1 2 c j st c j x t X s e ds j π + ∞ − ∞ = However, calculation of this integral is beyond the scope of this course. The use of the one-sided Laplace Transform pair rather than the two-sided Laplace transform pair constrains the transform pair such that the transform of one signal can yield only one result and allows us to ignore issues about the regions of convergence. The transform pair relationship is often represented by ( ) ( ) x t X s A common transform notation uses a scripted ‘L’ for the transform operator (see p. 340 in the text) Fourier vs. Laplace: “Regarding the difference between Laplace Transforms (LP) and Fourier Transforms (FT), FT is a specific case of LT, which is more general. In Laplace transform the argument s= v+wi, whereas Fourier transform is a special case of Laplace transform with v=0. That is why LT forces convergence as t approaches infinity. The key difference is that LP is better at handling problems with initial conditions, while FT is better at handling boundary value problems.” ~
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