ECE634S09_L4_LaplaceTransforms

# ECE634S09_L4_LaplaceTransforms - ECE634 Signals and Systems...

This preview shows pages 1–2. Sign up to view the full content.

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.” ~

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern