Latest revision as of 10:29, 16 September 2013

Homework 10, ECE301 Fall 2008, Prof. Boutin

Click here to return to the ECE301 Fall 2008 Course Page of Prof. Boutin.

Quick Links to homework assignments 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11

                       == Fundamentals of Laplace Transform ==

     Let the signal be:

     $ x(t) =e^ {-at} \mathit{u} (t). $
     
     Here is how to compute the Laplace Transform of $ x(t) $:

     $  \begin{align} X(s) &= \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt, \\      &= \int_{-\infty}^{\infty}{e^{-at}}{e^{-st}}dt   ,\text{ since }\mathit{u} (t)=1,\text{ for }t>0, \text{ else }\mathit{u} (t)=0, \\      &=\frac{1}{s+a}. ~^*  \end{align}  $

Note: the last equality (with a *) is untrue. Please do not write this on the test or you will get points marked off. I really appreciate this mistake being on Rhea, please do not erase it --Mboutin 11:58, 21 November 2008 (UTC)

Correction of above:

$ \begin{align} X(s) &= \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt, \\ &= \int_{0}^{\infty}{e^{-at}}{e^{-st}}dt ,\text{ let } s=b+j\omega, \\ &=\int_{0}^{\infty}{e^{-(a+b+j\omega)t}}dt, \\ \end{align} $

If $ a+b\leq 0 $, then the integral Diverges

Else,

$ \begin{align} X(s) &=\frac{e^{-(a+b)t}e^{-j\omega t}}{-(a+b+j\omega)}|_0^\infty, \\ &=0-\frac{-1}{s+a}, \\ &=\frac{1}{s+a} \end{align} $

• Homework _ECE301Fall2008mboutin#10 Daniel Morris: Properties of the Region of Convergence(ROC)
• HW10 Jun Hyeong Park_ECE301Fall2008mboutin
• HW10 Justin Kietzman- Properties of Laplace_ECE301Fall2008mboutin
• HW10 Brian Thomas- More Properties of Laplace_ECE301Fall2008mboutin
• HW10 Bavorndej Chanyasak_ECE301Fall2008mboutin
• HW10 Sangwan Han HW_ECE301Fall2008mboutin#10
• HW10 Sourabh Ranka_ECE301Fall2008mboutin
• HW10 Emily Blount_ECE301Fall2008mboutin
• HW10 Ananya Panja_ECE301Fall2008mboutin
• HW10 Derek Hopper_ECE301Fall2008mboutin
• HW10 Josh Long - Laplace Transform_ECE301Fall2008mboutin
• HW10 Monil Goklani_ECE301Fall2008mboutin
• HW10 David Record_ECE301Fall2008mboutin
• HW10 Xujun Huang_ECE301Fall2008mboutin
• HW10 Carlos Leon - Laplace transform table_ECE301Fall2008mboutin
• HW10 Ben Moeller - Bigger Laplace Transform Table & Trig Identities_ECE301Fall2008mboutin
• HW10 Zachary Curosh - Laplace Transform and Basic Properties_ECE301Fall2008mboutin
• HW10 SangMo je_ECE301Fall2008mboutin
• HW10 Seung Seob Lee_ECE301Fall2008mboutin
• HW10 Thomas Wroblewski_ECE301Fall2008mboutin
• HW10 Joseph Mazzei - Laplace Transform Pairs added to Table on Front Page_ECE301Fall2008mboutin
• HW10 Daniel Barjum: Example illustration relationship between Fourier and Laplace transform_ECE301Fall2008mboutin
• HW10 Vivek Ravi Region of Convergence for Laplace Transforms_ECE301Fall2008mboutin
• HW10 William Schmidt Initial and Final Value Theorem_ECE301Fall2008mboutin