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| − | + | [[Category:ECE 301]] | |
| + | [[Category:Fall 2008]] | ||
| + | [[Category:mboutin]] | ||
| + | [[Category:Homework]] | ||
| + | =Homework 10, [[ECE301]] Fall 2008, Prof. [[user:mboutin|Boutin]]= | ||
| + | ---- | ||
| + | Click [[Main_Page_ECE301Fall2008mboutin|here]] to return to the [[Main_Page_ECE301Fall2008mboutin|ECE301 Fall 2008 Course Page of Prof. Boutin]]. | ||
| − | <math>x(t) = | + | '''Quick Links to homework assignments''' |
| + | [[Homework 1_ECE301Fall2008mboutin| 1]] | ||
| + | |[[Homework 2_ECE301Fall2008mboutin| 2]] | ||
| + | |[[Homework 3_ECE301Fall2008mboutin| 3]] | ||
| + | |[[Homework 4_ECE301Fall2008mboutin| 4]] | ||
| + | |[[Homework 5_ECE301Fall2008mboutin| 5]] | ||
| + | |[[Homework 6_ECE301Fall2008mboutin| 6]] | ||
| + | |[[Homework 7_ECE301Fall2008mboutin| 7]] | ||
| + | |[[Homework 8_ECE301Fall2008mboutin| 8]] | ||
| + | |[[Homework 9_ECE301Fall2008mboutin| 9]] | ||
| + | |[[Homework 10_ECE301Fall2008mboutin| 10]] | ||
| + | |[[homework 11_ECE301Fall2008mboutin| 11]] | ||
| + | ---- | ||
| + | |||
| + | |||
| + | == '''Fundamentals of Laplace Transform''' == | ||
| + | |||
| + | Let the signal be: | ||
| + | |||
| + | <math>x(t) =e^ {-at} \mathit{u} (t).</math> | ||
| + | |||
| + | Here is how to compute the Laplace Transform of <math>x(t)</math>: | ||
| + | |||
| + | <math> | ||
| + | \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} | ||
| + | </math> | ||
| + | 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 --[[User:Mboutin|Mboutin]] 11:58, 21 November 2008 (UTC) | ||
| + | |||
| + | Correction of above: | ||
| + | |||
| + | <math> | ||
| + | \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} | ||
| + | </math> | ||
| + | |||
| + | If <math>a+b\leq 0</math>, then the integral Diverges | ||
| + | |||
| + | Else, | ||
| + | |||
| + | <math> | ||
| + | \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} | ||
| + | </math> | ||
| + | * [[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]] | ||
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
