| Line 1: | Line 1: | ||
| − | [[Category:ECE301 S11 Exam 3 more practice]][[Category: | + | [[Category:ECE301 S11 Exam 3 more practice]] |
| + | [[Category:ECE301Spring2011Boutin]] | ||
| + | [[Category:Problem_solving]] | ||
| − | = | + | = Problem = |
| + | Compute the convolution | ||
| + | <math>z(t)=x(t)*y(t) \ </math> | ||
| + | between | ||
| − | + | <math> x(t) = e^{jwt}u(t+2) \ </math> | |
| − | + | ||
| + | and | ||
| + | <math>y(t) = e^{jwt}u(t-2) \ </math>. | ||
| + | = My Solution= | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | z(t) &= e^{jwt}u(t+2) * e^{jwt}u(t-2) \\ | ||
| + | &= \int_{-\infty}^{\infty} e^{jw\tau}u(\tau+2) e^{jw(t- \tau)}u(t - \tau -2)\mathrm{d}\tau \\ | ||
| + | &= \int_{-2}^{t-2} e^{jw\tau} e^{jw(t- \tau)}\mathrm{d}\tau \\ | ||
| + | &= e^{jwt} \int_{-2}^{t-2} 1 \mathrm{d}\tau \\ | ||
| + | &= \begin{cases} | ||
| + | t e^{jwt} &, \text{when }t > 2 \\ | ||
| + | 0 &, \text{else}\end{cases} | ||
| + | \end{align} | ||
| + | </math> | ||
| + | ==Comments== | ||
| + | Write them here. | ||
| + | ---- | ||
[[ ECE301 S11 Exam 3 more practice|Back to ECE301 S11 Exam 3 more practice]] | [[ ECE301 S11 Exam 3 more practice|Back to ECE301 S11 Exam 3 more practice]] | ||
Latest revision as of 06:04, 6 May 2011
Problem
Compute the convolution
$ z(t)=x(t)*y(t) \ $
between
$ x(t) = e^{jwt}u(t+2) \ $
and
$ y(t) = e^{jwt}u(t-2) \ $.
My Solution
$ \begin{align} z(t) &= e^{jwt}u(t+2) * e^{jwt}u(t-2) \\ &= \int_{-\infty}^{\infty} e^{jw\tau}u(\tau+2) e^{jw(t- \tau)}u(t - \tau -2)\mathrm{d}\tau \\ &= \int_{-2}^{t-2} e^{jw\tau} e^{jw(t- \tau)}\mathrm{d}\tau \\ &= e^{jwt} \int_{-2}^{t-2} 1 \mathrm{d}\tau \\ &= \begin{cases} t e^{jwt} &, \text{when }t > 2 \\ 0 &, \text{else}\end{cases} \end{align} $
Comments
Write them here.
