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| − | [[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[n]=x[n]*y[n] \ </math> | ||
| + | between | ||
| − | + | <math> x[n] = e^{jwn}u[n] \ </math> | |
| + | |||
| + | and | ||
| + | <math> y[n] = e^{jwn}u[n-6] \ </math>. | ||
| + | = My Solution= | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | z[n] &= x[n]*y[n]\\ | ||
| + | &=\sum_{k=-\infty}^{\infty}x[k]y[n-k] \\ | ||
| + | &=\sum_{k=-\infty}^{\infty}e^{jwk}u[k]e^{jw(n-k)}u[n-k-6] \\ | ||
| + | &=\begin{cases} | ||
| + | \sum_{k=0}^{n-6}e^{jwk}e^{jwn}e^{-jwk}, & n \geq 6 \\ | ||
| + | 0, & n < 5 \\ | ||
| + | \end{cases}\\ | ||
| + | &=\begin{cases} | ||
| + | e^{jwn}\sum_{k=0}^{n-6}1, & n \geq 6 \\ | ||
| + | 0, & n < 5 \\ | ||
| + | \end{cases}\\ | ||
| + | |||
| + | &=\begin{cases} | ||
| + | e^{jwn}(n-5), & n \geq 6 \\ | ||
| + | 0, & n < 5 \\ | ||
| + | \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 07:29, 6 May 2011
Problem
Compute the convolution
$ z[n]=x[n]*y[n] \ $
between
$ x[n] = e^{jwn}u[n] \ $
and
$ y[n] = e^{jwn}u[n-6] \ $.
My Solution
$ \begin{align} z[n] &= x[n]*y[n]\\ &=\sum_{k=-\infty}^{\infty}x[k]y[n-k] \\ &=\sum_{k=-\infty}^{\infty}e^{jwk}u[k]e^{jw(n-k)}u[n-k-6] \\ &=\begin{cases} \sum_{k=0}^{n-6}e^{jwk}e^{jwn}e^{-jwk}, & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ &=\begin{cases} e^{jwn}\sum_{k=0}^{n-6}1, & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ &=\begin{cases} e^{jwn}(n-5), & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ \end{align} $
Comments
Write them here.
