(→EXERCISE) |
(→SOLUTION) |
||
| Line 6: | Line 6: | ||
== SOLUTION == | == SOLUTION == | ||
| − | <math> | + | <math>\,\mathcal{X}_1(\omega)=\mathcal{F}(x_1[n])=\sum_{n=-\infty}^{\infty}x_1[n]e^{-j\omega n}\,</math> |
| + | |||
| + | <math>\,=\sum_{n=-\infty}^{\infty}\alpha^{n}u[n]e^{-j\omega n}\,</math> | ||
| + | |||
| + | <math>\,=\sum_{n=0}^{\infty}\alpha^{n}e^{-j\omega n}\,</math> | ||
| + | |||
| + | <math>\,=\sum_{n=0}^{\infty}(\alpha e^{-j\omega })^{n}\,</math> | ||
| + | |||
| + | but <math>\,|\alpha e^{-j\omega }|<1\,</math> | ||
| + | |||
| + | <math>\,=\frac{1}{1-\alpha e^{-j\omega }}\,</math> | ||
Latest revision as of 16:39, 24 October 2008
EXERCISE
Assume $ |\alpha|<1 $
Compute the F.T. of $ x_1[n]=\alpha^{n}u[n] $
SOLUTION
$ \,\mathcal{X}_1(\omega)=\mathcal{F}(x_1[n])=\sum_{n=-\infty}^{\infty}x_1[n]e^{-j\omega n}\, $
$ \,=\sum_{n=-\infty}^{\infty}\alpha^{n}u[n]e^{-j\omega n}\, $
$ \,=\sum_{n=0}^{\infty}\alpha^{n}e^{-j\omega n}\, $
$ \,=\sum_{n=0}^{\infty}(\alpha e^{-j\omega })^{n}\, $
but $ \,|\alpha e^{-j\omega }|<1\, $
$ \,=\frac{1}{1-\alpha e^{-j\omega }}\, $
