(New page: == Question 1 == <math>\mathcal{F} (n^2u[n-2] - n^2 u[n+2]) = \sum^{\infty}_{n = -\infty}(n^2u[n-2] - n^2 u[n+2])e^{ j\omega n}\,</math> <math>=\sum^{\infty}_{n = -\infty}(n^2(u[n-2] - ...) |
(→Question 2) |
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== Question 2 == | == Question 2 == | ||
| + | <math>x(t) = \mathcal{F}^{-1} (\mathcal{X}(\omega)) \,</math> | ||
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| + | <math>= \frac{1}{2\pi} \pi \mathcal{F}^{-1} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o))</math> | ||
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| + | <math>= \frac{1}{2} \int^{\infty}_{-\infty} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o)) e^{j\omega t} d\omega</math> | ||
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| + | <math>= \frac{1}{2} e^{j\omega _o t} \int^{\infty}_{-\infty} \delta(\omega + \omega _o) d\omega + \frac{1}{2} e^{-j\omega _o t} \int^{\infty}_{-\infty}\delta(\omega - \omega _o)) d\omega</math> | ||
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| + | <math>= \frac{1}{2} e^{j\omega _o t} + \frac{1}{2} e^{-j\omega _o t} d\omega</math> | ||
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| + | <math>= \frac{1}{2} ( e^{j\omega _o t} + e^{-j\omega _o t} ) </math> | ||
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| + | <math>= \cos(\omega _o t) \,</math> | ||
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| + | == Question 3== | ||
Revision as of 16:15, 21 October 2008
Question 1
$ \mathcal{F} (n^2u[n-2] - n^2 u[n+2]) = \sum^{\infty}_{n = -\infty}(n^2u[n-2] - n^2 u[n+2])e^{ j\omega n}\, $
$ =\sum^{\infty}_{n = -\infty}(n^2(u[n-2] - u[n+2])e^{ j\omega n}\, $
$ = \sum^{2}_{n = -2}(n^2 e^{ j\omega n}\, $
$ = 4e^{2 j\omega } + 4e^{-2 j\omega } + e^{ j\omega } + e^{- j\omega }\, $
$ = 8\frac{e^{2 j\omega } + e^{-2 j\omega }}{2}+ 2\frac{e^{ j\omega } + e^{- j\omega }}{2} \, $
$ = 8\cos(2\omega) + 2\cos(-\omega)\, $
Question 2
$ x(t) = \mathcal{F}^{-1} (\mathcal{X}(\omega)) \, $
$ = \frac{1}{2\pi} \pi \mathcal{F}^{-1} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o)) $
$ = \frac{1}{2} \int^{\infty}_{-\infty} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o)) e^{j\omega t} d\omega $
$ = \frac{1}{2} e^{j\omega _o t} \int^{\infty}_{-\infty} \delta(\omega + \omega _o) d\omega + \frac{1}{2} e^{-j\omega _o t} \int^{\infty}_{-\infty}\delta(\omega - \omega _o)) d\omega $
$ = \frac{1}{2} e^{j\omega _o t} + \frac{1}{2} e^{-j\omega _o t} d\omega $
$ = \frac{1}{2} ( e^{j\omega _o t} + e^{-j\omega _o t} ) $
$ = \cos(\omega _o t) \, $
