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<math>\,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega \,</math> | <math>\,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega \,</math> | ||
| − | <math>\,x(t)=\int_{-\infty}^{\infty}e^{-|\omega +3|}e^{j\omega t}\,d\omega | + | <math>\,x(t)=\frac{1}{2\pi}\left( |
| − | + \int_{-\infty}^{\infty}e^{j(\omega + 5)}\delta(\omega - \pi)e^{j\omega t}\,d\omega\,</math> | + | \int_{-\infty}^{\infty}e^{-|\omega +3|}e^{j\omega t}\,d\omega |
| + | + \int_{-\infty}^{\infty}e^{j(\omega + 5)}\delta(\omega - \pi)e^{j\omega t}\,d\omega | ||
| + | \right)\,</math> | ||
| − | <math>\,x(t)=\int_{-\infty}^{-3}e^{\omega +3}e^{j\omega t}\,d\omega | + | <math>\,x(t)=\frac{1}{2\pi}\left( |
| + | \int_{-\infty}^{-3}e^{\omega +3}e^{j\omega t}\,d\omega | ||
+ \int_{-3}^{\infty}e^{-\omega -3}e^{j\omega t}\,d\omega + | + \int_{-3}^{\infty}e^{-\omega -3}e^{j\omega t}\,d\omega + | ||
| − | e^{j5}\int_{-\infty}^{\infty}e^{j(t+1)\omega}\delta(\omega - \pi)\,d\omega\,</math> | + | e^{j5}\int_{-\infty}^{\infty}e^{j(t+1)\omega}\delta(\omega - \pi)\,d\omega |
| + | \right)\,</math> | ||
| − | <math>\,x(t)=e^{3}\int_{-\infty}^{-3}e^{(jt+1)\omega}\,d\omega | + | <math>\,x(t)=\frac{1}{2\pi}\left( |
| + | e^{3}\int_{-\infty}^{-3}e^{(jt+1)\omega}\,d\omega | ||
+ e^{-3}\int_{-3}^{\infty}e^{(jt-1)\omega}\,d\omega | + e^{-3}\int_{-3}^{\infty}e^{(jt-1)\omega}\,d\omega | ||
| − | + e^{j5}e^{j(t+1)\pi}\,</math> | + | + e^{j5}e^{j(t+1)\pi} |
| + | \right)\,</math> | ||
| − | <math>\,x(t)=\frac{e^{3}}{jt+1}\left. e^{(jt+1)\omega}\right]_{-\infty}^{-3} | + | <math>\,x(t)=\frac{1}{2\pi}\left( |
| + | \frac{e^{3}}{jt+1}\left. e^{(jt+1)\omega}\right]_{-\infty}^{-3} | ||
+ \frac{e^{-3}}{jt-1}\left. e^{(jt-1)\omega}\right]_{-3}^{\infty} | + \frac{e^{-3}}{jt-1}\left. e^{(jt-1)\omega}\right]_{-3}^{\infty} | ||
| − | + e^{j(\pi(t+1)+5)}\,</math> | + | + e^{j(\pi(t+1)+5)} |
| + | \right)\,</math> | ||
| − | <math>\,x(t)=\frac{e^{3}}{jt+1}(e^{-3(jt+1)}-0) | + | <math>\,x(t)=\frac{1}{2\pi}\left( |
| + | \frac{e^{3}}{jt+1}(e^{-3(jt+1)}-0) | ||
+ \frac{e^{-3}}{jt-1}(0-e^{-3(jt-1)}) | + \frac{e^{-3}}{jt-1}(0-e^{-3(jt-1)}) | ||
| − | + e^{j(\pi(t+1)+5)}\,</math> | + | + e^{j(\pi(t+1)+5)} |
| + | \right)\,</math> | ||
Revision as of 21:17, 5 October 2008
Compute the inverse Fourier transform of the following signal using the integral formula:
$ \,\mathcal{X}(\omega)=e^{-|\omega +3|} + e^{j(\omega + 5)}\delta(\omega - \pi)\, $
Answer
$ \,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega \, $
$ \,x(t)=\frac{1}{2\pi}\left( \int_{-\infty}^{\infty}e^{-|\omega +3|}e^{j\omega t}\,d\omega + \int_{-\infty}^{\infty}e^{j(\omega + 5)}\delta(\omega - \pi)e^{j\omega t}\,d\omega \right)\, $
$ \,x(t)=\frac{1}{2\pi}\left( \int_{-\infty}^{-3}e^{\omega +3}e^{j\omega t}\,d\omega + \int_{-3}^{\infty}e^{-\omega -3}e^{j\omega t}\,d\omega + e^{j5}\int_{-\infty}^{\infty}e^{j(t+1)\omega}\delta(\omega - \pi)\,d\omega \right)\, $
$ \,x(t)=\frac{1}{2\pi}\left( e^{3}\int_{-\infty}^{-3}e^{(jt+1)\omega}\,d\omega + e^{-3}\int_{-3}^{\infty}e^{(jt-1)\omega}\,d\omega + e^{j5}e^{j(t+1)\pi} \right)\, $
$ \,x(t)=\frac{1}{2\pi}\left( \frac{e^{3}}{jt+1}\left. e^{(jt+1)\omega}\right]_{-\infty}^{-3} + \frac{e^{-3}}{jt-1}\left. e^{(jt-1)\omega}\right]_{-3}^{\infty} + e^{j(\pi(t+1)+5)} \right)\, $
$ \,x(t)=\frac{1}{2\pi}\left( \frac{e^{3}}{jt+1}(e^{-3(jt+1)}-0) + \frac{e^{-3}}{jt-1}(0-e^{-3(jt-1)}) + e^{j(\pi(t+1)+5)} \right)\, $
