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| + | [[Category:problem solving]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:inverse Fourier transform]] | ||
| + | [[Category:signals and systems]] | ||
| + | == Example of Computation of inverse Fourier transform (CT signals) == | ||
| + | A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]] | ||
| + | ---- | ||
| + | |||
Compute the inverse Fourier transform of the following signal using the integral formula: | Compute the inverse Fourier transform of the following signal using the integral formula: | ||
<math>\,\mathcal{X}(\omega)=e^{-|\omega +3|} + e^{j(\omega + 5)}\delta(\omega - \pi)\,</math> | <math>\,\mathcal{X}(\omega)=e^{-|\omega +3|} + e^{j(\omega + 5)}\delta(\omega - \pi)\,</math> | ||
| − | |||
== Answer == | == Answer == | ||
| − | <math>\,x(t)=\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)=\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)\,</math> | ||
| + | |||
| + | <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 + | ||
| + | e^{j5}\int_{-\infty}^{\infty}e^{j(t+1)\omega}\delta(\omega - \pi)\,d\omega | ||
| + | \right)\,</math> | ||
| + | |||
| + | <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^{j5}e^{j(t+1)\pi} | ||
| + | \right)\,</math> | ||
| + | |||
| + | <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} | ||
| + | + e^{j(\pi(t+1)+5)} | ||
| + | \right)\,</math> | ||
| + | |||
| + | <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)}) | ||
| + | + e^{j(\pi(t+1)+5)} | ||
| + | \right)\,</math> | ||
| + | |||
| + | <math>\,x(t)=\frac{1}{2\pi}\left( | ||
| + | \frac{1}{jt+1}e^{-j3t} | ||
| + | - \frac{1}{jt-1}e^{-j3t} | ||
| + | + e^{j(\pi(t+1)+5)} | ||
| + | \right)\,</math> | ||
| + | |||
| + | <math>\,x(t)= | ||
| + | \frac{e^{-j3t}}{2\pi} \left(\frac{1}{jt+1} - \frac{1}{jt-1} \right) | ||
| + | + \frac{e^{j(\pi(t+1)+5)}}{2\pi} | ||
| + | \,</math> | ||
| + | |||
| + | <math>\,x(t)= | ||
| + | \frac{e^{-j3t}}{2\pi}\frac{jt-1-(jt+1)}{-t^2-1} | ||
| + | + \frac{e^{j(\pi(t+1)+5)}}{2\pi} | ||
| + | \,</math> | ||
| + | |||
| + | <math>\,x(t)= | ||
| + | \frac{e^{-j3t}}{2\pi}\frac{2}{t^2+1} | ||
| + | + \frac{e^{j(\pi(t+1)+5)}}{2\pi} | ||
| + | \,</math> | ||
| + | |||
| + | <math>\,x(t)= | ||
| + | \frac{e^{-j3t}}{\pi(t^2+1)} | ||
| + | + \frac{e^{j(\pi(t+1)+5)}}{2\pi} | ||
| + | \,</math> | ||
| + | |||
| + | |||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:42, 16 September 2013
Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
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)\, $
$ \,x(t)=\frac{1}{2\pi}\left( \frac{1}{jt+1}e^{-j3t} - \frac{1}{jt-1}e^{-j3t} + e^{j(\pi(t+1)+5)} \right)\, $
$ \,x(t)= \frac{e^{-j3t}}{2\pi} \left(\frac{1}{jt+1} - \frac{1}{jt-1} \right) + \frac{e^{j(\pi(t+1)+5)}}{2\pi} \, $
$ \,x(t)= \frac{e^{-j3t}}{2\pi}\frac{jt-1-(jt+1)}{-t^2-1} + \frac{e^{j(\pi(t+1)+5)}}{2\pi} \, $
$ \,x(t)= \frac{e^{-j3t}}{2\pi}\frac{2}{t^2+1} + \frac{e^{j(\pi(t+1)+5)}}{2\pi} \, $
$ \,x(t)= \frac{e^{-j3t}}{\pi(t^2+1)} + \frac{e^{j(\pi(t+1)+5)}}{2\pi} \, $
