(New page: For the signal: <math>x(t) = e^{-8(t + 1)} u(t + 1)</math> <math>X(\omega) = \int_{-\infty}^\infty e^{-8(t + 1)} u(t + 1) e^{-j\omega t} \mathrm{d}t</math> when <math>t + 1 < 0 </math> ...) |
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| + | [[Category:problem solving]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:signals and systems]] | ||
| + | == Example of Computation of Fourier transform of a CT SIGNAL == | ||
| + | A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]] | ||
| + | ---- | ||
| + | |||
| + | |||
For the signal: | For the signal: | ||
| − | <math>x(t) = e^{-8(t + 1)} u(t + 1)</math> | + | <math>x(t) = e^{-8(t + 1)} u(t + 1) \ </math> |
<math>X(\omega) = \int_{-\infty}^\infty e^{-8(t + 1)} u(t + 1) e^{-j\omega t} \mathrm{d}t</math> | <math>X(\omega) = \int_{-\infty}^\infty e^{-8(t + 1)} u(t + 1) e^{-j\omega t} \mathrm{d}t</math> | ||
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<math> = e^{-8} \frac{ -e^{-(8+j\omega)t}}{(8+j\omega)t}\bigg|_{t=-1}^\infty</math> | <math> = e^{-8} \frac{ -e^{-(8+j\omega)t}}{(8+j\omega)t}\bigg|_{t=-1}^\infty</math> | ||
<math> = e^{-8} \frac{ e^{(8+j\omega)}}{(8+j\omega)}</math> | <math> = e^{-8} \frac{ e^{(8+j\omega)}}{(8+j\omega)}</math> | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:34, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
For the signal:
$ x(t) = e^{-8(t + 1)} u(t + 1) \ $
$ X(\omega) = \int_{-\infty}^\infty e^{-8(t + 1)} u(t + 1) e^{-j\omega t} \mathrm{d}t $
when $ t + 1 < 0 $ or $ t < -1 $, $ u(t + 1) = 0 $ so,
$ X(\omega) = \int_{-1}^\infty e^{-8(t + 1)} e^{-j\omega t}\mathrm{d}t $
$ = e^{-8} \int_{-1}^\infty e^{-8t} e^{-j \omega t}\mathrm{d}t $ $ = e^{-8} \int_{-1}^\infty e^{-(8+j\omega)t} \mathrm{d}t $
$ = e^{-8} \frac{ -e^{-(8+j\omega)t}}{(8+j\omega)t}\bigg|_{t=-1}^\infty $ $ = e^{-8} \frac{ e^{(8+j\omega)}}{(8+j\omega)} $
