(New page: == Fourier Transform == '''Signal:''' '''<big><big><math>x(t) = 2tjwe^{-t} u(t)</math></big></big>''' <math>\mathcal{F} = \int_{-\infty}^{+\infty}x(t) e^{-jwt} \, dt = \int_{-\infty}^{+\...) |
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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]] | ||
| + | ---- | ||
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== Fourier Transform == | == Fourier Transform == | ||
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<math>\Rightarrow \mathcal{F} = \int_{1}^{0}\frac{2tjwu}{-2tjwu} \, du = \int_{1}^{0} 1\, du = \left [u] \right ]_1^0 = \left [ e^{-jwt^2} \right ]_0^\infty = -1 </math> | <math>\Rightarrow \mathcal{F} = \int_{1}^{0}\frac{2tjwu}{-2tjwu} \, du = \int_{1}^{0} 1\, du = \left [u] \right ]_1^0 = \left [ e^{-jwt^2} \right ]_0^\infty = -1 </math> | ||
| + | |||
| + | |||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:37, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
Fourier Transform
Signal: $ x(t) = 2tjwe^{-t} u(t) $
$ \mathcal{F} = \int_{-\infty}^{+\infty}x(t) e^{-jwt} \, dt = \int_{-\infty}^{+\infty}2tjwe^{-t} u(t) e^{-jwt} \, dt = \int_{0}^{+\infty}2tjw e^{-jwt^2} \, dt $
let $ u = e^{-jwt^2}, du = -2tjwe^{-jwt^2} dt \Rightarrow dt = \frac{du}{-2tjwe^{-jwt^2}} = \frac{du}{{-2tjwu}} $
$ \Rightarrow \mathcal{F} = \int_{1}^{0}\frac{2tjwu}{-2tjwu} \, du = \int_{1}^{0} 1\, du = \left [u] \right ]_1^0 = \left [ e^{-jwt^2} \right ]_0^\infty = -1 $
