(New page: == Fourier Transform ==) |
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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]] | ||
| + | ---- | ||
== Fourier Transform == | == Fourier Transform == | ||
| + | Signal: x(t) = <math> e^{3|t-1|}</math> | ||
| + | |||
| + | <math>X(j \omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \!</math> | ||
| + | |||
| + | <math> = \int_{-\infty}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \!</math> | ||
| + | |||
| + | <math> = \int_{1}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \!</math> + <math> \int_{-\infty}^{1} e^{2|t-1|} e^{-j\omega t} dt \!</math> | ||
| + | |||
| + | |||
| + | = <math> \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} </math> | ||
| + | |||
| + | |||
| + | = <math> \frac{4e^{-j \omega}}{4 + \omega ^2} | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:31, 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) = $ e^{3|t-1|} $
$ X(j \omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \! $
$ = \int_{-\infty}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \! $
$ = \int_{1}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \! $ + $ \int_{-\infty}^{1} e^{2|t-1|} e^{-j\omega t} dt \! $
= $ \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} $
= $ \frac{4e^{-j \omega}}{4 + \omega ^2} ---- [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] $
