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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]] | ||
| + | ---- | ||
| + | |||
<math>x(t) = e^{-3t} , t>3 \,</math>, | <math>x(t) = e^{-3t} , t>3 \,</math>, | ||
<math>x(t)= e^{-6t} , 0 \le t \le 3</math>, | <math>x(t)= e^{-6t} , 0 \le t \le 3</math>, | ||
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<math>x(t)= e^{-3t} u(t-3) + e^{-6t}( u(t-3)-u(t))\,</math> | <math>x(t)= e^{-3t} u(t-3) + e^{-6t}( u(t-3)-u(t))\,</math> | ||
| + | |||
<math>X(\omega) = \int^\infty_\infty e^{-3t}e^{-j\omega t} dt + \int^2_0 e^{-6t}e^{-j\omega t} dt\,</math> | <math>X(\omega) = \int^\infty_\infty e^{-3t}e^{-j\omega t} dt + \int^2_0 e^{-6t}e^{-j\omega t} dt\,</math> | ||
| + | |||
<math>X(\omega) = \int^\infty_\infty e^{-(3+j\omega)t} dt + \int^3_0 e^{-(6+j\omega) t} dt\,</math> | <math>X(\omega) = \int^\infty_\infty e^{-(3+j\omega)t} dt + \int^3_0 e^{-(6+j\omega) t} dt\,</math> | ||
| − | <math>X(\omega) = {\left. \frac{e^{-(j\omega + 3)t}}{-(j\omega +3)} \right]^{\infty} | + | |
| + | <math>X(\omega) = {\left. \frac{e^{-(j\omega + 3)t}}{-(j\omega +3)} \right]^{\infty}_3 } + {\left. \frac{e^{-(j\omega + 6)t}}{-(j\omega +6)} \right]^3_0 }\,</math> | ||
| + | |||
<math>X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} - \frac{e^{-(3j\omega + 18)t}}{-j\omega +6} + \frac{1}{6+j\omega} \,</math> | <math>X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} - \frac{e^{-(3j\omega + 18)t}}{-j\omega +6} + \frac{1}{6+j\omega} \,</math> | ||
| + | |||
<math>X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} + \frac{1 - e^{-(3j\omega + 18)t}}{-j\omega +6} \,</math> | <math>X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} + \frac{1 - e^{-(3j\omega + 18)t}}{-j\omega +6} \,</math> | ||
| + | |||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:36, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
$ x(t) = e^{-3t} , t>3 \, $, $ x(t)= e^{-6t} , 0 \le t \le 3 $, $ x(t)= 0 , t < 0 \, $
$ x(t)= e^{-3t} u(t-3) + e^{-6t}( u(t-3)-u(t))\, $
$ X(\omega) = \int^\infty_\infty e^{-3t}e^{-j\omega t} dt + \int^2_0 e^{-6t}e^{-j\omega t} dt\, $
$ X(\omega) = \int^\infty_\infty e^{-(3+j\omega)t} dt + \int^3_0 e^{-(6+j\omega) t} dt\, $
$ X(\omega) = {\left. \frac{e^{-(j\omega + 3)t}}{-(j\omega +3)} \right]^{\infty}_3 } + {\left. \frac{e^{-(j\omega + 6)t}}{-(j\omega +6)} \right]^3_0 }\, $
$ X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} - \frac{e^{-(3j\omega + 18)t}}{-j\omega +6} + \frac{1}{6+j\omega} \, $
$ X(\omega) = \frac{e^{-(3j\omega + 9)}}{j\omega +3} + \frac{1 - e^{-(3j\omega + 18)t}}{-j\omega +6} \, $
