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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}_0 } + {\left. \frac{e^{-(j\omega + 6)t}}{-(j\omega +6)} \right]^3_0 }\,</math>
 
<math>X(\omega) = {\left. \frac{e^{-(j\omega + 3)t}}{-(j\omega +3)} \right]^{\infty}_0 } + {\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>

Revision as of 17:48, 8 October 2008

$ 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}_0 } + {\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} \, $

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva