View source for HW5.2 Jun Hyeong Park ECE301Fall2008mboutin - Rhea

[[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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<math> x(t) = e^{-|t-1|} \,</math><br><br>
<math> X(w) = \int_{-\infty}^{\infty}e^{-|t-1|}e^{-jwt}dt</math><br><br>
<math> X(w) = \int_{-\infty}^{1}e^{(t-1)}e^{-jwt}dt+\int_{1}^{\infty}e^{-(t-1)}e^{-jwt}dt</math><br><br>
<math> X(w) = \int_{-\infty}^{1}e^{-1}e^{(1-jw)t}dt+\int_{1}^{\infty}e^{1}e^{-(1+jw)t}dt</math><br><br>
<math> X(w) = {\left.\frac{e^{-1}e^{(1-jw)t}}{1-jw}\right]^1_{-\infty} }+{\left.\frac{-e^{1}e^{-(1+jw)t}}{1+jw}\right]^{\infty}_1 }</math><math>  = e^{-1}\frac{e^{(1-jw)}}{1-jw}+e^{1}\frac{e^{-(1+jw)}}{1+jw}</math><br><br>
<math> X(w) = \frac{e^{-jw}}{1-jw}+\frac{e^{-jw}}{1+jw}</math><br><br>
<math> X(w) = \frac{2e^{-jw}}{1+w^2}</math><br><br>
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[[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]]