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<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^{(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) = \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^{ | + | <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{e^{-jw}}{1-jw}+\frac{e^{-jw}}{1+jw}</math><br><br> | ||
<math> X(w) = \frac{2e^{-jw}}{1+w^2}</math><br><br> | <math> X(w) = \frac{2e^{-jw}}{1+w^2}</math><br><br> | ||
Revision as of 17:07, 7 October 2008
$ x(t) = e^{-|t-1|} \, $
$ X(w) = \int_{-\infty}^{\infty}e^{-|t-1|}e^{-jwt}dt $
$ X(w) = \int_{-\infty}^{1}e^{(t-1)}e^{-jwt}dt+\int_{1}^{\infty}e^{-(t-1)}e^{-jwt}dt $
$ X(w) = \int_{-\infty}^{1}e^{-1}e^{(1-jw)t}dt+\int_{1}^{\infty}e^{1}e^{-(1+jw)t}dt $
$ 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 } $$ = e^{-1}\frac{e^{(1-jw)}}{1-jw}+e^{1}\frac{e^{-(1+jw)}}{1+jw} $
$ X(w) = \frac{e^{-jw}}{1-jw}+\frac{e^{-jw}}{1+jw} $
$ X(w) = \frac{2e^{-jw}}{1+w^2} $
