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Specify a signal x(t) and compute its Fourier transform using the integral formula.( Make a hard one) | Specify a signal x(t) and compute its Fourier transform using the integral formula.( Make a hard one) | ||
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| + | <math>e^{-2(t-1)}u(t-1)\,</math> | ||
<math>\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\,</math> | <math>\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)= \int_{1}^{ \infty} e^{2-t(2+jw)}dt\,</math> | ||
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| + | integrating and putting in limits | ||
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| + | <math>\,\mathcal{X}(\omega)= \frac{e^{2-(2+jw)}}{2+jw} \,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)= \frac{e^{2-2-jw}}{2+jw} \,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)= \frac{e^{-jw}}{2+jw} \,</math> | ||
Revision as of 18:08, 7 October 2008
Specify a signal x(t) and compute its Fourier transform using the integral formula.( Make a hard one)
$ e^{-2(t-1)}u(t-1)\, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)= \int_{1}^{ \infty} e^{2-t(2+jw)}dt\, $
integrating and putting in limits
$ \,\mathcal{X}(\omega)= \frac{e^{2-(2+jw)}}{2+jw} \, $
$ \,\mathcal{X}(\omega)= \frac{e^{2-2-jw}}{2+jw} \, $
$ \,\mathcal{X}(\omega)= \frac{e^{-jw}}{2+jw} \, $
