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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]] | ||
| + | ---- | ||
Compute the fourier transform of this signal below: | Compute the fourier transform of this signal below: | ||
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<math>\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty} e^{-7t} u(t+1) e^{-j\omega t} dt + \int_{-\infty}^{\infty} e^{-23t} u(t-1) e^{-j\omega t}dt\,</math> | <math>\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty} e^{-7t} u(t+1) e^{-j\omega t} dt + \int_{-\infty}^{\infty} e^{-23t} u(t-1) e^{-j\omega t}dt\,</math> | ||
| − | <math>\,\mathcal{X}(\omega)=\int_{-1}^{\infty} e^{-(7+j\omega )t} dt + \int_{1}^{\infty} e^{-(23+j\omega )t} | + | <math>\,\mathcal{X}(\omega)=\int_{-1}^{\infty} e^{-(7+j\omega )t} dt + \int_{1}^{\infty} e^{-(23+j\omega )t} dt\,</math> |
| − | <math>\left. \frac{e^{- | + | <math>\,\mathcal{X}(\omega)={\left. \frac{e^{-(7+j\omega )t}}{-(7+j\omega )}\right]_{-1}^{\infty}} + {\left. \frac{e^{-(23+j\omega )t}}{-(23+j\omega )}\right]_{1}^{\infty}}\,</math> |
| + | |||
| + | <math>\,\mathcal{X}(\omega)=\frac{-e^{7+j\omega}}{-(7+j\omega )} - \frac{-e^{-(23+j\omega)}}{-(23+j\omega )} \,</math> | ||
| + | |||
| + | <math>\,\mathcal{X}(\omega)=\frac{e^{7+j\omega}}{7+j\omega} - \frac{e^{-(23+j\omega)}}{23+j\omega} \,</math> | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:26, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
Compute the fourier transform of this signal below:
$ \,x(t)=e^{-7t}u(t+1) + e^{23t}u(t-1)\, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{\infty} e^{-7t} u(t+1) e^{-j\omega t} dt + \int_{-\infty}^{\infty} e^{-23t} u(t-1) e^{-j\omega t}dt\, $
$ \,\mathcal{X}(\omega)=\int_{-1}^{\infty} e^{-(7+j\omega )t} dt + \int_{1}^{\infty} e^{-(23+j\omega )t} dt\, $
$ \,\mathcal{X}(\omega)={\left. \frac{e^{-(7+j\omega )t}}{-(7+j\omega )}\right]_{-1}^{\infty}} + {\left. \frac{e^{-(23+j\omega )t}}{-(23+j\omega )}\right]_{1}^{\infty}}\, $
$ \,\mathcal{X}(\omega)=\frac{-e^{7+j\omega}}{-(7+j\omega )} - \frac{-e^{-(23+j\omega)}}{-(23+j\omega )} \, $
$ \,\mathcal{X}(\omega)=\frac{e^{7+j\omega}}{7+j\omega} - \frac{e^{-(23+j\omega)}}{23+j\omega} \, $
