(New page: <math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math>) |
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| + | [[Category:problem solving]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:inverse Fourier transform]] | ||
| + | [[Category:signals and systems]] | ||
| + | == Example of Computation of inverse Fourier transform (CT signals) == | ||
| + | A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]] | ||
| + | ---- | ||
| + | Compute the Inverse Fourier Transform of: | ||
| + | |||
<math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math> | <math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math> | ||
| + | |||
| + | <math>x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \mathcal{X} (\omega) e^{jwt}dw</math> | ||
| + | |||
| + | <math>x(t) = \frac{1}{2\pi} \frac{\pi}{j}\int^{\infty}_{-\infty} \delta(w-2\pi)e^{jwt} dw + \frac{1}{2\pi}\frac{2\pi}{j} \int_{-\infty}^{\infty} \delta(w+2\pi)e^{jwt} dw]</math> | ||
| + | |||
| + | <math>x(t) = \frac{1}{2j}e^{j2\pi t} + \frac{1}{j}e^{-j2\pi t} | ||
| + | |||
| + | |||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:52, 16 September 2013
Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
Compute the Inverse Fourier Transform of:
$ \,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi) $
$ x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \mathcal{X} (\omega) e^{jwt}dw $
$ x(t) = \frac{1}{2\pi} \frac{\pi}{j}\int^{\infty}_{-\infty} \delta(w-2\pi)e^{jwt} dw + \frac{1}{2\pi}\frac{2\pi}{j} \int_{-\infty}^{\infty} \delta(w+2\pi)e^{jwt} dw] $
$ x(t) = \frac{1}{2j}e^{j2\pi t} + \frac{1}{j}e^{-j2\pi t} ---- [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] $
