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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]] $

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Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010