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)=3\pi \delta (\omega-\pi) $

Using the Formula for Inverse Fourier Transforms:

$ \mathcal{F}^{-1}(\mathcal{X}(\omega))= x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} \,d\omega $

So:

$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}3\pi \delta (\omega-\pi)e^{j\omega t} \,d\omega $ Using sifting property:

$ =\frac{3\pi}{2\pi}e^{j\pi t}=1.5e^{j\pi t} $


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Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood