Example of Computation of inverse Fourier transform (CT signals)

A practice problem on CT Fourier transform


Inverse F.T'ing

Given

$ X(\omega) = 3\pi\delta(\omega-\pi)+\delta(\omega-2\pi)-2\pi\delta(\omega-3\pi)\! $


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$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{jwt}dw $


$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}[3\pi\delta(\omega-\pi)+\delta(\omega-2\pi)-2\pi\delta(\omega-3\pi)\!]e^{jwt}dw $


$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}3\pi\delta(\omega-\pi)e^{jwt}dw+\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega-2\pi)e^{jwt}dw-\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi\delta(\omega-3\pi)\!e^{jwt}dw $


$ x(t)= \frac{3}{2}\int_{-\infty}^{\infty}\delta(\omega-\pi)e^{jwt}dw+\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega-2\pi)e^{jwt}dw-\int_{-\infty}^{\infty}\delta(\omega-3\pi)\!e^{jwt}dw $

$ x(t)= \frac{3*e^{j\pi*t}}{2}e^{jwt}+\frac{e^{j2\pi*t}}{2\pi}e^{jwt}-e^{j3*\pi*t}e^{jwt} $


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