$ \chi(\omega) = 2 \pi \sigma (\omega - \pi) $
$ x[n] = \frac{1}{2\pi}\int_{-\infty}^{\infty} 2\pi \delta (\omega - \pi)e^{j\omega t} dw $
$ x[n] = \int_{-\infty}^\infty \delta (\omega - \pi)e^{j\omega t} dw $
