$ \mathcal{X}(\omega) = \left\{ {\begin{array}{*{20}c} {1,} & {-2 \le \omega \le 0} \\ { -1,} & {0 \le g(x) \ge 2} \\ {0,} & {| \omega| > 2} \\ \end{array}} \right. $
$ \ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega $
$ \ x(t)=\frac{1}{2\pi}\int_{-2}^{0}e^{j\omega t}\,d\omega + \frac{1}{2\pi}\int_{0}^{2}-e^{j\omega t}\,d\omega $
$ x(t)=\frac{1}{2\pi}(\frac{1}{jt} - \frac{e^{-j 2 t}}{jt}) + \frac{1}{2\pi}(-\frac{e^{j 2 t}}{jt} + \frac{1}{jt}) $
