f[x_] := 
        Which[-∞ < x < -(π/2), 
          0, -(π/2) < x < (π/2), 1, (π/2) < x < ∞, 
          0];
(Sqrt[2*Pi])*FourierTransform[f[x], x, w]

g[
          t_] := ((2* Sin[((π* t)/2)])/t);
(Sqrt[2*Pi])*FourierTransform[g[t], t, w] // FullSimplify

Plot[(Sign[π - 2 w] + Sign[(π/2) + w]), {w, -10, 
        10}]