Suppose I have a signal $u(x)$ where $x\in[0,2\pi]$ and the signal has following form in Fourier space,

$\hat{u}(\kappa)=\begin{cases} \frac{1+i\tan{(\varphi_u(\kappa))}}{\sqrt{1+\tan^2{(\varphi_u(\kappa))}}}\kappa^{\alpha/2},\ \ 0<\kappa\le\kappa_c \\ 0+i0,\ \ \ else \end{cases}$

where $\varphi_u(\kappa)$ is a random number for each $\kappa$, and $\alpha$ is a constant.

How can I get $u(x)$ using inverse Fourier transform?