I have a frequency-domain representation $X(e^{i\omega})$ of the complex discrete one-dimensional signal $x[n]$: $X(e^{i\omega})=\mathcal{F}\{x[n]\}$. Is there a frequency-domain transformation of $X(e^{i\omega})$ into $\hat{X}(e^{i\omega})=\mathcal{F}(|x[n]|)$?
Obviously, $\hat{X}(e^{i\omega})=\mathcal{F}\left\{\left|\mathcal{F}^{-1}\{X(e^{i\omega})\}\right|\right\}$ will do the trick, but I am interested if there is a dual to the absolute value function $|x|$ in the frequency domain.