I need to perform a Fourier transform of the function which has the following form $$\hat{f}(\omega)=\mathcal{F}(\mathbb{e}^{i S(x)}R(x)),$$ where both $R(x)$ and $S(x)$ are real. I am wondering if it is possible to keep the polar representation of the resulting function, namely $$\hat{f}(\omega)=\mathbb{e}^{i \hat{S}(\omega)} \hat{R}(\omega).$$ I am looking for a numerical algorithm which allows to do such a transformation.

Of course, one can first obtain result in the coordinate representation $$\hat{f}(\omega) = \Re\{\hat{f}(\omega\} + i\Im\{\hat{f}(\omega\},$$ and then convert it to the polar form. But I am looking for a direct method since the latter transformation can be numerically unstable.


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