I've come to realize that the Fourier transform of an already Fourier transformed signal gives the time-reversal signal.

$$\mathcal F(\mathcal F(x(t)))=x(−t)$$

ref 1,ref 2

However, my question is, if Fourier transformation is applied to the magnitude of the Fourier transformed signal (excluding phase information), is there a way to interpret the returned signal?

$$\mathcal F\{|\mathcal F\{x\}|\}=?$$


If it were the squared magnitude, the result would be the auto-correlation function, apart from a time inversion and possibly a scaling factor. If it's just the magnitude, then the result can be characterized as the only even function that has the same Fourier transform magnitude as the original function.


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