# Absolute values of two functions and absolute values of their Fourier transform coincides

Let $$f, g in L^2(mathbb{R})$$.

Is it true that if both $$|f|=|g|$$ and $$|hat f|=|hat g|$$ hold, then there exists $$theta in mathbb{R}$$ such that $$f=ge^{itheta}$$?

I am not able to prove it or disprove it. I suspect that this is true.