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I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function $$f = e^{i \phi(x)}. $$

I am interested in the following problem. If I know the function/distribution $B(x): \mathbb{R} \to \mathbb{C}$ $$f(x) * f(-x)^* = \int_{-\infty}^{\infty} e^{i (\phi(\xi) - \phi(\xi-x))}d\xi = B(x)$$ can I recover $\phi(x)$?

My initial idea was to figure out the correspondence between $\phi(x)$ and $\hat{f}(k)$ since I know $\hat{f}(k) \hat{f}(k)^*$, where $$\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi ikx} dx$$ but according to What are the properties of the fourier transform of a phase-only function? $\hat{f}(k)$ is a pretty hard to characterize distribution. This is also related to When I know self convolution of the complex function can I recover function itself or its modulus?, where I got negative answer for more general problem. Physical reasoning of this is to recover the intensity of the pure phase object from Fresnel integral, see Wikipedia knowing $I_z(x,y) = E(x,y,z)E(x,y,z)^*$ how to recover $I_0(x,y) = E(x,y,0)E(x,y,0)^*$ in the special case of $E(x,y,0) = e^{i\psi(x,y)}$. There is some theory that relies upon the linearization of exponential function but I am interested if I can do something without it.

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    $\begingroup$ If $\phi$ is a solution and $0\le\phi<\pi$, then $\phi+c$ is also a solution for any $c\in(0,\pi]$. $\endgroup$
    – Iosif Pinelis
    Commented Feb 7, 2023 at 17:46
  • $\begingroup$ Yes, I need to fix it. Usually I know $x_0$ s.t. $\phi(x) = 0$ for $|x|>x_0$ so I can say $\phi(x_0)=0$. $\endgroup$
    – VojtaK
    Commented Feb 7, 2023 at 18:03
  • $\begingroup$ For small $x$ upto first order $B(x)≈\int e^{i\phi'(t)x}dt=\int g(z) e^{izx} dz$ where $z=\phi'(t), g(z)=\frac{1}{\phi''(t)}$. By inverse Fourier transform we can approximately obtain $g(z)$. Explicitly, $\phi(t)=\int \psi^{-1}(t)dt$ where $\psi(z)=\int g(z)dz$. $\endgroup$
    – Alapan Das
    Commented Feb 8, 2023 at 9:53
  • $\begingroup$ Since the time of pulse compression phase modulated radars have been invented, at least 85 years ago, the only useful function that has been found is the linear FM chirp, that is $\phi(\xi)=a+b\xi+c\xi^2$ in which case $B(x)\propto \delta(x)$. As far as I know all other phase functions $\phi$ applied to pulse compression just try to approximate $\delta(x)$ for obvious simplicity in detection practice. $\endgroup$
    – hyportnex
    Commented Feb 8, 2023 at 13:13

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