Phase Retrieval for Wide Band Signals

This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given $f\in L^2(\mathbb{R})$ with Fourier transform in $L^2(\mathbb{R},e^{2c|x|} dx)$, we find all functions $g \in L^2 (\mathbb{R})$ with Fourier transform in $L^2(\mathbb{R}, e^{2c|x|} dx)$, such that $|f(x)|=|g(x)|$ for all $x \in \mathbb{R}$. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of $f$ and $g$, and determine if these constraints force uniqueness of the solution. Joint work with Ph. Jaming and K. Kellay