This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given $f\in L^2(\R)$ with Fourier transform in $L^2(\R,e^{2c|x|}\,\mbox{d}x)$, we find all functions $g\in L^2(\R)$ with Fourier transform in $L^2(\R,e^{2c|x|}\,\mbox{d}x)$, such that $|f(x)|=|g(x)|$ for all $x\in \R$. 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. This was a joint work with Philippe Jaming and Karim Kellay.