# Phase retrieval for wide-band signals

## TL;DR

This paper addresses the phase retrieval problem for wide-band signals by transforming it into a Hardy space problem, providing conditions for uniqueness and solutions using inner-outer factorization.

## Contribution

It introduces a novel approach to phase retrieval for wide-band signals by leveraging Hardy space techniques and explores conditions for solution uniqueness.

## Key findings

- Successfully translated the phase retrieval problem into Hardy spaces.
- Derived conditions under which solutions are unique.
- Extended analysis to include additional transform constraints.

## Abstract

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|} dx), we find all functions g $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x| dx}), 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.