# Sparse phase retrieval of one-dimensional signals by Prony's method

## TL;DR

This paper demonstrates that sparse signals composed of spikes or B-splines can be almost surely recovered from Fourier intensity measurements using Prony's method, providing a constructive approach and numerical algorithms.

## Contribution

It introduces a novel application of Prony's method for sparse phase retrieval from Fourier intensities, including an explicit algorithm for signal reconstruction.

## Key findings

- Recovery is possible from O(N^2) measurements.
- The method applies to signals with arbitrary real spike locations or B-spline knots.
- Numerical examples validate the proposed algorithm.

## Abstract

In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from O(N^2) Fourier intensity measurements up to trivial ambiguities. The constructive proof consists of two steps, where in the first step the Prony method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.