An elementary approach for the phase retrieval problem

TL;DR

This paper introduces a recursive, non-linear method for phase retrieval using Fourier spectrum analysis, achieving an $O(N^2)$ complexity for 1D problems, applicable to higher dimensions with potential noise limitations.

Contribution

It presents a novel recursive approach for phase retrieval that simplifies computation and extends to multi-dimensional Fourier analysis.

Findings

Achieves $O(N^2)$ complexity for 1D phase retrieval.

Applicable to higher dimensions without changing complexity.

Discusses performance limitations under noisy conditions.

Abstract

If the phase retrieval problem can be solved by a method similar to that of solving a system of linear equations under the context of FFT, the time complexity of computer based phase retrieval algorithm would be reduced. Here I present such a method which is recursive but highly non-linear in nature, based on a close look at the Fourier spectrum of the square of the function norm. In a one dimensional problem it takes $O(N^2)$ steps of calculation to recover the phases of an N component complex vector. This method could work in 1, 2 or even higher dimensional finite Fourier analysis without changes in the behavior of time complexity. For one dimensional problem the performance of an algorithm based on this method is shown, where the limitations are discussed too, especially when subject to random noises which contains significant high frequency components.