Sparse Phase Retrieval from Short-Time Fourier Measurements

TL;DR

This paper addresses 1D phase retrieval by leveraging the redundancy of the short-time Fourier transform (STFT) to enable unique recovery of signals, especially sparse ones, and proposes an efficient algorithm with improved performance.

Contribution

It introduces a novel approach for phase retrieval using STFT measurements and adapts the GESPAR algorithm for sparse signal recovery, demonstrating enhanced results.

Findings

Unique recovery for nonvanishing inputs using STFT

Efficient sparse recovery algorithm based on GESPAR

Improved performance over traditional Fourier magnitude methods

Abstract

We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements.