Phase Retrieval for $L^2([-\pi,\pi])$ via the Provably Accurate and Noise Robust Numerical Inversion of Spectrogram Measurements

TL;DR

This paper introduces two algorithms for reconstructing smooth functions from spectrogram measurements, providing theoretical guarantees and demonstrating practical effectiveness and robustness in noisy conditions.

Contribution

The paper presents two novel algorithms for phase retrieval from spectrograms with provable error bounds and practical numerical validation.

Findings

Algorithms achieve accurate reconstruction with theoretical error guarantees

Both methods demonstrate good numerical convergence in practice

Algorithms are robust to noise in spectrogram measurements

Abstract

In this paper, we focus on the approximation of smooth functions $f: [-\pi, \pi] \rightarrow \mathbb{C}$, up to an unresolvable global phase ambiguity, from a finite set of Short Time Fourier Transform (STFT) magnitude (i.e., spectrogram) measurements. Two algorithms are developed for approximately inverting such measurements, each with theoretical error guarantees establishing their correctness. A detailed numerical study also demonstrates that both algorithms work well in practice and have good numerical convergence behavior.